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ckpts/universal/global_step40/zero/1.word_embeddings.weight/fp32.pt

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+size 415237325

ckpts/universal/global_step40/zero/6.attention.query_key_value.weight/exp_avg.pt

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+version https://git-lfs.github.com/spec/v1
+oid sha256:ce25be7492d38d2401e256ae8089ef032633f16bfa045b27e9a74b1a60828331
+size 50332828

venv/lib/python3.10/site-packages/scipy/integrate/_ivp/__init__.py

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+"""Suite of ODE solvers implemented in Python."""
+from .ivp import solve_ivp
+from .rk import RK23, RK45, DOP853
+from .radau import Radau
+from .bdf import BDF
+from .lsoda import LSODA
+from .common import OdeSolution
+from .base import DenseOutput, OdeSolver

venv/lib/python3.10/site-packages/scipy/integrate/_ivp/base.py

@@ -0,0 +1,290 @@
+import numpy as np
+
+
+def check_arguments(fun, y0, support_complex):
+    """Helper function for checking arguments common to all solvers."""
+    y0 = np.asarray(y0)
+    if np.issubdtype(y0.dtype, np.complexfloating):
+        if not support_complex:
+            raise ValueError("`y0` is complex, but the chosen solver does "
+                             "not support integration in a complex domain.")
+        dtype = complex
+    else:
+        dtype = float
+    y0 = y0.astype(dtype, copy=False)
+
+    if y0.ndim != 1:
+        raise ValueError("`y0` must be 1-dimensional.")
+
+    if not np.isfinite(y0).all():
+        raise ValueError("All components of the initial state `y0` must be finite.")
+
+    def fun_wrapped(t, y):
+        return np.asarray(fun(t, y), dtype=dtype)
+
+    return fun_wrapped, y0
+
+
+class OdeSolver:
+    """Base class for ODE solvers.
+
+    In order to implement a new solver you need to follow the guidelines:
+
+        1. A constructor must accept parameters presented in the base class
+           (listed below) along with any other parameters specific to a solver.
+        2. A constructor must accept arbitrary extraneous arguments
+           ``**extraneous``, but warn that these arguments are irrelevant
+           using `common.warn_extraneous` function. Do not pass these
+           arguments to the base class.
+        3. A solver must implement a private method `_step_impl(self)` which
+           propagates a solver one step further. It must return tuple
+           ``(success, message)``, where ``success`` is a boolean indicating
+           whether a step was successful, and ``message`` is a string
+           containing description of a failure if a step failed or None
+           otherwise.
+        4. A solver must implement a private method `_dense_output_impl(self)`,
+           which returns a `DenseOutput` object covering the last successful
+           step.
+        5. A solver must have attributes listed below in Attributes section.
+           Note that ``t_old`` and ``step_size`` are updated automatically.
+        6. Use `fun(self, t, y)` method for the system rhs evaluation, this
+           way the number of function evaluations (`nfev`) will be tracked
+           automatically.
+        7. For convenience, a base class provides `fun_single(self, t, y)` and
+           `fun_vectorized(self, t, y)` for evaluating the rhs in
+           non-vectorized and vectorized fashions respectively (regardless of
+           how `fun` from the constructor is implemented). These calls don't
+           increment `nfev`.
+        8. If a solver uses a Jacobian matrix and LU decompositions, it should
+           track the number of Jacobian evaluations (`njev`) and the number of
+           LU decompositions (`nlu`).
+        9. By convention, the function evaluations used to compute a finite
+           difference approximation of the Jacobian should not be counted in
+           `nfev`, thus use `fun_single(self, t, y)` or
+           `fun_vectorized(self, t, y)` when computing a finite difference
+           approximation of the Jacobian.
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system: the time derivative of the state ``y``
+        at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a
+        scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must
+        return an array of the same shape as ``y``. See `vectorized` for more
+        information.
+    t0 : float
+        Initial time.
+    y0 : array_like, shape (n,)
+        Initial state.
+    t_bound : float
+        Boundary time --- the integration won't continue beyond it. It also
+        determines the direction of the integration.
+    vectorized : bool
+        Whether `fun` can be called in a vectorized fashion. Default is False.
+
+        If ``vectorized`` is False, `fun` will always be called with ``y`` of
+        shape ``(n,)``, where ``n = len(y0)``.
+
+        If ``vectorized`` is True, `fun` may be called with ``y`` of shape
+        ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave
+        such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of
+        the returned array is the time derivative of the state corresponding
+        with a column of ``y``).
+
+        Setting ``vectorized=True`` allows for faster finite difference
+        approximation of the Jacobian by methods 'Radau' and 'BDF', but
+        will result in slower execution for other methods. It can also
+        result in slower overall execution for 'Radau' and 'BDF' in some
+        circumstances (e.g. small ``len(y0)``).
+    support_complex : bool, optional
+        Whether integration in a complex domain should be supported.
+        Generally determined by a derived solver class capabilities.
+        Default is False.
+
+    Attributes
+    ----------
+    n : int
+        Number of equations.
+    status : string
+        Current status of the solver: 'running', 'finished' or 'failed'.
+    t_bound : float
+        Boundary time.
+    direction : float
+        Integration direction: +1 or -1.
+    t : float
+        Current time.
+    y : ndarray
+        Current state.
+    t_old : float
+        Previous time. None if no steps were made yet.
+    step_size : float
+        Size of the last successful step. None if no steps were made yet.
+    nfev : int
+        Number of the system's rhs evaluations.
+    njev : int
+        Number of the Jacobian evaluations.
+    nlu : int
+        Number of LU decompositions.
+    """
+    TOO_SMALL_STEP = "Required step size is less than spacing between numbers."
+
+    def __init__(self, fun, t0, y0, t_bound, vectorized,
+                 support_complex=False):
+        self.t_old = None
+        self.t = t0
+        self._fun, self.y = check_arguments(fun, y0, support_complex)
+        self.t_bound = t_bound
+        self.vectorized = vectorized
+
+        if vectorized:
+            def fun_single(t, y):
+                return self._fun(t, y[:, None]).ravel()
+            fun_vectorized = self._fun
+        else:
+            fun_single = self._fun
+
+            def fun_vectorized(t, y):
+                f = np.empty_like(y)
+                for i, yi in enumerate(y.T):
+                    f[:, i] = self._fun(t, yi)
+                return f
+
+        def fun(t, y):
+            self.nfev += 1
+            return self.fun_single(t, y)
+
+        self.fun = fun
+        self.fun_single = fun_single
+        self.fun_vectorized = fun_vectorized
+
+        self.direction = np.sign(t_bound - t0) if t_bound != t0 else 1
+        self.n = self.y.size
+        self.status = 'running'
+
+        self.nfev = 0
+        self.njev = 0
+        self.nlu = 0
+
+    @property
+    def step_size(self):
+        if self.t_old is None:
+            return None
+        else:
+            return np.abs(self.t - self.t_old)
+
+    def step(self):
+        """Perform one integration step.
+
+        Returns
+        -------
+        message : string or None
+            Report from the solver. Typically a reason for a failure if
+            `self.status` is 'failed' after the step was taken or None
+            otherwise.
+        """
+        if self.status != 'running':
+            raise RuntimeError("Attempt to step on a failed or finished "
+                               "solver.")
+
+        if self.n == 0 or self.t == self.t_bound:
+            # Handle corner cases of empty solver or no integration.
+            self.t_old = self.t
+            self.t = self.t_bound
+            message = None
+            self.status = 'finished'
+        else:
+            t = self.t
+            success, message = self._step_impl()
+
+            if not success:
+                self.status = 'failed'
+            else:
+                self.t_old = t
+                if self.direction * (self.t - self.t_bound) >= 0:
+                    self.status = 'finished'
+
+        return message
+
+    def dense_output(self):
+        """Compute a local interpolant over the last successful step.
+
+        Returns
+        -------
+        sol : `DenseOutput`
+            Local interpolant over the last successful step.
+        """
+        if self.t_old is None:
+            raise RuntimeError("Dense output is available after a successful "
+                               "step was made.")
+
+        if self.n == 0 or self.t == self.t_old:
+            # Handle corner cases of empty solver and no integration.
+            return ConstantDenseOutput(self.t_old, self.t, self.y)
+        else:
+            return self._dense_output_impl()
+
+    def _step_impl(self):
+        raise NotImplementedError
+
+    def _dense_output_impl(self):
+        raise NotImplementedError
+
+
+class DenseOutput:
+    """Base class for local interpolant over step made by an ODE solver.
+
+    It interpolates between `t_min` and `t_max` (see Attributes below).
+    Evaluation outside this interval is not forbidden, but the accuracy is not
+    guaranteed.
+
+    Attributes
+    ----------
+    t_min, t_max : float
+        Time range of the interpolation.
+    """
+    def __init__(self, t_old, t):
+        self.t_old = t_old
+        self.t = t
+        self.t_min = min(t, t_old)
+        self.t_max = max(t, t_old)
+
+    def __call__(self, t):
+        """Evaluate the interpolant.
+
+        Parameters
+        ----------
+        t : float or array_like with shape (n_points,)
+            Points to evaluate the solution at.
+
+        Returns
+        -------
+        y : ndarray, shape (n,) or (n, n_points)
+            Computed values. Shape depends on whether `t` was a scalar or a
+            1-D array.
+        """
+        t = np.asarray(t)
+        if t.ndim > 1:
+            raise ValueError("`t` must be a float or a 1-D array.")
+        return self._call_impl(t)
+
+    def _call_impl(self, t):
+        raise NotImplementedError
+
+
+class ConstantDenseOutput(DenseOutput):
+    """Constant value interpolator.
+
+    This class used for degenerate integration cases: equal integration limits
+    or a system with 0 equations.
+    """
+    def __init__(self, t_old, t, value):
+        super().__init__(t_old, t)
+        self.value = value
+
+    def _call_impl(self, t):
+        if t.ndim == 0:
+            return self.value
+        else:
+            ret = np.empty((self.value.shape[0], t.shape[0]))
+            ret[:] = self.value[:, None]
+            return ret

venv/lib/python3.10/site-packages/scipy/integrate/_ivp/bdf.py

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+import numpy as np
+from scipy.linalg import lu_factor, lu_solve
+from scipy.sparse import issparse, csc_matrix, eye
+from scipy.sparse.linalg import splu
+from scipy.optimize._numdiff import group_columns
+from .common import (validate_max_step, validate_tol, select_initial_step,
+                     norm, EPS, num_jac, validate_first_step,
+                     warn_extraneous)
+from .base import OdeSolver, DenseOutput
+
+
+MAX_ORDER = 5
+NEWTON_MAXITER = 4
+MIN_FACTOR = 0.2
+MAX_FACTOR = 10
+
+
+def compute_R(order, factor):
+    """Compute the matrix for changing the differences array."""
+    I = np.arange(1, order + 1)[:, None]
+    J = np.arange(1, order + 1)
+    M = np.zeros((order + 1, order + 1))
+    M[1:, 1:] = (I - 1 - factor * J) / I
+    M[0] = 1
+    return np.cumprod(M, axis=0)
+
+
+def change_D(D, order, factor):
+    """Change differences array in-place when step size is changed."""
+    R = compute_R(order, factor)
+    U = compute_R(order, 1)
+    RU = R.dot(U)
+    D[:order + 1] = np.dot(RU.T, D[:order + 1])
+
+
+def solve_bdf_system(fun, t_new, y_predict, c, psi, LU, solve_lu, scale, tol):
+    """Solve the algebraic system resulting from BDF method."""
+    d = 0
+    y = y_predict.copy()
+    dy_norm_old = None
+    converged = False
+    for k in range(NEWTON_MAXITER):
+        f = fun(t_new, y)
+        if not np.all(np.isfinite(f)):
+            break
+
+        dy = solve_lu(LU, c * f - psi - d)
+        dy_norm = norm(dy / scale)
+
+        if dy_norm_old is None:
+            rate = None
+        else:
+            rate = dy_norm / dy_norm_old
+
+        if (rate is not None and (rate >= 1 or
+                rate ** (NEWTON_MAXITER - k) / (1 - rate) * dy_norm > tol)):
+            break
+
+        y += dy
+        d += dy
+
+        if (dy_norm == 0 or
+                rate is not None and rate / (1 - rate) * dy_norm < tol):
+            converged = True
+            break
+
+        dy_norm_old = dy_norm
+
+    return converged, k + 1, y, d
+
+
+class BDF(OdeSolver):
+    """Implicit method based on backward-differentiation formulas.
+
+    This is a variable order method with the order varying automatically from
+    1 to 5. The general framework of the BDF algorithm is described in [1]_.
+    This class implements a quasi-constant step size as explained in [2]_.
+    The error estimation strategy for the constant-step BDF is derived in [3]_.
+    An accuracy enhancement using modified formulas (NDF) [2]_ is also implemented.
+
+    Can be applied in the complex domain.
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system: the time derivative of the state ``y``
+        at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a
+        scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must
+        return an array of the same shape as ``y``. See `vectorized` for more
+        information.
+    t0 : float
+        Initial time.
+    y0 : array_like, shape (n,)
+        Initial state.
+    t_bound : float
+        Boundary time - the integration won't continue beyond it. It also
+        determines the direction of the integration.
+    first_step : float or None, optional
+        Initial step size. Default is ``None`` which means that the algorithm
+        should choose.
+    max_step : float, optional
+        Maximum allowed step size. Default is np.inf, i.e., the step size is not
+        bounded and determined solely by the solver.
+    rtol, atol : float and array_like, optional
+        Relative and absolute tolerances. The solver keeps the local error
+        estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
+        relative accuracy (number of correct digits), while `atol` controls
+        absolute accuracy (number of correct decimal places). To achieve the
+        desired `rtol`, set `atol` to be smaller than the smallest value that
+        can be expected from ``rtol * abs(y)`` so that `rtol` dominates the
+        allowable error. If `atol` is larger than ``rtol * abs(y)`` the
+        number of correct digits is not guaranteed. Conversely, to achieve the
+        desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller
+        than `atol`. If components of y have different scales, it might be
+        beneficial to set different `atol` values for different components by
+        passing array_like with shape (n,) for `atol`. Default values are
+        1e-3 for `rtol` and 1e-6 for `atol`.
+    jac : {None, array_like, sparse_matrix, callable}, optional
+        Jacobian matrix of the right-hand side of the system with respect to y,
+        required by this method. The Jacobian matrix has shape (n, n) and its
+        element (i, j) is equal to ``d f_i / d y_j``.
+        There are three ways to define the Jacobian:
+
+            * If array_like or sparse_matrix, the Jacobian is assumed to
+              be constant.
+            * If callable, the Jacobian is assumed to depend on both
+              t and y; it will be called as ``jac(t, y)`` as necessary.
+              For the 'Radau' and 'BDF' methods, the return value might be a
+              sparse matrix.
+            * If None (default), the Jacobian will be approximated by
+              finite differences.
+
+        It is generally recommended to provide the Jacobian rather than
+        relying on a finite-difference approximation.
+    jac_sparsity : {None, array_like, sparse matrix}, optional
+        Defines a sparsity structure of the Jacobian matrix for a
+        finite-difference approximation. Its shape must be (n, n). This argument
+        is ignored if `jac` is not `None`. If the Jacobian has only few non-zero
+        elements in *each* row, providing the sparsity structure will greatly
+        speed up the computations [4]_. A zero entry means that a corresponding
+        element in the Jacobian is always zero. If None (default), the Jacobian
+        is assumed to be dense.
+    vectorized : bool, optional
+        Whether `fun` can be called in a vectorized fashion. Default is False.
+
+        If ``vectorized`` is False, `fun` will always be called with ``y`` of
+        shape ``(n,)``, where ``n = len(y0)``.
+
+        If ``vectorized`` is True, `fun` may be called with ``y`` of shape
+        ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave
+        such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of
+        the returned array is the time derivative of the state corresponding
+        with a column of ``y``).
+
+        Setting ``vectorized=True`` allows for faster finite difference
+        approximation of the Jacobian by this method, but may result in slower
+        execution overall in some circumstances (e.g. small ``len(y0)``).
+
+    Attributes
+    ----------
+    n : int
+        Number of equations.
+    status : string
+        Current status of the solver: 'running', 'finished' or 'failed'.
+    t_bound : float
+        Boundary time.
+    direction : float
+        Integration direction: +1 or -1.
+    t : float
+        Current time.
+    y : ndarray
+        Current state.
+    t_old : float
+        Previous time. None if no steps were made yet.
+    step_size : float
+        Size of the last successful step. None if no steps were made yet.
+    nfev : int
+        Number of evaluations of the right-hand side.
+    njev : int
+        Number of evaluations of the Jacobian.
+    nlu : int
+        Number of LU decompositions.
+
+    References
+    ----------
+    .. [1] G. D. Byrne, A. C. Hindmarsh, "A Polyalgorithm for the Numerical
+           Solution of Ordinary Differential Equations", ACM Transactions on
+           Mathematical Software, Vol. 1, No. 1, pp. 71-96, March 1975.
+    .. [2] L. F. Shampine, M. W. Reichelt, "THE MATLAB ODE SUITE", SIAM J. SCI.
+           COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997.
+    .. [3] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations I:
+           Nonstiff Problems", Sec. III.2.
+    .. [4] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
+           sparse Jacobian matrices", Journal of the Institute of Mathematics
+           and its Applications, 13, pp. 117-120, 1974.
+    """
+    def __init__(self, fun, t0, y0, t_bound, max_step=np.inf,
+                 rtol=1e-3, atol=1e-6, jac=None, jac_sparsity=None,
+                 vectorized=False, first_step=None, **extraneous):
+        warn_extraneous(extraneous)
+        super().__init__(fun, t0, y0, t_bound, vectorized,
+                         support_complex=True)
+        self.max_step = validate_max_step(max_step)
+        self.rtol, self.atol = validate_tol(rtol, atol, self.n)
+        f = self.fun(self.t, self.y)
+        if first_step is None:
+            self.h_abs = select_initial_step(self.fun, self.t, self.y, f,
+                                             self.direction, 1,
+                                             self.rtol, self.atol)
+        else:
+            self.h_abs = validate_first_step(first_step, t0, t_bound)
+        self.h_abs_old = None
+        self.error_norm_old = None
+
+        self.newton_tol = max(10 * EPS / rtol, min(0.03, rtol ** 0.5))
+
+        self.jac_factor = None
+        self.jac, self.J = self._validate_jac(jac, jac_sparsity)
+        if issparse(self.J):
+            def lu(A):
+                self.nlu += 1
+                return splu(A)
+
+            def solve_lu(LU, b):
+                return LU.solve(b)
+
+            I = eye(self.n, format='csc', dtype=self.y.dtype)
+        else:
+            def lu(A):
+                self.nlu += 1
+                return lu_factor(A, overwrite_a=True)
+
+            def solve_lu(LU, b):
+                return lu_solve(LU, b, overwrite_b=True)
+
+            I = np.identity(self.n, dtype=self.y.dtype)
+
+        self.lu = lu
+        self.solve_lu = solve_lu
+        self.I = I
+
+        kappa = np.array([0, -0.1850, -1/9, -0.0823, -0.0415, 0])
+        self.gamma = np.hstack((0, np.cumsum(1 / np.arange(1, MAX_ORDER + 1))))
+        self.alpha = (1 - kappa) * self.gamma
+        self.error_const = kappa * self.gamma + 1 / np.arange(1, MAX_ORDER + 2)
+
+        D = np.empty((MAX_ORDER + 3, self.n), dtype=self.y.dtype)
+        D[0] = self.y
+        D[1] = f * self.h_abs * self.direction
+        self.D = D
+
+        self.order = 1
+        self.n_equal_steps = 0
+        self.LU = None
+
+    def _validate_jac(self, jac, sparsity):
+        t0 = self.t
+        y0 = self.y
+
+        if jac is None:
+            if sparsity is not None:
+                if issparse(sparsity):
+                    sparsity = csc_matrix(sparsity)
+                groups = group_columns(sparsity)
+                sparsity = (sparsity, groups)
+
+            def jac_wrapped(t, y):
+                self.njev += 1
+                f = self.fun_single(t, y)
+                J, self.jac_factor = num_jac(self.fun_vectorized, t, y, f,
+                                             self.atol, self.jac_factor,
+                                             sparsity)
+                return J
+            J = jac_wrapped(t0, y0)
+        elif callable(jac):
+            J = jac(t0, y0)
+            self.njev += 1
+            if issparse(J):
+                J = csc_matrix(J, dtype=y0.dtype)
+
+                def jac_wrapped(t, y):
+                    self.njev += 1
+                    return csc_matrix(jac(t, y), dtype=y0.dtype)
+            else:
+                J = np.asarray(J, dtype=y0.dtype)
+
+                def jac_wrapped(t, y):
+                    self.njev += 1
+                    return np.asarray(jac(t, y), dtype=y0.dtype)
+
+            if J.shape != (self.n, self.n):
+                raise ValueError("`jac` is expected to have shape {}, but "
+                                 "actually has {}."
+                                 .format((self.n, self.n), J.shape))
+        else:
+            if issparse(jac):
+                J = csc_matrix(jac, dtype=y0.dtype)
+            else:
+                J = np.asarray(jac, dtype=y0.dtype)
+
+            if J.shape != (self.n, self.n):
+                raise ValueError("`jac` is expected to have shape {}, but "
+                                 "actually has {}."
+                                 .format((self.n, self.n), J.shape))
+            jac_wrapped = None
+
+        return jac_wrapped, J
+
+    def _step_impl(self):
+        t = self.t
+        D = self.D
+
+        max_step = self.max_step
+        min_step = 10 * np.abs(np.nextafter(t, self.direction * np.inf) - t)
+        if self.h_abs > max_step:
+            h_abs = max_step
+            change_D(D, self.order, max_step / self.h_abs)
+            self.n_equal_steps = 0
+        elif self.h_abs < min_step:
+            h_abs = min_step
+            change_D(D, self.order, min_step / self.h_abs)
+            self.n_equal_steps = 0
+        else:
+            h_abs = self.h_abs
+
+        atol = self.atol
+        rtol = self.rtol
+        order = self.order
+
+        alpha = self.alpha
+        gamma = self.gamma
+        error_const = self.error_const
+
+        J = self.J
+        LU = self.LU
+        current_jac = self.jac is None
+
+        step_accepted = False
+        while not step_accepted:
+            if h_abs < min_step:
+                return False, self.TOO_SMALL_STEP
+
+            h = h_abs * self.direction
+            t_new = t + h
+
+            if self.direction * (t_new - self.t_bound) > 0:
+                t_new = self.t_bound
+                change_D(D, order, np.abs(t_new - t) / h_abs)
+                self.n_equal_steps = 0
+                LU = None
+
+            h = t_new - t
+            h_abs = np.abs(h)
+
+            y_predict = np.sum(D[:order + 1], axis=0)
+
+            scale = atol + rtol * np.abs(y_predict)
+            psi = np.dot(D[1: order + 1].T, gamma[1: order + 1]) / alpha[order]
+
+            converged = False
+            c = h / alpha[order]
+            while not converged:
+                if LU is None:
+                    LU = self.lu(self.I - c * J)
+
+                converged, n_iter, y_new, d = solve_bdf_system(
+                    self.fun, t_new, y_predict, c, psi, LU, self.solve_lu,
+                    scale, self.newton_tol)
+
+                if not converged:
+                    if current_jac:
+                        break
+                    J = self.jac(t_new, y_predict)
+                    LU = None
+                    current_jac = True
+
+            if not converged:
+                factor = 0.5
+                h_abs *= factor
+                change_D(D, order, factor)
+                self.n_equal_steps = 0
+                LU = None
+                continue
+
+            safety = 0.9 * (2 * NEWTON_MAXITER + 1) / (2 * NEWTON_MAXITER
+                                                       + n_iter)
+
+            scale = atol + rtol * np.abs(y_new)
+            error = error_const[order] * d
+            error_norm = norm(error / scale)
+
+            if error_norm > 1:
+                factor = max(MIN_FACTOR,
+                             safety * error_norm ** (-1 / (order + 1)))
+                h_abs *= factor
+                change_D(D, order, factor)
+                self.n_equal_steps = 0
+                # As we didn't have problems with convergence, we don't
+                # reset LU here.
+            else:
+                step_accepted = True
+
+        self.n_equal_steps += 1
+
+        self.t = t_new
+        self.y = y_new
+
+        self.h_abs = h_abs
+        self.J = J
+        self.LU = LU
+
+        # Update differences. The principal relation here is
+        # D^{j + 1} y_n = D^{j} y_n - D^{j} y_{n - 1}. Keep in mind that D
+        # contained difference for previous interpolating polynomial and
+        # d = D^{k + 1} y_n. Thus this elegant code follows.
+        D[order + 2] = d - D[order + 1]
+        D[order + 1] = d
+        for i in reversed(range(order + 1)):
+            D[i] += D[i + 1]
+
+        if self.n_equal_steps < order + 1:
+            return True, None
+
+        if order > 1:
+            error_m = error_const[order - 1] * D[order]
+            error_m_norm = norm(error_m / scale)
+        else:
+            error_m_norm = np.inf
+
+        if order < MAX_ORDER:
+            error_p = error_const[order + 1] * D[order + 2]
+            error_p_norm = norm(error_p / scale)
+        else:
+            error_p_norm = np.inf
+
+        error_norms = np.array([error_m_norm, error_norm, error_p_norm])
+        with np.errstate(divide='ignore'):
+            factors = error_norms ** (-1 / np.arange(order, order + 3))
+
+        delta_order = np.argmax(factors) - 1
+        order += delta_order
+        self.order = order
+
+        factor = min(MAX_FACTOR, safety * np.max(factors))
+        self.h_abs *= factor
+        change_D(D, order, factor)
+        self.n_equal_steps = 0
+        self.LU = None
+
+        return True, None
+
+    def _dense_output_impl(self):
+        return BdfDenseOutput(self.t_old, self.t, self.h_abs * self.direction,
+                              self.order, self.D[:self.order + 1].copy())
+
+
+class BdfDenseOutput(DenseOutput):
+    def __init__(self, t_old, t, h, order, D):
+        super().__init__(t_old, t)
+        self.order = order
+        self.t_shift = self.t - h * np.arange(self.order)
+        self.denom = h * (1 + np.arange(self.order))
+        self.D = D
+
+    def _call_impl(self, t):
+        if t.ndim == 0:
+            x = (t - self.t_shift) / self.denom
+            p = np.cumprod(x)
+        else:
+            x = (t - self.t_shift[:, None]) / self.denom[:, None]
+            p = np.cumprod(x, axis=0)
+
+        y = np.dot(self.D[1:].T, p)
+        if y.ndim == 1:
+            y += self.D[0]
+        else:
+            y += self.D[0, :, None]
+
+        return y

venv/lib/python3.10/site-packages/scipy/integrate/_ivp/common.py

@@ -0,0 +1,440 @@
+from itertools import groupby
+from warnings import warn
+import numpy as np
+from scipy.sparse import find, coo_matrix
+
+
+EPS = np.finfo(float).eps
+
+
+def validate_first_step(first_step, t0, t_bound):
+    """Assert that first_step is valid and return it."""
+    if first_step <= 0:
+        raise ValueError("`first_step` must be positive.")
+    if first_step > np.abs(t_bound - t0):
+        raise ValueError("`first_step` exceeds bounds.")
+    return first_step
+
+
+def validate_max_step(max_step):
+    """Assert that max_Step is valid and return it."""
+    if max_step <= 0:
+        raise ValueError("`max_step` must be positive.")
+    return max_step
+
+
+def warn_extraneous(extraneous):
+    """Display a warning for extraneous keyword arguments.
+
+    The initializer of each solver class is expected to collect keyword
+    arguments that it doesn't understand and warn about them. This function
+    prints a warning for each key in the supplied dictionary.
+
+    Parameters
+    ----------
+    extraneous : dict
+        Extraneous keyword arguments
+    """
+    if extraneous:
+        warn("The following arguments have no effect for a chosen solver: {}."
+             .format(", ".join(f"`{x}`" for x in extraneous)),
+             stacklevel=3)
+
+
+def validate_tol(rtol, atol, n):
+    """Validate tolerance values."""
+
+    if np.any(rtol < 100 * EPS):
+        warn("At least one element of `rtol` is too small. "
+             f"Setting `rtol = np.maximum(rtol, {100 * EPS})`.",
+             stacklevel=3)
+        rtol = np.maximum(rtol, 100 * EPS)
+
+    atol = np.asarray(atol)
+    if atol.ndim > 0 and atol.shape != (n,):
+        raise ValueError("`atol` has wrong shape.")
+
+    if np.any(atol < 0):
+        raise ValueError("`atol` must be positive.")
+
+    return rtol, atol
+
+
+def norm(x):
+    """Compute RMS norm."""
+    return np.linalg.norm(x) / x.size ** 0.5
+
+
+def select_initial_step(fun, t0, y0, f0, direction, order, rtol, atol):
+    """Empirically select a good initial step.
+
+    The algorithm is described in [1]_.
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system.
+    t0 : float
+        Initial value of the independent variable.
+    y0 : ndarray, shape (n,)
+        Initial value of the dependent variable.
+    f0 : ndarray, shape (n,)
+        Initial value of the derivative, i.e., ``fun(t0, y0)``.
+    direction : float
+        Integration direction.
+    order : float
+        Error estimator order. It means that the error controlled by the
+        algorithm is proportional to ``step_size ** (order + 1)`.
+    rtol : float
+        Desired relative tolerance.
+    atol : float
+        Desired absolute tolerance.
+
+    Returns
+    -------
+    h_abs : float
+        Absolute value of the suggested initial step.
+
+    References
+    ----------
+    .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
+           Equations I: Nonstiff Problems", Sec. II.4.
+    """
+    if y0.size == 0:
+        return np.inf
+
+    scale = atol + np.abs(y0) * rtol
+    d0 = norm(y0 / scale)
+    d1 = norm(f0 / scale)
+    if d0 < 1e-5 or d1 < 1e-5:
+        h0 = 1e-6
+    else:
+        h0 = 0.01 * d0 / d1
+
+    y1 = y0 + h0 * direction * f0
+    f1 = fun(t0 + h0 * direction, y1)
+    d2 = norm((f1 - f0) / scale) / h0
+
+    if d1 <= 1e-15 and d2 <= 1e-15:
+        h1 = max(1e-6, h0 * 1e-3)
+    else:
+        h1 = (0.01 / max(d1, d2)) ** (1 / (order + 1))
+
+    return min(100 * h0, h1)
+
+
+class OdeSolution:
+    """Continuous ODE solution.
+
+    It is organized as a collection of `DenseOutput` objects which represent
+    local interpolants. It provides an algorithm to select a right interpolant
+    for each given point.
+
+    The interpolants cover the range between `t_min` and `t_max` (see
+    Attributes below). Evaluation outside this interval is not forbidden, but
+    the accuracy is not guaranteed.
+
+    When evaluating at a breakpoint (one of the values in `ts`) a segment with
+    the lower index is selected.
+
+    Parameters
+    ----------
+    ts : array_like, shape (n_segments + 1,)
+        Time instants between which local interpolants are defined. Must
+        be strictly increasing or decreasing (zero segment with two points is
+        also allowed).
+    interpolants : list of DenseOutput with n_segments elements
+        Local interpolants. An i-th interpolant is assumed to be defined
+        between ``ts[i]`` and ``ts[i + 1]``.
+    alt_segment : boolean
+        Requests the alternative interpolant segment selection scheme. At each
+        solver integration point, two interpolant segments are available. The
+        default (False) and alternative (True) behaviours select the segment
+        for which the requested time corresponded to ``t`` and ``t_old``,
+        respectively. This functionality is only relevant for testing the
+        interpolants' accuracy: different integrators use different
+        construction strategies.
+
+    Attributes
+    ----------
+    t_min, t_max : float
+        Time range of the interpolation.
+    """
+    def __init__(self, ts, interpolants, alt_segment=False):
+        ts = np.asarray(ts)
+        d = np.diff(ts)
+        # The first case covers integration on zero segment.
+        if not ((ts.size == 2 and ts[0] == ts[-1])
+                or np.all(d > 0) or np.all(d < 0)):
+            raise ValueError("`ts` must be strictly increasing or decreasing.")
+
+        self.n_segments = len(interpolants)
+        if ts.shape != (self.n_segments + 1,):
+            raise ValueError("Numbers of time stamps and interpolants "
+                             "don't match.")
+
+        self.ts = ts
+        self.interpolants = interpolants
+        if ts[-1] >= ts[0]:
+            self.t_min = ts[0]
+            self.t_max = ts[-1]
+            self.ascending = True
+            self.side = "right" if alt_segment else "left"
+            self.ts_sorted = ts
+        else:
+            self.t_min = ts[-1]
+            self.t_max = ts[0]
+            self.ascending = False
+            self.side = "left" if alt_segment else "right"
+            self.ts_sorted = ts[::-1]
+
+    def _call_single(self, t):
+        # Here we preserve a certain symmetry that when t is in self.ts,
+        # if alt_segment=False, then we prioritize a segment with a lower
+        # index.
+        ind = np.searchsorted(self.ts_sorted, t, side=self.side)
+
+        segment = min(max(ind - 1, 0), self.n_segments - 1)
+        if not self.ascending:
+            segment = self.n_segments - 1 - segment
+
+        return self.interpolants[segment](t)
+
+    def __call__(self, t):
+        """Evaluate the solution.
+
+        Parameters
+        ----------
+        t : float or array_like with shape (n_points,)
+            Points to evaluate at.
+
+        Returns
+        -------
+        y : ndarray, shape (n_states,) or (n_states, n_points)
+            Computed values. Shape depends on whether `t` is a scalar or a
+            1-D array.
+        """
+        t = np.asarray(t)
+
+        if t.ndim == 0:
+            return self._call_single(t)
+
+        order = np.argsort(t)
+        reverse = np.empty_like(order)
+        reverse[order] = np.arange(order.shape[0])
+        t_sorted = t[order]
+
+        # See comment in self._call_single.
+        segments = np.searchsorted(self.ts_sorted, t_sorted, side=self.side)
+        segments -= 1
+        segments[segments < 0] = 0
+        segments[segments > self.n_segments - 1] = self.n_segments - 1
+        if not self.ascending:
+            segments = self.n_segments - 1 - segments
+
+        ys = []
+        group_start = 0
+        for segment, group in groupby(segments):
+            group_end = group_start + len(list(group))
+            y = self.interpolants[segment](t_sorted[group_start:group_end])
+            ys.append(y)
+            group_start = group_end
+
+        ys = np.hstack(ys)
+        ys = ys[:, reverse]
+
+        return ys
+
+
+NUM_JAC_DIFF_REJECT = EPS ** 0.875
+NUM_JAC_DIFF_SMALL = EPS ** 0.75
+NUM_JAC_DIFF_BIG = EPS ** 0.25
+NUM_JAC_MIN_FACTOR = 1e3 * EPS
+NUM_JAC_FACTOR_INCREASE = 10
+NUM_JAC_FACTOR_DECREASE = 0.1
+
+
+def num_jac(fun, t, y, f, threshold, factor, sparsity=None):
+    """Finite differences Jacobian approximation tailored for ODE solvers.
+
+    This function computes finite difference approximation to the Jacobian
+    matrix of `fun` with respect to `y` using forward differences.
+    The Jacobian matrix has shape (n, n) and its element (i, j) is equal to
+    ``d f_i / d y_j``.
+
+    A special feature of this function is the ability to correct the step
+    size from iteration to iteration. The main idea is to keep the finite
+    difference significantly separated from its round-off error which
+    approximately equals ``EPS * np.abs(f)``. It reduces a possibility of a
+    huge error and assures that the estimated derivative are reasonably close
+    to the true values (i.e., the finite difference approximation is at least
+    qualitatively reflects the structure of the true Jacobian).
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system implemented in a vectorized fashion.
+    t : float
+        Current time.
+    y : ndarray, shape (n,)
+        Current state.
+    f : ndarray, shape (n,)
+        Value of the right hand side at (t, y).
+    threshold : float
+        Threshold for `y` value used for computing the step size as
+        ``factor * np.maximum(np.abs(y), threshold)``. Typically, the value of
+        absolute tolerance (atol) for a solver should be passed as `threshold`.
+    factor : ndarray with shape (n,) or None
+        Factor to use for computing the step size. Pass None for the very
+        evaluation, then use the value returned from this function.
+    sparsity : tuple (structure, groups) or None
+        Sparsity structure of the Jacobian, `structure` must be csc_matrix.
+
+    Returns
+    -------
+    J : ndarray or csc_matrix, shape (n, n)
+        Jacobian matrix.
+    factor : ndarray, shape (n,)
+        Suggested `factor` for the next evaluation.
+    """
+    y = np.asarray(y)
+    n = y.shape[0]
+    if n == 0:
+        return np.empty((0, 0)), factor
+
+    if factor is None:
+        factor = np.full(n, EPS ** 0.5)
+    else:
+        factor = factor.copy()
+
+    # Direct the step as ODE dictates, hoping that such a step won't lead to
+    # a problematic region. For complex ODEs it makes sense to use the real
+    # part of f as we use steps along real axis.
+    f_sign = 2 * (np.real(f) >= 0).astype(float) - 1
+    y_scale = f_sign * np.maximum(threshold, np.abs(y))
+    h = (y + factor * y_scale) - y
+
+    # Make sure that the step is not 0 to start with. Not likely it will be
+    # executed often.
+    for i in np.nonzero(h == 0)[0]:
+        while h[i] == 0:
+            factor[i] *= 10
+            h[i] = (y[i] + factor[i] * y_scale[i]) - y[i]
+
+    if sparsity is None:
+        return _dense_num_jac(fun, t, y, f, h, factor, y_scale)
+    else:
+        structure, groups = sparsity
+        return _sparse_num_jac(fun, t, y, f, h, factor, y_scale,
+                               structure, groups)
+
+
+def _dense_num_jac(fun, t, y, f, h, factor, y_scale):
+    n = y.shape[0]
+    h_vecs = np.diag(h)
+    f_new = fun(t, y[:, None] + h_vecs)
+    diff = f_new - f[:, None]
+    max_ind = np.argmax(np.abs(diff), axis=0)
+    r = np.arange(n)
+    max_diff = np.abs(diff[max_ind, r])
+    scale = np.maximum(np.abs(f[max_ind]), np.abs(f_new[max_ind, r]))
+
+    diff_too_small = max_diff < NUM_JAC_DIFF_REJECT * scale
+    if np.any(diff_too_small):
+        ind, = np.nonzero(diff_too_small)
+        new_factor = NUM_JAC_FACTOR_INCREASE * factor[ind]
+        h_new = (y[ind] + new_factor * y_scale[ind]) - y[ind]
+        h_vecs[ind, ind] = h_new
+        f_new = fun(t, y[:, None] + h_vecs[:, ind])
+        diff_new = f_new - f[:, None]
+        max_ind = np.argmax(np.abs(diff_new), axis=0)
+        r = np.arange(ind.shape[0])
+        max_diff_new = np.abs(diff_new[max_ind, r])
+        scale_new = np.maximum(np.abs(f[max_ind]), np.abs(f_new[max_ind, r]))
+
+        update = max_diff[ind] * scale_new < max_diff_new * scale[ind]
+        if np.any(update):
+            update, = np.nonzero(update)
+            update_ind = ind[update]
+            factor[update_ind] = new_factor[update]
+            h[update_ind] = h_new[update]
+            diff[:, update_ind] = diff_new[:, update]
+            scale[update_ind] = scale_new[update]
+            max_diff[update_ind] = max_diff_new[update]
+
+    diff /= h
+
+    factor[max_diff < NUM_JAC_DIFF_SMALL * scale] *= NUM_JAC_FACTOR_INCREASE
+    factor[max_diff > NUM_JAC_DIFF_BIG * scale] *= NUM_JAC_FACTOR_DECREASE
+    factor = np.maximum(factor, NUM_JAC_MIN_FACTOR)
+
+    return diff, factor
+
+
+def _sparse_num_jac(fun, t, y, f, h, factor, y_scale, structure, groups):
+    n = y.shape[0]
+    n_groups = np.max(groups) + 1
+    h_vecs = np.empty((n_groups, n))
+    for group in range(n_groups):
+        e = np.equal(group, groups)
+        h_vecs[group] = h * e
+    h_vecs = h_vecs.T
+
+    f_new = fun(t, y[:, None] + h_vecs)
+    df = f_new - f[:, None]
+
+    i, j, _ = find(structure)
+    diff = coo_matrix((df[i, groups[j]], (i, j)), shape=(n, n)).tocsc()
+    max_ind = np.array(abs(diff).argmax(axis=0)).ravel()
+    r = np.arange(n)
+    max_diff = np.asarray(np.abs(diff[max_ind, r])).ravel()
+    scale = np.maximum(np.abs(f[max_ind]),
+                       np.abs(f_new[max_ind, groups[r]]))
+
+    diff_too_small = max_diff < NUM_JAC_DIFF_REJECT * scale
+    if np.any(diff_too_small):
+        ind, = np.nonzero(diff_too_small)
+        new_factor = NUM_JAC_FACTOR_INCREASE * factor[ind]
+        h_new = (y[ind] + new_factor * y_scale[ind]) - y[ind]
+        h_new_all = np.zeros(n)
+        h_new_all[ind] = h_new
+
+        groups_unique = np.unique(groups[ind])
+        groups_map = np.empty(n_groups, dtype=int)
+        h_vecs = np.empty((groups_unique.shape[0], n))
+        for k, group in enumerate(groups_unique):
+            e = np.equal(group, groups)
+            h_vecs[k] = h_new_all * e
+            groups_map[group] = k
+        h_vecs = h_vecs.T
+
+        f_new = fun(t, y[:, None] + h_vecs)
+        df = f_new - f[:, None]
+        i, j, _ = find(structure[:, ind])
+        diff_new = coo_matrix((df[i, groups_map[groups[ind[j]]]],
+                               (i, j)), shape=(n, ind.shape[0])).tocsc()
+
+        max_ind_new = np.array(abs(diff_new).argmax(axis=0)).ravel()
+        r = np.arange(ind.shape[0])
+        max_diff_new = np.asarray(np.abs(diff_new[max_ind_new, r])).ravel()
+        scale_new = np.maximum(
+            np.abs(f[max_ind_new]),
+            np.abs(f_new[max_ind_new, groups_map[groups[ind]]]))
+
+        update = max_diff[ind] * scale_new < max_diff_new * scale[ind]
+        if np.any(update):
+            update, = np.nonzero(update)
+            update_ind = ind[update]
+            factor[update_ind] = new_factor[update]
+            h[update_ind] = h_new[update]
+            diff[:, update_ind] = diff_new[:, update]
+            scale[update_ind] = scale_new[update]
+            max_diff[update_ind] = max_diff_new[update]
+
+    diff.data /= np.repeat(h, np.diff(diff.indptr))
+
+    factor[max_diff < NUM_JAC_DIFF_SMALL * scale] *= NUM_JAC_FACTOR_INCREASE
+    factor[max_diff > NUM_JAC_DIFF_BIG * scale] *= NUM_JAC_FACTOR_DECREASE
+    factor = np.maximum(factor, NUM_JAC_MIN_FACTOR)
+
+    return diff, factor

venv/lib/python3.10/site-packages/scipy/integrate/_ivp/dop853_coefficients.py

@@ -0,0 +1,193 @@
+import numpy as np
+
+N_STAGES = 12
+N_STAGES_EXTENDED = 16
+INTERPOLATOR_POWER = 7
+
+C = np.array([0.0,
+              0.526001519587677318785587544488e-01,
+              0.789002279381515978178381316732e-01,
+              0.118350341907227396726757197510,
+              0.281649658092772603273242802490,
+              0.333333333333333333333333333333,
+              0.25,
+              0.307692307692307692307692307692,
+              0.651282051282051282051282051282,
+              0.6,
+              0.857142857142857142857142857142,
+              1.0,
+              1.0,
+              0.1,
+              0.2,
+              0.777777777777777777777777777778])
+
+A = np.zeros((N_STAGES_EXTENDED, N_STAGES_EXTENDED))
+A[1, 0] = 5.26001519587677318785587544488e-2
+
+A[2, 0] = 1.97250569845378994544595329183e-2
+A[2, 1] = 5.91751709536136983633785987549e-2
+
+A[3, 0] = 2.95875854768068491816892993775e-2
+A[3, 2] = 8.87627564304205475450678981324e-2
+
+A[4, 0] = 2.41365134159266685502369798665e-1
+A[4, 2] = -8.84549479328286085344864962717e-1
+A[4, 3] = 9.24834003261792003115737966543e-1
+
+A[5, 0] = 3.7037037037037037037037037037e-2
+A[5, 3] = 1.70828608729473871279604482173e-1
+A[5, 4] = 1.25467687566822425016691814123e-1
+
+A[6, 0] = 3.7109375e-2
+A[6, 3] = 1.70252211019544039314978060272e-1
+A[6, 4] = 6.02165389804559606850219397283e-2
+A[6, 5] = -1.7578125e-2
+
+A[7, 0] = 3.70920001185047927108779319836e-2
+A[7, 3] = 1.70383925712239993810214054705e-1
+A[7, 4] = 1.07262030446373284651809199168e-1
+A[7, 5] = -1.53194377486244017527936158236e-2
+A[7, 6] = 8.27378916381402288758473766002e-3
+
+A[8, 0] = 6.24110958716075717114429577812e-1
+A[8, 3] = -3.36089262944694129406857109825
+A[8, 4] = -8.68219346841726006818189891453e-1
+A[8, 5] = 2.75920996994467083049415600797e1
+A[8, 6] = 2.01540675504778934086186788979e1
+A[8, 7] = -4.34898841810699588477366255144e1
+
+A[9, 0] = 4.77662536438264365890433908527e-1
+A[9, 3] = -2.48811461997166764192642586468
+A[9, 4] = -5.90290826836842996371446475743e-1
+A[9, 5] = 2.12300514481811942347288949897e1
+A[9, 6] = 1.52792336328824235832596922938e1
+A[9, 7] = -3.32882109689848629194453265587e1
+A[9, 8] = -2.03312017085086261358222928593e-2
+
+A[10, 0] = -9.3714243008598732571704021658e-1
+A[10, 3] = 5.18637242884406370830023853209
+A[10, 4] = 1.09143734899672957818500254654
+A[10, 5] = -8.14978701074692612513997267357
+A[10, 6] = -1.85200656599969598641566180701e1
+A[10, 7] = 2.27394870993505042818970056734e1
+A[10, 8] = 2.49360555267965238987089396762
+A[10, 9] = -3.0467644718982195003823669022
+
+A[11, 0] = 2.27331014751653820792359768449
+A[11, 3] = -1.05344954667372501984066689879e1
+A[11, 4] = -2.00087205822486249909675718444
+A[11, 5] = -1.79589318631187989172765950534e1
+A[11, 6] = 2.79488845294199600508499808837e1
+A[11, 7] = -2.85899827713502369474065508674
+A[11, 8] = -8.87285693353062954433549289258
+A[11, 9] = 1.23605671757943030647266201528e1
+A[11, 10] = 6.43392746015763530355970484046e-1
+
+A[12, 0] = 5.42937341165687622380535766363e-2
+A[12, 5] = 4.45031289275240888144113950566
+A[12, 6] = 1.89151789931450038304281599044
+A[12, 7] = -5.8012039600105847814672114227
+A[12, 8] = 3.1116436695781989440891606237e-1
+A[12, 9] = -1.52160949662516078556178806805e-1
+A[12, 10] = 2.01365400804030348374776537501e-1
+A[12, 11] = 4.47106157277725905176885569043e-2
+
+A[13, 0] = 5.61675022830479523392909219681e-2
+A[13, 6] = 2.53500210216624811088794765333e-1
+A[13, 7] = -2.46239037470802489917441475441e-1
+A[13, 8] = -1.24191423263816360469010140626e-1
+A[13, 9] = 1.5329179827876569731206322685e-1
+A[13, 10] = 8.20105229563468988491666602057e-3
+A[13, 11] = 7.56789766054569976138603589584e-3
+A[13, 12] = -8.298e-3
+
+A[14, 0] = 3.18346481635021405060768473261e-2
+A[14, 5] = 2.83009096723667755288322961402e-2
+A[14, 6] = 5.35419883074385676223797384372e-2
+A[14, 7] = -5.49237485713909884646569340306e-2
+A[14, 10] = -1.08347328697249322858509316994e-4
+A[14, 11] = 3.82571090835658412954920192323e-4
+A[14, 12] = -3.40465008687404560802977114492e-4
+A[14, 13] = 1.41312443674632500278074618366e-1
+
+A[15, 0] = -4.28896301583791923408573538692e-1
+A[15, 5] = -4.69762141536116384314449447206
+A[15, 6] = 7.68342119606259904184240953878
+A[15, 7] = 4.06898981839711007970213554331
+A[15, 8] = 3.56727187455281109270669543021e-1
+A[15, 12] = -1.39902416515901462129418009734e-3
+A[15, 13] = 2.9475147891527723389556272149
+A[15, 14] = -9.15095847217987001081870187138
+
+
+B = A[N_STAGES, :N_STAGES]
+
+E3 = np.zeros(N_STAGES + 1)
+E3[:-1] = B.copy()
+E3[0] -= 0.244094488188976377952755905512
+E3[8] -= 0.733846688281611857341361741547
+E3[11] -= 0.220588235294117647058823529412e-1
+
+E5 = np.zeros(N_STAGES + 1)
+E5[0] = 0.1312004499419488073250102996e-1
+E5[5] = -0.1225156446376204440720569753e+1
+E5[6] = -0.4957589496572501915214079952
+E5[7] = 0.1664377182454986536961530415e+1
+E5[8] = -0.3503288487499736816886487290
+E5[9] = 0.3341791187130174790297318841
+E5[10] = 0.8192320648511571246570742613e-1
+E5[11] = -0.2235530786388629525884427845e-1
+
+# First 3 coefficients are computed separately.
+D = np.zeros((INTERPOLATOR_POWER - 3, N_STAGES_EXTENDED))
+D[0, 0] = -0.84289382761090128651353491142e+1
+D[0, 5] = 0.56671495351937776962531783590
+D[0, 6] = -0.30689499459498916912797304727e+1
+D[0, 7] = 0.23846676565120698287728149680e+1
+D[0, 8] = 0.21170345824450282767155149946e+1
+D[0, 9] = -0.87139158377797299206789907490
+D[0, 10] = 0.22404374302607882758541771650e+1
+D[0, 11] = 0.63157877876946881815570249290
+D[0, 12] = -0.88990336451333310820698117400e-1
+D[0, 13] = 0.18148505520854727256656404962e+2
+D[0, 14] = -0.91946323924783554000451984436e+1
+D[0, 15] = -0.44360363875948939664310572000e+1
+
+D[1, 0] = 0.10427508642579134603413151009e+2
+D[1, 5] = 0.24228349177525818288430175319e+3
+D[1, 6] = 0.16520045171727028198505394887e+3
+D[1, 7] = -0.37454675472269020279518312152e+3
+D[1, 8] = -0.22113666853125306036270938578e+2
+D[1, 9] = 0.77334326684722638389603898808e+1
+D[1, 10] = -0.30674084731089398182061213626e+2
+D[1, 11] = -0.93321305264302278729567221706e+1
+D[1, 12] = 0.15697238121770843886131091075e+2
+D[1, 13] = -0.31139403219565177677282850411e+2
+D[1, 14] = -0.93529243588444783865713862664e+1
+D[1, 15] = 0.35816841486394083752465898540e+2
+
+D[2, 0] = 0.19985053242002433820987653617e+2
+D[2, 5] = -0.38703730874935176555105901742e+3
+D[2, 6] = -0.18917813819516756882830838328e+3
+D[2, 7] = 0.52780815920542364900561016686e+3
+D[2, 8] = -0.11573902539959630126141871134e+2
+D[2, 9] = 0.68812326946963000169666922661e+1
+D[2, 10] = -0.10006050966910838403183860980e+1
+D[2, 11] = 0.77771377980534432092869265740
+D[2, 12] = -0.27782057523535084065932004339e+1
+D[2, 13] = -0.60196695231264120758267380846e+2
+D[2, 14] = 0.84320405506677161018159903784e+2
+D[2, 15] = 0.11992291136182789328035130030e+2
+
+D[3, 0] = -0.25693933462703749003312586129e+2
+D[3, 5] = -0.15418974869023643374053993627e+3
+D[3, 6] = -0.23152937917604549567536039109e+3
+D[3, 7] = 0.35763911791061412378285349910e+3
+D[3, 8] = 0.93405324183624310003907691704e+2
+D[3, 9] = -0.37458323136451633156875139351e+2
+D[3, 10] = 0.10409964950896230045147246184e+3
+D[3, 11] = 0.29840293426660503123344363579e+2
+D[3, 12] = -0.43533456590011143754432175058e+2
+D[3, 13] = 0.96324553959188282948394950600e+2
+D[3, 14] = -0.39177261675615439165231486172e+2
+D[3, 15] = -0.14972683625798562581422125276e+3

venv/lib/python3.10/site-packages/scipy/integrate/_ivp/ivp.py

@@ -0,0 +1,748 @@
+import inspect
+import numpy as np
+from .bdf import BDF
+from .radau import Radau
+from .rk import RK23, RK45, DOP853
+from .lsoda import LSODA
+from scipy.optimize import OptimizeResult
+from .common import EPS, OdeSolution
+from .base import OdeSolver
+
+
+METHODS = {'RK23': RK23,
+           'RK45': RK45,
+           'DOP853': DOP853,
+           'Radau': Radau,
+           'BDF': BDF,
+           'LSODA': LSODA}
+
+
+MESSAGES = {0: "The solver successfully reached the end of the integration interval.",
+            1: "A termination event occurred."}
+
+
+class OdeResult(OptimizeResult):
+    pass
+
+
+def prepare_events(events):
+    """Standardize event functions and extract attributes."""
+    if callable(events):
+        events = (events,)
+
+    max_events = np.empty(len(events))
+    direction = np.empty(len(events))
+    for i, event in enumerate(events):
+        terminal = getattr(event, 'terminal', None)
+        direction[i] = getattr(event, 'direction', 0)
+
+        message = ('The `terminal` attribute of each event '
+                   'must be a boolean or positive integer.')
+        if terminal is None or terminal == 0:
+            max_events[i] = np.inf
+        elif int(terminal) == terminal and terminal > 0:
+            max_events[i] = terminal
+        else:
+            raise ValueError(message)
+
+    return events, max_events, direction
+
+
+def solve_event_equation(event, sol, t_old, t):
+    """Solve an equation corresponding to an ODE event.
+
+    The equation is ``event(t, y(t)) = 0``, here ``y(t)`` is known from an
+    ODE solver using some sort of interpolation. It is solved by
+    `scipy.optimize.brentq` with xtol=atol=4*EPS.
+
+    Parameters
+    ----------
+    event : callable
+        Function ``event(t, y)``.
+    sol : callable
+        Function ``sol(t)`` which evaluates an ODE solution between `t_old`
+        and  `t`.
+    t_old, t : float
+        Previous and new values of time. They will be used as a bracketing
+        interval.
+
+    Returns
+    -------
+    root : float
+        Found solution.
+    """
+    from scipy.optimize import brentq
+    return brentq(lambda t: event(t, sol(t)), t_old, t,
+                  xtol=4 * EPS, rtol=4 * EPS)
+
+
+def handle_events(sol, events, active_events, event_count, max_events,
+                  t_old, t):
+    """Helper function to handle events.
+
+    Parameters
+    ----------
+    sol : DenseOutput
+        Function ``sol(t)`` which evaluates an ODE solution between `t_old`
+        and  `t`.
+    events : list of callables, length n_events
+        Event functions with signatures ``event(t, y)``.
+    active_events : ndarray
+        Indices of events which occurred.
+    event_count : ndarray
+        Current number of occurrences for each event.
+    max_events : ndarray, shape (n_events,)
+        Number of occurrences allowed for each event before integration
+        termination is issued.
+    t_old, t : float
+        Previous and new values of time.
+
+    Returns
+    -------
+    root_indices : ndarray
+        Indices of events which take zero between `t_old` and `t` and before
+        a possible termination.
+    roots : ndarray
+        Values of t at which events occurred.
+    terminate : bool
+        Whether a terminal event occurred.
+    """
+    roots = [solve_event_equation(events[event_index], sol, t_old, t)
+             for event_index in active_events]
+
+    roots = np.asarray(roots)
+
+    if np.any(event_count[active_events] >= max_events[active_events]):
+        if t > t_old:
+            order = np.argsort(roots)
+        else:
+            order = np.argsort(-roots)
+        active_events = active_events[order]
+        roots = roots[order]
+        t = np.nonzero(event_count[active_events]
+                       >= max_events[active_events])[0][0]
+        active_events = active_events[:t + 1]
+        roots = roots[:t + 1]
+        terminate = True
+    else:
+        terminate = False
+
+    return active_events, roots, terminate
+
+
+def find_active_events(g, g_new, direction):
+    """Find which event occurred during an integration step.
+
+    Parameters
+    ----------
+    g, g_new : array_like, shape (n_events,)
+        Values of event functions at a current and next points.
+    direction : ndarray, shape (n_events,)
+        Event "direction" according to the definition in `solve_ivp`.
+
+    Returns
+    -------
+    active_events : ndarray
+        Indices of events which occurred during the step.
+    """
+    g, g_new = np.asarray(g), np.asarray(g_new)
+    up = (g <= 0) & (g_new >= 0)
+    down = (g >= 0) & (g_new <= 0)
+    either = up | down
+    mask = (up & (direction > 0) |
+            down & (direction < 0) |
+            either & (direction == 0))
+
+    return np.nonzero(mask)[0]
+
+
+def solve_ivp(fun, t_span, y0, method='RK45', t_eval=None, dense_output=False,
+              events=None, vectorized=False, args=None, **options):
+    """Solve an initial value problem for a system of ODEs.
+
+    This function numerically integrates a system of ordinary differential
+    equations given an initial value::
+
+        dy / dt = f(t, y)
+        y(t0) = y0
+
+    Here t is a 1-D independent variable (time), y(t) is an
+    N-D vector-valued function (state), and an N-D
+    vector-valued function f(t, y) determines the differential equations.
+    The goal is to find y(t) approximately satisfying the differential
+    equations, given an initial value y(t0)=y0.
+
+    Some of the solvers support integration in the complex domain, but note
+    that for stiff ODE solvers, the right-hand side must be
+    complex-differentiable (satisfy Cauchy-Riemann equations [11]_).
+    To solve a problem in the complex domain, pass y0 with a complex data type.
+    Another option always available is to rewrite your problem for real and
+    imaginary parts separately.
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system: the time derivative of the state ``y``
+        at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a
+        scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. Additional
+        arguments need to be passed if ``args`` is used (see documentation of
+        ``args`` argument). ``fun`` must return an array of the same shape as
+        ``y``. See `vectorized` for more information.
+    t_span : 2-member sequence
+        Interval of integration (t0, tf). The solver starts with t=t0 and
+        integrates until it reaches t=tf. Both t0 and tf must be floats
+        or values interpretable by the float conversion function.
+    y0 : array_like, shape (n,)
+        Initial state. For problems in the complex domain, pass `y0` with a
+        complex data type (even if the initial value is purely real).
+    method : string or `OdeSolver`, optional
+        Integration method to use:
+
+            * 'RK45' (default): Explicit Runge-Kutta method of order 5(4) [1]_.
+              The error is controlled assuming accuracy of the fourth-order
+              method, but steps are taken using the fifth-order accurate
+              formula (local extrapolation is done). A quartic interpolation
+              polynomial is used for the dense output [2]_. Can be applied in
+              the complex domain.
+            * 'RK23': Explicit Runge-Kutta method of order 3(2) [3]_. The error
+              is controlled assuming accuracy of the second-order method, but
+              steps are taken using the third-order accurate formula (local
+              extrapolation is done). A cubic Hermite polynomial is used for the
+              dense output. Can be applied in the complex domain.
+            * 'DOP853': Explicit Runge-Kutta method of order 8 [13]_.
+              Python implementation of the "DOP853" algorithm originally
+              written in Fortran [14]_. A 7-th order interpolation polynomial
+              accurate to 7-th order is used for the dense output.
+              Can be applied in the complex domain.
+            * 'Radau': Implicit Runge-Kutta method of the Radau IIA family of
+              order 5 [4]_. The error is controlled with a third-order accurate
+              embedded formula. A cubic polynomial which satisfies the
+              collocation conditions is used for the dense output.
+            * 'BDF': Implicit multi-step variable-order (1 to 5) method based
+              on a backward differentiation formula for the derivative
+              approximation [5]_. The implementation follows the one described
+              in [6]_. A quasi-constant step scheme is used and accuracy is
+              enhanced using the NDF modification. Can be applied in the
+              complex domain.
+            * 'LSODA': Adams/BDF method with automatic stiffness detection and
+              switching [7]_, [8]_. This is a wrapper of the Fortran solver
+              from ODEPACK.
+
+        Explicit Runge-Kutta methods ('RK23', 'RK45', 'DOP853') should be used
+        for non-stiff problems and implicit methods ('Radau', 'BDF') for
+        stiff problems [9]_. Among Runge-Kutta methods, 'DOP853' is recommended
+        for solving with high precision (low values of `rtol` and `atol`).
+
+        If not sure, first try to run 'RK45'. If it makes unusually many
+        iterations, diverges, or fails, your problem is likely to be stiff and
+        you should use 'Radau' or 'BDF'. 'LSODA' can also be a good universal
+        choice, but it might be somewhat less convenient to work with as it
+        wraps old Fortran code.
+
+        You can also pass an arbitrary class derived from `OdeSolver` which
+        implements the solver.
+    t_eval : array_like or None, optional
+        Times at which to store the computed solution, must be sorted and lie
+        within `t_span`. If None (default), use points selected by the solver.
+    dense_output : bool, optional
+        Whether to compute a continuous solution. Default is False.
+    events : callable, or list of callables, optional
+        Events to track. If None (default), no events will be tracked.
+        Each event occurs at the zeros of a continuous function of time and
+        state. Each function must have the signature ``event(t, y)`` where
+        additional argument have to be passed if ``args`` is used (see
+        documentation of ``args`` argument). Each function must return a
+        float. The solver will find an accurate value of `t` at which
+        ``event(t, y(t)) = 0`` using a root-finding algorithm. By default,
+        all zeros will be found. The solver looks for a sign change over
+        each step, so if multiple zero crossings occur within one step,
+        events may be missed. Additionally each `event` function might
+        have the following attributes:
+
+            terminal: bool or int, optional
+                When boolean, whether to terminate integration if this event occurs.
+                When integral, termination occurs after the specified the number of
+                occurences of this event.
+                Implicitly False if not assigned.
+            direction: float, optional
+                Direction of a zero crossing. If `direction` is positive,
+                `event` will only trigger when going from negative to positive,
+                and vice versa if `direction` is negative. If 0, then either
+                direction will trigger event. Implicitly 0 if not assigned.
+
+        You can assign attributes like ``event.terminal = True`` to any
+        function in Python.
+    vectorized : bool, optional
+        Whether `fun` can be called in a vectorized fashion. Default is False.
+
+        If ``vectorized`` is False, `fun` will always be called with ``y`` of
+        shape ``(n,)``, where ``n = len(y0)``.
+
+        If ``vectorized`` is True, `fun` may be called with ``y`` of shape
+        ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave
+        such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of
+        the returned array is the time derivative of the state corresponding
+        with a column of ``y``).
+
+        Setting ``vectorized=True`` allows for faster finite difference
+        approximation of the Jacobian by methods 'Radau' and 'BDF', but
+        will result in slower execution for other methods and for 'Radau' and
+        'BDF' in some circumstances (e.g. small ``len(y0)``).
+    args : tuple, optional
+        Additional arguments to pass to the user-defined functions.  If given,
+        the additional arguments are passed to all user-defined functions.
+        So if, for example, `fun` has the signature ``fun(t, y, a, b, c)``,
+        then `jac` (if given) and any event functions must have the same
+        signature, and `args` must be a tuple of length 3.
+    **options
+        Options passed to a chosen solver. All options available for already
+        implemented solvers are listed below.
+    first_step : float or None, optional
+        Initial step size. Default is `None` which means that the algorithm
+        should choose.
+    max_step : float, optional
+        Maximum allowed step size. Default is np.inf, i.e., the step size is not
+        bounded and determined solely by the solver.
+    rtol, atol : float or array_like, optional
+        Relative and absolute tolerances. The solver keeps the local error
+        estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
+        relative accuracy (number of correct digits), while `atol` controls
+        absolute accuracy (number of correct decimal places). To achieve the
+        desired `rtol`, set `atol` to be smaller than the smallest value that
+        can be expected from ``rtol * abs(y)`` so that `rtol` dominates the
+        allowable error. If `atol` is larger than ``rtol * abs(y)`` the
+        number of correct digits is not guaranteed. Conversely, to achieve the
+        desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller
+        than `atol`. If components of y have different scales, it might be
+        beneficial to set different `atol` values for different components by
+        passing array_like with shape (n,) for `atol`. Default values are
+        1e-3 for `rtol` and 1e-6 for `atol`.
+    jac : array_like, sparse_matrix, callable or None, optional
+        Jacobian matrix of the right-hand side of the system with respect
+        to y, required by the 'Radau', 'BDF' and 'LSODA' method. The
+        Jacobian matrix has shape (n, n) and its element (i, j) is equal to
+        ``d f_i / d y_j``.  There are three ways to define the Jacobian:
+
+            * If array_like or sparse_matrix, the Jacobian is assumed to
+              be constant. Not supported by 'LSODA'.
+            * If callable, the Jacobian is assumed to depend on both
+              t and y; it will be called as ``jac(t, y)``, as necessary.
+              Additional arguments have to be passed if ``args`` is
+              used (see documentation of ``args`` argument).
+              For 'Radau' and 'BDF' methods, the return value might be a
+              sparse matrix.
+            * If None (default), the Jacobian will be approximated by
+              finite differences.
+
+        It is generally recommended to provide the Jacobian rather than
+        relying on a finite-difference approximation.
+    jac_sparsity : array_like, sparse matrix or None, optional
+        Defines a sparsity structure of the Jacobian matrix for a finite-
+        difference approximation. Its shape must be (n, n). This argument
+        is ignored if `jac` is not `None`. If the Jacobian has only few
+        non-zero elements in *each* row, providing the sparsity structure
+        will greatly speed up the computations [10]_. A zero entry means that
+        a corresponding element in the Jacobian is always zero. If None
+        (default), the Jacobian is assumed to be dense.
+        Not supported by 'LSODA', see `lband` and `uband` instead.
+    lband, uband : int or None, optional
+        Parameters defining the bandwidth of the Jacobian for the 'LSODA'
+        method, i.e., ``jac[i, j] != 0 only for i - lband <= j <= i + uband``.
+        Default is None. Setting these requires your jac routine to return the
+        Jacobian in the packed format: the returned array must have ``n``
+        columns and ``uband + lband + 1`` rows in which Jacobian diagonals are
+        written. Specifically ``jac_packed[uband + i - j , j] = jac[i, j]``.
+        The same format is used in `scipy.linalg.solve_banded` (check for an
+        illustration).  These parameters can be also used with ``jac=None`` to
+        reduce the number of Jacobian elements estimated by finite differences.
+    min_step : float, optional
+        The minimum allowed step size for 'LSODA' method.
+        By default `min_step` is zero.
+
+    Returns
+    -------
+    Bunch object with the following fields defined:
+    t : ndarray, shape (n_points,)
+        Time points.
+    y : ndarray, shape (n, n_points)
+        Values of the solution at `t`.
+    sol : `OdeSolution` or None
+        Found solution as `OdeSolution` instance; None if `dense_output` was
+        set to False.
+    t_events : list of ndarray or None
+        Contains for each event type a list of arrays at which an event of
+        that type event was detected. None if `events` was None.
+    y_events : list of ndarray or None
+        For each value of `t_events`, the corresponding value of the solution.
+        None if `events` was None.
+    nfev : int
+        Number of evaluations of the right-hand side.
+    njev : int
+        Number of evaluations of the Jacobian.
+    nlu : int
+        Number of LU decompositions.
+    status : int
+        Reason for algorithm termination:
+
+            * -1: Integration step failed.
+            *  0: The solver successfully reached the end of `tspan`.
+            *  1: A termination event occurred.
+
+    message : string
+        Human-readable description of the termination reason.
+    success : bool
+        True if the solver reached the interval end or a termination event
+        occurred (``status >= 0``).
+
+    References
+    ----------
+    .. [1] J. R. Dormand, P. J. Prince, "A family of embedded Runge-Kutta
+           formulae", Journal of Computational and Applied Mathematics, Vol. 6,
+           No. 1, pp. 19-26, 1980.
+    .. [2] L. W. Shampine, "Some Practical Runge-Kutta Formulas", Mathematics
+           of Computation,, Vol. 46, No. 173, pp. 135-150, 1986.
+    .. [3] P. Bogacki, L.F. Shampine, "A 3(2) Pair of Runge-Kutta Formulas",
+           Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989.
+    .. [4] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II:
+           Stiff and Differential-Algebraic Problems", Sec. IV.8.
+    .. [5] `Backward Differentiation Formula
+            <https://en.wikipedia.org/wiki/Backward_differentiation_formula>`_
+            on Wikipedia.
+    .. [6] L. F. Shampine, M. W. Reichelt, "THE MATLAB ODE SUITE", SIAM J. SCI.
+           COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997.
+    .. [7] A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE
+           Solvers," IMACS Transactions on Scientific Computation, Vol 1.,
+           pp. 55-64, 1983.
+    .. [8] L. Petzold, "Automatic selection of methods for solving stiff and
+           nonstiff systems of ordinary differential equations", SIAM Journal
+           on Scientific and Statistical Computing, Vol. 4, No. 1, pp. 136-148,
+           1983.
+    .. [9] `Stiff equation <https://en.wikipedia.org/wiki/Stiff_equation>`_ on
+           Wikipedia.
+    .. [10] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
+            sparse Jacobian matrices", Journal of the Institute of Mathematics
+            and its Applications, 13, pp. 117-120, 1974.
+    .. [11] `Cauchy-Riemann equations
+             <https://en.wikipedia.org/wiki/Cauchy-Riemann_equations>`_ on
+             Wikipedia.
+    .. [12] `Lotka-Volterra equations
+            <https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations>`_
+            on Wikipedia.
+    .. [13] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
+            Equations I: Nonstiff Problems", Sec. II.
+    .. [14] `Page with original Fortran code of DOP853
+            <http://www.unige.ch/~hairer/software.html>`_.
+
+    Examples
+    --------
+    Basic exponential decay showing automatically chosen time points.
+
+    >>> import numpy as np
+    >>> from scipy.integrate import solve_ivp
+    >>> def exponential_decay(t, y): return -0.5 * y
+    >>> sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8])
+    >>> print(sol.t)
+    [ 0.          0.11487653  1.26364188  3.06061781  4.81611105  6.57445806
+      8.33328988 10.        ]
+    >>> print(sol.y)
+    [[2.         1.88836035 1.06327177 0.43319312 0.18017253 0.07483045
+      0.03107158 0.01350781]
+     [4.         3.7767207  2.12654355 0.86638624 0.36034507 0.14966091
+      0.06214316 0.02701561]
+     [8.         7.5534414  4.25308709 1.73277247 0.72069014 0.29932181
+      0.12428631 0.05403123]]
+
+    Specifying points where the solution is desired.
+
+    >>> sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8],
+    ...                 t_eval=[0, 1, 2, 4, 10])
+    >>> print(sol.t)
+    [ 0  1  2  4 10]
+    >>> print(sol.y)
+    [[2.         1.21305369 0.73534021 0.27066736 0.01350938]
+     [4.         2.42610739 1.47068043 0.54133472 0.02701876]
+     [8.         4.85221478 2.94136085 1.08266944 0.05403753]]
+
+    Cannon fired upward with terminal event upon impact. The ``terminal`` and
+    ``direction`` fields of an event are applied by monkey patching a function.
+    Here ``y[0]`` is position and ``y[1]`` is velocity. The projectile starts
+    at position 0 with velocity +10. Note that the integration never reaches
+    t=100 because the event is terminal.
+
+    >>> def upward_cannon(t, y): return [y[1], -0.5]
+    >>> def hit_ground(t, y): return y[0]
+    >>> hit_ground.terminal = True
+    >>> hit_ground.direction = -1
+    >>> sol = solve_ivp(upward_cannon, [0, 100], [0, 10], events=hit_ground)
+    >>> print(sol.t_events)
+    [array([40.])]
+    >>> print(sol.t)
+    [0.00000000e+00 9.99900010e-05 1.09989001e-03 1.10988901e-02
+     1.11088891e-01 1.11098890e+00 1.11099890e+01 4.00000000e+01]
+
+    Use `dense_output` and `events` to find position, which is 100, at the apex
+    of the cannonball's trajectory. Apex is not defined as terminal, so both
+    apex and hit_ground are found. There is no information at t=20, so the sol
+    attribute is used to evaluate the solution. The sol attribute is returned
+    by setting ``dense_output=True``. Alternatively, the `y_events` attribute
+    can be used to access the solution at the time of the event.
+
+    >>> def apex(t, y): return y[1]
+    >>> sol = solve_ivp(upward_cannon, [0, 100], [0, 10],
+    ...                 events=(hit_ground, apex), dense_output=True)
+    >>> print(sol.t_events)
+    [array([40.]), array([20.])]
+    >>> print(sol.t)
+    [0.00000000e+00 9.99900010e-05 1.09989001e-03 1.10988901e-02
+     1.11088891e-01 1.11098890e+00 1.11099890e+01 4.00000000e+01]
+    >>> print(sol.sol(sol.t_events[1][0]))
+    [100.   0.]
+    >>> print(sol.y_events)
+    [array([[-5.68434189e-14, -1.00000000e+01]]),
+     array([[1.00000000e+02, 1.77635684e-15]])]
+
+    As an example of a system with additional parameters, we'll implement
+    the Lotka-Volterra equations [12]_.
+
+    >>> def lotkavolterra(t, z, a, b, c, d):
+    ...     x, y = z
+    ...     return [a*x - b*x*y, -c*y + d*x*y]
+    ...
+
+    We pass in the parameter values a=1.5, b=1, c=3 and d=1 with the `args`
+    argument.
+
+    >>> sol = solve_ivp(lotkavolterra, [0, 15], [10, 5], args=(1.5, 1, 3, 1),
+    ...                 dense_output=True)
+
+    Compute a dense solution and plot it.
+
+    >>> t = np.linspace(0, 15, 300)
+    >>> z = sol.sol(t)
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(t, z.T)
+    >>> plt.xlabel('t')
+    >>> plt.legend(['x', 'y'], shadow=True)
+    >>> plt.title('Lotka-Volterra System')
+    >>> plt.show()
+
+    A couple examples of using solve_ivp to solve the differential
+    equation ``y' = Ay`` with complex matrix ``A``.
+
+    >>> A = np.array([[-0.25 + 0.14j, 0, 0.33 + 0.44j],
+    ...               [0.25 + 0.58j, -0.2 + 0.14j, 0],
+    ...               [0, 0.2 + 0.4j, -0.1 + 0.97j]])
+
+    Solving an IVP with ``A`` from above and ``y`` as 3x1 vector:
+
+    >>> def deriv_vec(t, y):
+    ...     return A @ y
+    >>> result = solve_ivp(deriv_vec, [0, 25],
+    ...                    np.array([10 + 0j, 20 + 0j, 30 + 0j]),
+    ...                    t_eval=np.linspace(0, 25, 101))
+    >>> print(result.y[:, 0])
+    [10.+0.j 20.+0.j 30.+0.j]
+    >>> print(result.y[:, -1])
+    [18.46291039+45.25653651j 10.01569306+36.23293216j
+     -4.98662741+80.07360388j]
+
+    Solving an IVP with ``A`` from above with ``y`` as 3x3 matrix :
+
+    >>> def deriv_mat(t, y):
+    ...     return (A @ y.reshape(3, 3)).flatten()
+    >>> y0 = np.array([[2 + 0j, 3 + 0j, 4 + 0j],
+    ...                [5 + 0j, 6 + 0j, 7 + 0j],
+    ...                [9 + 0j, 34 + 0j, 78 + 0j]])
+
+    >>> result = solve_ivp(deriv_mat, [0, 25], y0.flatten(),
+    ...                    t_eval=np.linspace(0, 25, 101))
+    >>> print(result.y[:, 0].reshape(3, 3))
+    [[ 2.+0.j  3.+0.j  4.+0.j]
+     [ 5.+0.j  6.+0.j  7.+0.j]
+     [ 9.+0.j 34.+0.j 78.+0.j]]
+    >>> print(result.y[:, -1].reshape(3, 3))
+    [[  5.67451179 +12.07938445j  17.2888073  +31.03278837j
+        37.83405768 +63.25138759j]
+     [  3.39949503 +11.82123994j  21.32530996 +44.88668871j
+        53.17531184+103.80400411j]
+     [ -2.26105874 +22.19277664j -15.1255713  +70.19616341j
+       -38.34616845+153.29039931j]]
+
+
+    """
+    if method not in METHODS and not (
+            inspect.isclass(method) and issubclass(method, OdeSolver)):
+        raise ValueError(f"`method` must be one of {METHODS} or OdeSolver class.")
+
+    t0, tf = map(float, t_span)
+
+    if args is not None:
+        # Wrap the user's fun (and jac, if given) in lambdas to hide the
+        # additional parameters.  Pass in the original fun as a keyword
+        # argument to keep it in the scope of the lambda.
+        try:
+            _ = [*(args)]
+        except TypeError as exp:
+            suggestion_tuple = (
+                "Supplied 'args' cannot be unpacked. Please supply `args`"
+                f" as a tuple (e.g. `args=({args},)`)"
+            )
+            raise TypeError(suggestion_tuple) from exp
+
+        def fun(t, x, fun=fun):
+            return fun(t, x, *args)
+        jac = options.get('jac')
+        if callable(jac):
+            options['jac'] = lambda t, x: jac(t, x, *args)
+
+    if t_eval is not None:
+        t_eval = np.asarray(t_eval)
+        if t_eval.ndim != 1:
+            raise ValueError("`t_eval` must be 1-dimensional.")
+
+        if np.any(t_eval < min(t0, tf)) or np.any(t_eval > max(t0, tf)):
+            raise ValueError("Values in `t_eval` are not within `t_span`.")
+
+        d = np.diff(t_eval)
+        if tf > t0 and np.any(d <= 0) or tf < t0 and np.any(d >= 0):
+            raise ValueError("Values in `t_eval` are not properly sorted.")
+
+        if tf > t0:
+            t_eval_i = 0
+        else:
+            # Make order of t_eval decreasing to use np.searchsorted.
+            t_eval = t_eval[::-1]
+            # This will be an upper bound for slices.
+            t_eval_i = t_eval.shape[0]
+
+    if method in METHODS:
+        method = METHODS[method]
+
+    solver = method(fun, t0, y0, tf, vectorized=vectorized, **options)
+
+    if t_eval is None:
+        ts = [t0]
+        ys = [y0]
+    elif t_eval is not None and dense_output:
+        ts = []
+        ti = [t0]
+        ys = []
+    else:
+        ts = []
+        ys = []
+
+    interpolants = []
+
+    if events is not None:
+        events, max_events, event_dir = prepare_events(events)
+        event_count = np.zeros(len(events))
+        if args is not None:
+            # Wrap user functions in lambdas to hide the additional parameters.
+            # The original event function is passed as a keyword argument to the
+            # lambda to keep the original function in scope (i.e., avoid the
+            # late binding closure "gotcha").
+            events = [lambda t, x, event=event: event(t, x, *args)
+                      for event in events]
+        g = [event(t0, y0) for event in events]
+        t_events = [[] for _ in range(len(events))]
+        y_events = [[] for _ in range(len(events))]
+    else:
+        t_events = None
+        y_events = None
+
+    status = None
+    while status is None:
+        message = solver.step()
+
+        if solver.status == 'finished':
+            status = 0
+        elif solver.status == 'failed':
+            status = -1
+            break
+
+        t_old = solver.t_old
+        t = solver.t
+        y = solver.y
+
+        if dense_output:
+            sol = solver.dense_output()
+            interpolants.append(sol)
+        else:
+            sol = None
+
+        if events is not None:
+            g_new = [event(t, y) for event in events]
+            active_events = find_active_events(g, g_new, event_dir)
+            if active_events.size > 0:
+                if sol is None:
+                    sol = solver.dense_output()
+
+                event_count[active_events] += 1
+                root_indices, roots, terminate = handle_events(
+                    sol, events, active_events, event_count, max_events,
+                    t_old, t)
+
+                for e, te in zip(root_indices, roots):
+                    t_events[e].append(te)
+                    y_events[e].append(sol(te))
+
+                if terminate:
+                    status = 1
+                    t = roots[-1]
+                    y = sol(t)
+
+            g = g_new
+
+        if t_eval is None:
+            ts.append(t)
+            ys.append(y)
+        else:
+            # The value in t_eval equal to t will be included.
+            if solver.direction > 0:
+                t_eval_i_new = np.searchsorted(t_eval, t, side='right')
+                t_eval_step = t_eval[t_eval_i:t_eval_i_new]
+            else:
+                t_eval_i_new = np.searchsorted(t_eval, t, side='left')
+                # It has to be done with two slice operations, because
+                # you can't slice to 0th element inclusive using backward
+                # slicing.
+                t_eval_step = t_eval[t_eval_i_new:t_eval_i][::-1]
+
+            if t_eval_step.size > 0:
+                if sol is None:
+                    sol = solver.dense_output()
+                ts.append(t_eval_step)
+                ys.append(sol(t_eval_step))
+                t_eval_i = t_eval_i_new
+
+        if t_eval is not None and dense_output:
+            ti.append(t)
+
+    message = MESSAGES.get(status, message)
+
+    if t_events is not None:
+        t_events = [np.asarray(te) for te in t_events]
+        y_events = [np.asarray(ye) for ye in y_events]
+
+    if t_eval is None:
+        ts = np.array(ts)
+        ys = np.vstack(ys).T
+    elif ts:
+        ts = np.hstack(ts)
+        ys = np.hstack(ys)
+
+    if dense_output:
+        if t_eval is None:
+            sol = OdeSolution(
+                ts, interpolants, alt_segment=True if method in [BDF, LSODA] else False
+            )
+        else:
+            sol = OdeSolution(
+                ti, interpolants, alt_segment=True if method in [BDF, LSODA] else False
+            )
+    else:
+        sol = None
+
+    return OdeResult(t=ts, y=ys, sol=sol, t_events=t_events, y_events=y_events,
+                     nfev=solver.nfev, njev=solver.njev, nlu=solver.nlu,
+                     status=status, message=message, success=status >= 0)

venv/lib/python3.10/site-packages/scipy/integrate/_ivp/lsoda.py

@@ -0,0 +1,224 @@
+import numpy as np
+from scipy.integrate import ode
+from .common import validate_tol, validate_first_step, warn_extraneous
+from .base import OdeSolver, DenseOutput
+
+
+class LSODA(OdeSolver):
+    """Adams/BDF method with automatic stiffness detection and switching.
+
+    This is a wrapper to the Fortran solver from ODEPACK [1]_. It switches
+    automatically between the nonstiff Adams method and the stiff BDF method.
+    The method was originally detailed in [2]_.
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system: the time derivative of the state ``y``
+        at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a
+        scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must
+        return an array of the same shape as ``y``. See `vectorized` for more
+        information.
+    t0 : float
+        Initial time.
+    y0 : array_like, shape (n,)
+        Initial state.
+    t_bound : float
+        Boundary time - the integration won't continue beyond it. It also
+        determines the direction of the integration.
+    first_step : float or None, optional
+        Initial step size. Default is ``None`` which means that the algorithm
+        should choose.
+    min_step : float, optional
+        Minimum allowed step size. Default is 0.0, i.e., the step size is not
+        bounded and determined solely by the solver.
+    max_step : float, optional
+        Maximum allowed step size. Default is np.inf, i.e., the step size is not
+        bounded and determined solely by the solver.
+    rtol, atol : float and array_like, optional
+        Relative and absolute tolerances. The solver keeps the local error
+        estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
+        relative accuracy (number of correct digits), while `atol` controls
+        absolute accuracy (number of correct decimal places). To achieve the
+        desired `rtol`, set `atol` to be smaller than the smallest value that
+        can be expected from ``rtol * abs(y)`` so that `rtol` dominates the
+        allowable error. If `atol` is larger than ``rtol * abs(y)`` the
+        number of correct digits is not guaranteed. Conversely, to achieve the
+        desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller
+        than `atol`. If components of y have different scales, it might be
+        beneficial to set different `atol` values for different components by
+        passing array_like with shape (n,) for `atol`. Default values are
+        1e-3 for `rtol` and 1e-6 for `atol`.
+    jac : None or callable, optional
+        Jacobian matrix of the right-hand side of the system with respect to
+        ``y``. The Jacobian matrix has shape (n, n) and its element (i, j) is
+        equal to ``d f_i / d y_j``. The function will be called as
+        ``jac(t, y)``. If None (default), the Jacobian will be
+        approximated by finite differences. It is generally recommended to
+        provide the Jacobian rather than relying on a finite-difference
+        approximation.
+    lband, uband : int or None
+        Parameters defining the bandwidth of the Jacobian,
+        i.e., ``jac[i, j] != 0 only for i - lband <= j <= i + uband``. Setting
+        these requires your jac routine to return the Jacobian in the packed format:
+        the returned array must have ``n`` columns and ``uband + lband + 1``
+        rows in which Jacobian diagonals are written. Specifically
+        ``jac_packed[uband + i - j , j] = jac[i, j]``. The same format is used
+        in `scipy.linalg.solve_banded` (check for an illustration).
+        These parameters can be also used with ``jac=None`` to reduce the
+        number of Jacobian elements estimated by finite differences.
+    vectorized : bool, optional
+        Whether `fun` may be called in a vectorized fashion. False (default)
+        is recommended for this solver.
+
+        If ``vectorized`` is False, `fun` will always be called with ``y`` of
+        shape ``(n,)``, where ``n = len(y0)``.
+
+        If ``vectorized`` is True, `fun` may be called with ``y`` of shape
+        ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave
+        such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of
+        the returned array is the time derivative of the state corresponding
+        with a column of ``y``).
+
+        Setting ``vectorized=True`` allows for faster finite difference
+        approximation of the Jacobian by methods 'Radau' and 'BDF', but
+        will result in slower execution for this solver.
+
+    Attributes
+    ----------
+    n : int
+        Number of equations.
+    status : string
+        Current status of the solver: 'running', 'finished' or 'failed'.
+    t_bound : float
+        Boundary time.
+    direction : float
+        Integration direction: +1 or -1.
+    t : float
+        Current time.
+    y : ndarray
+        Current state.
+    t_old : float
+        Previous time. None if no steps were made yet.
+    nfev : int
+        Number of evaluations of the right-hand side.
+    njev : int
+        Number of evaluations of the Jacobian.
+
+    References
+    ----------
+    .. [1] A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE
+           Solvers," IMACS Transactions on Scientific Computation, Vol 1.,
+           pp. 55-64, 1983.
+    .. [2] L. Petzold, "Automatic selection of methods for solving stiff and
+           nonstiff systems of ordinary differential equations", SIAM Journal
+           on Scientific and Statistical Computing, Vol. 4, No. 1, pp. 136-148,
+           1983.
+    """
+    def __init__(self, fun, t0, y0, t_bound, first_step=None, min_step=0.0,
+                 max_step=np.inf, rtol=1e-3, atol=1e-6, jac=None, lband=None,
+                 uband=None, vectorized=False, **extraneous):
+        warn_extraneous(extraneous)
+        super().__init__(fun, t0, y0, t_bound, vectorized)
+
+        if first_step is None:
+            first_step = 0  # LSODA value for automatic selection.
+        else:
+            first_step = validate_first_step(first_step, t0, t_bound)
+
+        first_step *= self.direction
+
+        if max_step == np.inf:
+            max_step = 0  # LSODA value for infinity.
+        elif max_step <= 0:
+            raise ValueError("`max_step` must be positive.")
+
+        if min_step < 0:
+            raise ValueError("`min_step` must be nonnegative.")
+
+        rtol, atol = validate_tol(rtol, atol, self.n)
+
+        solver = ode(self.fun, jac)
+        solver.set_integrator('lsoda', rtol=rtol, atol=atol, max_step=max_step,
+                              min_step=min_step, first_step=first_step,
+                              lband=lband, uband=uband)
+        solver.set_initial_value(y0, t0)
+
+        # Inject t_bound into rwork array as needed for itask=5.
+        solver._integrator.rwork[0] = self.t_bound
+        solver._integrator.call_args[4] = solver._integrator.rwork
+
+        self._lsoda_solver = solver
+
+    def _step_impl(self):
+        solver = self._lsoda_solver
+        integrator = solver._integrator
+
+        # From lsoda.step and lsoda.integrate itask=5 means take a single
+        # step and do not go past t_bound.
+        itask = integrator.call_args[2]
+        integrator.call_args[2] = 5
+        solver._y, solver.t = integrator.run(
+            solver.f, solver.jac or (lambda: None), solver._y, solver.t,
+            self.t_bound, solver.f_params, solver.jac_params)
+        integrator.call_args[2] = itask
+
+        if solver.successful():
+            self.t = solver.t
+            self.y = solver._y
+            # From LSODA Fortran source njev is equal to nlu.
+            self.njev = integrator.iwork[12]
+            self.nlu = integrator.iwork[12]
+            return True, None
+        else:
+            return False, 'Unexpected istate in LSODA.'
+
+    def _dense_output_impl(self):
+        iwork = self._lsoda_solver._integrator.iwork
+        rwork = self._lsoda_solver._integrator.rwork
+
+        # We want to produce the Nordsieck history array, yh, up to the order
+        # used in the last successful iteration. The step size is unimportant
+        # because it will be scaled out in LsodaDenseOutput. Some additional
+        # work may be required because ODEPACK's LSODA implementation produces
+        # the Nordsieck history in the state needed for the next iteration.
+
+        # iwork[13] contains order from last successful iteration, while
+        # iwork[14] contains order to be attempted next.
+        order = iwork[13]
+
+        # rwork[11] contains the step size to be attempted next, while
+        # rwork[10] contains step size from last successful iteration.
+        h = rwork[11]
+
+        # rwork[20:20 + (iwork[14] + 1) * self.n] contains entries of the
+        # Nordsieck array in state needed for next iteration. We want
+        # the entries up to order for the last successful step so use the 
+        # following.
+        yh = np.reshape(rwork[20:20 + (order + 1) * self.n],
+                        (self.n, order + 1), order='F').copy()
+        if iwork[14] < order:
+            # If the order is set to decrease then the final column of yh
+            # has not been updated within ODEPACK's LSODA
+            # implementation because this column will not be used in the
+            # next iteration. We must rescale this column to make the
+            # associated step size consistent with the other columns.
+            yh[:, -1] *= (h / rwork[10]) ** order
+
+        return LsodaDenseOutput(self.t_old, self.t, h, order, yh)
+
+
+class LsodaDenseOutput(DenseOutput):
+    def __init__(self, t_old, t, h, order, yh):
+        super().__init__(t_old, t)
+        self.h = h
+        self.yh = yh
+        self.p = np.arange(order + 1)
+
+    def _call_impl(self, t):
+        if t.ndim == 0:
+            x = ((t - self.t) / self.h) ** self.p
+        else:
+            x = ((t - self.t) / self.h) ** self.p[:, None]
+
+        return np.dot(self.yh, x)

venv/lib/python3.10/site-packages/scipy/integrate/_ivp/radau.py

@@ -0,0 +1,574 @@
+import numpy as np
+from scipy.linalg import lu_factor, lu_solve
+from scipy.sparse import csc_matrix, issparse, eye
+from scipy.sparse.linalg import splu
+from scipy.optimize._numdiff import group_columns
+from .common import (validate_max_step, validate_tol, select_initial_step,
+                     norm, num_jac, EPS, warn_extraneous,
+                     validate_first_step)
+from .base import OdeSolver, DenseOutput
+
+S6 = 6 ** 0.5
+
+# Butcher tableau. A is not used directly, see below.
+C = np.array([(4 - S6) / 10, (4 + S6) / 10, 1])
+E = np.array([-13 - 7 * S6, -13 + 7 * S6, -1]) / 3
+
+# Eigendecomposition of A is done: A = T L T**-1. There is 1 real eigenvalue
+# and a complex conjugate pair. They are written below.
+MU_REAL = 3 + 3 ** (2 / 3) - 3 ** (1 / 3)
+MU_COMPLEX = (3 + 0.5 * (3 ** (1 / 3) - 3 ** (2 / 3))
+              - 0.5j * (3 ** (5 / 6) + 3 ** (7 / 6)))
+
+# These are transformation matrices.
+T = np.array([
+    [0.09443876248897524, -0.14125529502095421, 0.03002919410514742],
+    [0.25021312296533332, 0.20412935229379994, -0.38294211275726192],
+    [1, 1, 0]])
+TI = np.array([
+    [4.17871859155190428, 0.32768282076106237, 0.52337644549944951],
+    [-4.17871859155190428, -0.32768282076106237, 0.47662355450055044],
+    [0.50287263494578682, -2.57192694985560522, 0.59603920482822492]])
+# These linear combinations are used in the algorithm.
+TI_REAL = TI[0]
+TI_COMPLEX = TI[1] + 1j * TI[2]
+
+# Interpolator coefficients.
+P = np.array([
+    [13/3 + 7*S6/3, -23/3 - 22*S6/3, 10/3 + 5 * S6],
+    [13/3 - 7*S6/3, -23/3 + 22*S6/3, 10/3 - 5 * S6],
+    [1/3, -8/3, 10/3]])
+
+
+NEWTON_MAXITER = 6  # Maximum number of Newton iterations.
+MIN_FACTOR = 0.2  # Minimum allowed decrease in a step size.
+MAX_FACTOR = 10  # Maximum allowed increase in a step size.
+
+
+def solve_collocation_system(fun, t, y, h, Z0, scale, tol,
+                             LU_real, LU_complex, solve_lu):
+    """Solve the collocation system.
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system.
+    t : float
+        Current time.
+    y : ndarray, shape (n,)
+        Current state.
+    h : float
+        Step to try.
+    Z0 : ndarray, shape (3, n)
+        Initial guess for the solution. It determines new values of `y` at
+        ``t + h * C`` as ``y + Z0``, where ``C`` is the Radau method constants.
+    scale : ndarray, shape (n)
+        Problem tolerance scale, i.e. ``rtol * abs(y) + atol``.
+    tol : float
+        Tolerance to which solve the system. This value is compared with
+        the normalized by `scale` error.
+    LU_real, LU_complex
+        LU decompositions of the system Jacobians.
+    solve_lu : callable
+        Callable which solves a linear system given a LU decomposition. The
+        signature is ``solve_lu(LU, b)``.
+
+    Returns
+    -------
+    converged : bool
+        Whether iterations converged.
+    n_iter : int
+        Number of completed iterations.
+    Z : ndarray, shape (3, n)
+        Found solution.
+    rate : float
+        The rate of convergence.
+    """
+    n = y.shape[0]
+    M_real = MU_REAL / h
+    M_complex = MU_COMPLEX / h
+
+    W = TI.dot(Z0)
+    Z = Z0
+
+    F = np.empty((3, n))
+    ch = h * C
+
+    dW_norm_old = None
+    dW = np.empty_like(W)
+    converged = False
+    rate = None
+    for k in range(NEWTON_MAXITER):
+        for i in range(3):
+            F[i] = fun(t + ch[i], y + Z[i])
+
+        if not np.all(np.isfinite(F)):
+            break
+
+        f_real = F.T.dot(TI_REAL) - M_real * W[0]
+        f_complex = F.T.dot(TI_COMPLEX) - M_complex * (W[1] + 1j * W[2])
+
+        dW_real = solve_lu(LU_real, f_real)
+        dW_complex = solve_lu(LU_complex, f_complex)
+
+        dW[0] = dW_real
+        dW[1] = dW_complex.real
+        dW[2] = dW_complex.imag
+
+        dW_norm = norm(dW / scale)
+        if dW_norm_old is not None:
+            rate = dW_norm / dW_norm_old
+
+        if (rate is not None and (rate >= 1 or
+                rate ** (NEWTON_MAXITER - k) / (1 - rate) * dW_norm > tol)):
+            break
+
+        W += dW
+        Z = T.dot(W)
+
+        if (dW_norm == 0 or
+                rate is not None and rate / (1 - rate) * dW_norm < tol):
+            converged = True
+            break
+
+        dW_norm_old = dW_norm
+
+    return converged, k + 1, Z, rate
+
+
+def predict_factor(h_abs, h_abs_old, error_norm, error_norm_old):
+    """Predict by which factor to increase/decrease the step size.
+
+    The algorithm is described in [1]_.
+
+    Parameters
+    ----------
+    h_abs, h_abs_old : float
+        Current and previous values of the step size, `h_abs_old` can be None
+        (see Notes).
+    error_norm, error_norm_old : float
+        Current and previous values of the error norm, `error_norm_old` can
+        be None (see Notes).
+
+    Returns
+    -------
+    factor : float
+        Predicted factor.
+
+    Notes
+    -----
+    If `h_abs_old` and `error_norm_old` are both not None then a two-step
+    algorithm is used, otherwise a one-step algorithm is used.
+
+    References
+    ----------
+    .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
+           Equations II: Stiff and Differential-Algebraic Problems", Sec. IV.8.
+    """
+    if error_norm_old is None or h_abs_old is None or error_norm == 0:
+        multiplier = 1
+    else:
+        multiplier = h_abs / h_abs_old * (error_norm_old / error_norm) ** 0.25
+
+    with np.errstate(divide='ignore'):
+        factor = min(1, multiplier) * error_norm ** -0.25
+
+    return factor
+
+
+class Radau(OdeSolver):
+    """Implicit Runge-Kutta method of Radau IIA family of order 5.
+
+    The implementation follows [1]_. The error is controlled with a
+    third-order accurate embedded formula. A cubic polynomial which satisfies
+    the collocation conditions is used for the dense output.
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system: the time derivative of the state ``y``
+        at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a
+        scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must
+        return an array of the same shape as ``y``. See `vectorized` for more
+        information.
+    t0 : float
+        Initial time.
+    y0 : array_like, shape (n,)
+        Initial state.
+    t_bound : float
+        Boundary time - the integration won't continue beyond it. It also
+        determines the direction of the integration.
+    first_step : float or None, optional
+        Initial step size. Default is ``None`` which means that the algorithm
+        should choose.
+    max_step : float, optional
+        Maximum allowed step size. Default is np.inf, i.e., the step size is not
+        bounded and determined solely by the solver.
+    rtol, atol : float and array_like, optional
+        Relative and absolute tolerances. The solver keeps the local error
+        estimates less than ``atol + rtol * abs(y)``. HHere `rtol` controls a
+        relative accuracy (number of correct digits), while `atol` controls
+        absolute accuracy (number of correct decimal places). To achieve the
+        desired `rtol`, set `atol` to be smaller than the smallest value that
+        can be expected from ``rtol * abs(y)`` so that `rtol` dominates the
+        allowable error. If `atol` is larger than ``rtol * abs(y)`` the
+        number of correct digits is not guaranteed. Conversely, to achieve the
+        desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller
+        than `atol`. If components of y have different scales, it might be
+        beneficial to set different `atol` values for different components by
+        passing array_like with shape (n,) for `atol`. Default values are
+        1e-3 for `rtol` and 1e-6 for `atol`.
+    jac : {None, array_like, sparse_matrix, callable}, optional
+        Jacobian matrix of the right-hand side of the system with respect to
+        y, required by this method. The Jacobian matrix has shape (n, n) and
+        its element (i, j) is equal to ``d f_i / d y_j``.
+        There are three ways to define the Jacobian:
+
+            * If array_like or sparse_matrix, the Jacobian is assumed to
+              be constant.
+            * If callable, the Jacobian is assumed to depend on both
+              t and y; it will be called as ``jac(t, y)`` as necessary.
+              For the 'Radau' and 'BDF' methods, the return value might be a
+              sparse matrix.
+            * If None (default), the Jacobian will be approximated by
+              finite differences.
+
+        It is generally recommended to provide the Jacobian rather than
+        relying on a finite-difference approximation.
+    jac_sparsity : {None, array_like, sparse matrix}, optional
+        Defines a sparsity structure of the Jacobian matrix for a
+        finite-difference approximation. Its shape must be (n, n). This argument
+        is ignored if `jac` is not `None`. If the Jacobian has only few non-zero
+        elements in *each* row, providing the sparsity structure will greatly
+        speed up the computations [2]_. A zero entry means that a corresponding
+        element in the Jacobian is always zero. If None (default), the Jacobian
+        is assumed to be dense.
+    vectorized : bool, optional
+        Whether `fun` can be called in a vectorized fashion. Default is False.
+
+        If ``vectorized`` is False, `fun` will always be called with ``y`` of
+        shape ``(n,)``, where ``n = len(y0)``.
+
+        If ``vectorized`` is True, `fun` may be called with ``y`` of shape
+        ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave
+        such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of
+        the returned array is the time derivative of the state corresponding
+        with a column of ``y``).
+
+        Setting ``vectorized=True`` allows for faster finite difference
+        approximation of the Jacobian by this method, but may result in slower
+        execution overall in some circumstances (e.g. small ``len(y0)``).
+
+    Attributes
+    ----------
+    n : int
+        Number of equations.
+    status : string
+        Current status of the solver: 'running', 'finished' or 'failed'.
+    t_bound : float
+        Boundary time.
+    direction : float
+        Integration direction: +1 or -1.
+    t : float
+        Current time.
+    y : ndarray
+        Current state.
+    t_old : float
+        Previous time. None if no steps were made yet.
+    step_size : float
+        Size of the last successful step. None if no steps were made yet.
+    nfev : int
+        Number of evaluations of the right-hand side.
+    njev : int
+        Number of evaluations of the Jacobian.
+    nlu : int
+        Number of LU decompositions.
+
+    References
+    ----------
+    .. [1] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II:
+           Stiff and Differential-Algebraic Problems", Sec. IV.8.
+    .. [2] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
+           sparse Jacobian matrices", Journal of the Institute of Mathematics
+           and its Applications, 13, pp. 117-120, 1974.
+    """
+    def __init__(self, fun, t0, y0, t_bound, max_step=np.inf,
+                 rtol=1e-3, atol=1e-6, jac=None, jac_sparsity=None,
+                 vectorized=False, first_step=None, **extraneous):
+        warn_extraneous(extraneous)
+        super().__init__(fun, t0, y0, t_bound, vectorized)
+        self.y_old = None
+        self.max_step = validate_max_step(max_step)
+        self.rtol, self.atol = validate_tol(rtol, atol, self.n)
+        self.f = self.fun(self.t, self.y)
+        # Select initial step assuming the same order which is used to control
+        # the error.
+        if first_step is None:
+            self.h_abs = select_initial_step(
+                self.fun, self.t, self.y, self.f, self.direction,
+                3, self.rtol, self.atol)
+        else:
+            self.h_abs = validate_first_step(first_step, t0, t_bound)
+        self.h_abs_old = None
+        self.error_norm_old = None
+
+        self.newton_tol = max(10 * EPS / rtol, min(0.03, rtol ** 0.5))
+        self.sol = None
+
+        self.jac_factor = None
+        self.jac, self.J = self._validate_jac(jac, jac_sparsity)
+        if issparse(self.J):
+            def lu(A):
+                self.nlu += 1
+                return splu(A)
+
+            def solve_lu(LU, b):
+                return LU.solve(b)
+
+            I = eye(self.n, format='csc')
+        else:
+            def lu(A):
+                self.nlu += 1
+                return lu_factor(A, overwrite_a=True)
+
+            def solve_lu(LU, b):
+                return lu_solve(LU, b, overwrite_b=True)
+
+            I = np.identity(self.n)
+
+        self.lu = lu
+        self.solve_lu = solve_lu
+        self.I = I
+
+        self.current_jac = True
+        self.LU_real = None
+        self.LU_complex = None
+        self.Z = None
+
+    def _validate_jac(self, jac, sparsity):
+        t0 = self.t
+        y0 = self.y
+
+        if jac is None:
+            if sparsity is not None:
+                if issparse(sparsity):
+                    sparsity = csc_matrix(sparsity)
+                groups = group_columns(sparsity)
+                sparsity = (sparsity, groups)
+
+            def jac_wrapped(t, y, f):
+                self.njev += 1
+                J, self.jac_factor = num_jac(self.fun_vectorized, t, y, f,
+                                             self.atol, self.jac_factor,
+                                             sparsity)
+                return J
+            J = jac_wrapped(t0, y0, self.f)
+        elif callable(jac):
+            J = jac(t0, y0)
+            self.njev = 1
+            if issparse(J):
+                J = csc_matrix(J)
+
+                def jac_wrapped(t, y, _=None):
+                    self.njev += 1
+                    return csc_matrix(jac(t, y), dtype=float)
+
+            else:
+                J = np.asarray(J, dtype=float)
+
+                def jac_wrapped(t, y, _=None):
+                    self.njev += 1
+                    return np.asarray(jac(t, y), dtype=float)
+
+            if J.shape != (self.n, self.n):
+                raise ValueError("`jac` is expected to have shape {}, but "
+                                 "actually has {}."
+                                 .format((self.n, self.n), J.shape))
+        else:
+            if issparse(jac):
+                J = csc_matrix(jac)
+            else:
+                J = np.asarray(jac, dtype=float)
+
+            if J.shape != (self.n, self.n):
+                raise ValueError("`jac` is expected to have shape {}, but "
+                                 "actually has {}."
+                                 .format((self.n, self.n), J.shape))
+            jac_wrapped = None
+
+        return jac_wrapped, J
+
+    def _step_impl(self):
+        t = self.t
+        y = self.y
+        f = self.f
+
+        max_step = self.max_step
+        atol = self.atol
+        rtol = self.rtol
+
+        min_step = 10 * np.abs(np.nextafter(t, self.direction * np.inf) - t)
+        if self.h_abs > max_step:
+            h_abs = max_step
+            h_abs_old = None
+            error_norm_old = None
+        elif self.h_abs < min_step:
+            h_abs = min_step
+            h_abs_old = None
+            error_norm_old = None
+        else:
+            h_abs = self.h_abs
+            h_abs_old = self.h_abs_old
+            error_norm_old = self.error_norm_old
+
+        J = self.J
+        LU_real = self.LU_real
+        LU_complex = self.LU_complex
+
+        current_jac = self.current_jac
+        jac = self.jac
+
+        rejected = False
+        step_accepted = False
+        message = None
+        while not step_accepted:
+            if h_abs < min_step:
+                return False, self.TOO_SMALL_STEP
+
+            h = h_abs * self.direction
+            t_new = t + h
+
+            if self.direction * (t_new - self.t_bound) > 0:
+                t_new = self.t_bound
+
+            h = t_new - t
+            h_abs = np.abs(h)
+
+            if self.sol is None:
+                Z0 = np.zeros((3, y.shape[0]))
+            else:
+                Z0 = self.sol(t + h * C).T - y
+
+            scale = atol + np.abs(y) * rtol
+
+            converged = False
+            while not converged:
+                if LU_real is None or LU_complex is None:
+                    LU_real = self.lu(MU_REAL / h * self.I - J)
+                    LU_complex = self.lu(MU_COMPLEX / h * self.I - J)
+
+                converged, n_iter, Z, rate = solve_collocation_system(
+                    self.fun, t, y, h, Z0, scale, self.newton_tol,
+                    LU_real, LU_complex, self.solve_lu)
+
+                if not converged:
+                    if current_jac:
+                        break
+
+                    J = self.jac(t, y, f)
+                    current_jac = True
+                    LU_real = None
+                    LU_complex = None
+
+            if not converged:
+                h_abs *= 0.5
+                LU_real = None
+                LU_complex = None
+                continue
+
+            y_new = y + Z[-1]
+            ZE = Z.T.dot(E) / h
+            error = self.solve_lu(LU_real, f + ZE)
+            scale = atol + np.maximum(np.abs(y), np.abs(y_new)) * rtol
+            error_norm = norm(error / scale)
+            safety = 0.9 * (2 * NEWTON_MAXITER + 1) / (2 * NEWTON_MAXITER
+                                                       + n_iter)
+
+            if rejected and error_norm > 1:
+                error = self.solve_lu(LU_real, self.fun(t, y + error) + ZE)
+                error_norm = norm(error / scale)
+
+            if error_norm > 1:
+                factor = predict_factor(h_abs, h_abs_old,
+                                        error_norm, error_norm_old)
+                h_abs *= max(MIN_FACTOR, safety * factor)
+
+                LU_real = None
+                LU_complex = None
+                rejected = True
+            else:
+                step_accepted = True
+
+        recompute_jac = jac is not None and n_iter > 2 and rate > 1e-3
+
+        factor = predict_factor(h_abs, h_abs_old, error_norm, error_norm_old)
+        factor = min(MAX_FACTOR, safety * factor)
+
+        if not recompute_jac and factor < 1.2:
+            factor = 1
+        else:
+            LU_real = None
+            LU_complex = None
+
+        f_new = self.fun(t_new, y_new)
+        if recompute_jac:
+            J = jac(t_new, y_new, f_new)
+            current_jac = True
+        elif jac is not None:
+            current_jac = False
+
+        self.h_abs_old = self.h_abs
+        self.error_norm_old = error_norm
+
+        self.h_abs = h_abs * factor
+
+        self.y_old = y
+
+        self.t = t_new
+        self.y = y_new
+        self.f = f_new
+
+        self.Z = Z
+
+        self.LU_real = LU_real
+        self.LU_complex = LU_complex
+        self.current_jac = current_jac
+        self.J = J
+
+        self.t_old = t
+        self.sol = self._compute_dense_output()
+
+        return step_accepted, message
+
+    def _compute_dense_output(self):
+        Q = np.dot(self.Z.T, P)
+        return RadauDenseOutput(self.t_old, self.t, self.y_old, Q)
+
+    def _dense_output_impl(self):
+        return self.sol
+
+
+class RadauDenseOutput(DenseOutput):
+    def __init__(self, t_old, t, y_old, Q):
+        super().__init__(t_old, t)
+        self.h = t - t_old
+        self.Q = Q
+        self.order = Q.shape[1] - 1
+        self.y_old = y_old
+
+    def _call_impl(self, t):
+        x = (t - self.t_old) / self.h
+        if t.ndim == 0:
+            p = np.tile(x, self.order + 1)
+            p = np.cumprod(p)
+        else:
+            p = np.tile(x, (self.order + 1, 1))
+            p = np.cumprod(p, axis=0)
+        # Here we don't multiply by h, not a mistake.
+        y = np.dot(self.Q, p)
+        if y.ndim == 2:
+            y += self.y_old[:, None]
+        else:
+            y += self.y_old
+
+        return y

venv/lib/python3.10/site-packages/scipy/integrate/_ivp/rk.py

@@ -0,0 +1,601 @@
+import numpy as np
+from .base import OdeSolver, DenseOutput
+from .common import (validate_max_step, validate_tol, select_initial_step,
+                     norm, warn_extraneous, validate_first_step)
+from . import dop853_coefficients
+
+# Multiply steps computed from asymptotic behaviour of errors by this.
+SAFETY = 0.9
+
+MIN_FACTOR = 0.2  # Minimum allowed decrease in a step size.
+MAX_FACTOR = 10  # Maximum allowed increase in a step size.
+
+
+def rk_step(fun, t, y, f, h, A, B, C, K):
+    """Perform a single Runge-Kutta step.
+
+    This function computes a prediction of an explicit Runge-Kutta method and
+    also estimates the error of a less accurate method.
+
+    Notation for Butcher tableau is as in [1]_.
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system.
+    t : float
+        Current time.
+    y : ndarray, shape (n,)
+        Current state.
+    f : ndarray, shape (n,)
+        Current value of the derivative, i.e., ``fun(x, y)``.
+    h : float
+        Step to use.
+    A : ndarray, shape (n_stages, n_stages)
+        Coefficients for combining previous RK stages to compute the next
+        stage. For explicit methods the coefficients at and above the main
+        diagonal are zeros.
+    B : ndarray, shape (n_stages,)
+        Coefficients for combining RK stages for computing the final
+        prediction.
+    C : ndarray, shape (n_stages,)
+        Coefficients for incrementing time for consecutive RK stages.
+        The value for the first stage is always zero.
+    K : ndarray, shape (n_stages + 1, n)
+        Storage array for putting RK stages here. Stages are stored in rows.
+        The last row is a linear combination of the previous rows with
+        coefficients
+
+    Returns
+    -------
+    y_new : ndarray, shape (n,)
+        Solution at t + h computed with a higher accuracy.
+    f_new : ndarray, shape (n,)
+        Derivative ``fun(t + h, y_new)``.
+
+    References
+    ----------
+    .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
+           Equations I: Nonstiff Problems", Sec. II.4.
+    """
+    K[0] = f
+    for s, (a, c) in enumerate(zip(A[1:], C[1:]), start=1):
+        dy = np.dot(K[:s].T, a[:s]) * h
+        K[s] = fun(t + c * h, y + dy)
+
+    y_new = y + h * np.dot(K[:-1].T, B)
+    f_new = fun(t + h, y_new)
+
+    K[-1] = f_new
+
+    return y_new, f_new
+
+
+class RungeKutta(OdeSolver):
+    """Base class for explicit Runge-Kutta methods."""
+    C: np.ndarray = NotImplemented
+    A: np.ndarray = NotImplemented
+    B: np.ndarray = NotImplemented
+    E: np.ndarray = NotImplemented
+    P: np.ndarray = NotImplemented
+    order: int = NotImplemented
+    error_estimator_order: int = NotImplemented
+    n_stages: int = NotImplemented
+
+    def __init__(self, fun, t0, y0, t_bound, max_step=np.inf,
+                 rtol=1e-3, atol=1e-6, vectorized=False,
+                 first_step=None, **extraneous):
+        warn_extraneous(extraneous)
+        super().__init__(fun, t0, y0, t_bound, vectorized,
+                         support_complex=True)
+        self.y_old = None
+        self.max_step = validate_max_step(max_step)
+        self.rtol, self.atol = validate_tol(rtol, atol, self.n)
+        self.f = self.fun(self.t, self.y)
+        if first_step is None:
+            self.h_abs = select_initial_step(
+                self.fun, self.t, self.y, self.f, self.direction,
+                self.error_estimator_order, self.rtol, self.atol)
+        else:
+            self.h_abs = validate_first_step(first_step, t0, t_bound)
+        self.K = np.empty((self.n_stages + 1, self.n), dtype=self.y.dtype)
+        self.error_exponent = -1 / (self.error_estimator_order + 1)
+        self.h_previous = None
+
+    def _estimate_error(self, K, h):
+        return np.dot(K.T, self.E) * h
+
+    def _estimate_error_norm(self, K, h, scale):
+        return norm(self._estimate_error(K, h) / scale)
+
+    def _step_impl(self):
+        t = self.t
+        y = self.y
+
+        max_step = self.max_step
+        rtol = self.rtol
+        atol = self.atol
+
+        min_step = 10 * np.abs(np.nextafter(t, self.direction * np.inf) - t)
+
+        if self.h_abs > max_step:
+            h_abs = max_step
+        elif self.h_abs < min_step:
+            h_abs = min_step
+        else:
+            h_abs = self.h_abs
+
+        step_accepted = False
+        step_rejected = False
+
+        while not step_accepted:
+            if h_abs < min_step:
+                return False, self.TOO_SMALL_STEP
+
+            h = h_abs * self.direction
+            t_new = t + h
+
+            if self.direction * (t_new - self.t_bound) > 0:
+                t_new = self.t_bound
+
+            h = t_new - t
+            h_abs = np.abs(h)
+
+            y_new, f_new = rk_step(self.fun, t, y, self.f, h, self.A,
+                                   self.B, self.C, self.K)
+            scale = atol + np.maximum(np.abs(y), np.abs(y_new)) * rtol
+            error_norm = self._estimate_error_norm(self.K, h, scale)
+
+            if error_norm < 1:
+                if error_norm == 0:
+                    factor = MAX_FACTOR
+                else:
+                    factor = min(MAX_FACTOR,
+                                 SAFETY * error_norm ** self.error_exponent)
+
+                if step_rejected:
+                    factor = min(1, factor)
+
+                h_abs *= factor
+
+                step_accepted = True
+            else:
+                h_abs *= max(MIN_FACTOR,
+                             SAFETY * error_norm ** self.error_exponent)
+                step_rejected = True
+
+        self.h_previous = h
+        self.y_old = y
+
+        self.t = t_new
+        self.y = y_new
+
+        self.h_abs = h_abs
+        self.f = f_new
+
+        return True, None
+
+    def _dense_output_impl(self):
+        Q = self.K.T.dot(self.P)
+        return RkDenseOutput(self.t_old, self.t, self.y_old, Q)
+
+
+class RK23(RungeKutta):
+    """Explicit Runge-Kutta method of order 3(2).
+
+    This uses the Bogacki-Shampine pair of formulas [1]_. The error is controlled
+    assuming accuracy of the second-order method, but steps are taken using the
+    third-order accurate formula (local extrapolation is done). A cubic Hermite
+    polynomial is used for the dense output.
+
+    Can be applied in the complex domain.
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system: the time derivative of the state ``y``
+        at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a
+        scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must
+        return an array of the same shape as ``y``. See `vectorized` for more
+        information.
+    t0 : float
+        Initial time.
+    y0 : array_like, shape (n,)
+        Initial state.
+    t_bound : float
+        Boundary time - the integration won't continue beyond it. It also
+        determines the direction of the integration.
+    first_step : float or None, optional
+        Initial step size. Default is ``None`` which means that the algorithm
+        should choose.
+    max_step : float, optional
+        Maximum allowed step size. Default is np.inf, i.e., the step size is not
+        bounded and determined solely by the solver.
+    rtol, atol : float and array_like, optional
+        Relative and absolute tolerances. The solver keeps the local error
+        estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
+        relative accuracy (number of correct digits), while `atol` controls
+        absolute accuracy (number of correct decimal places). To achieve the
+        desired `rtol`, set `atol` to be smaller than the smallest value that
+        can be expected from ``rtol * abs(y)`` so that `rtol` dominates the
+        allowable error. If `atol` is larger than ``rtol * abs(y)`` the
+        number of correct digits is not guaranteed. Conversely, to achieve the
+        desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller
+        than `atol`. If components of y have different scales, it might be
+        beneficial to set different `atol` values for different components by
+        passing array_like with shape (n,) for `atol`. Default values are
+        1e-3 for `rtol` and 1e-6 for `atol`.
+    vectorized : bool, optional
+        Whether `fun` may be called in a vectorized fashion. False (default)
+        is recommended for this solver.
+
+        If ``vectorized`` is False, `fun` will always be called with ``y`` of
+        shape ``(n,)``, where ``n = len(y0)``.
+
+        If ``vectorized`` is True, `fun` may be called with ``y`` of shape
+        ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave
+        such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of
+        the returned array is the time derivative of the state corresponding
+        with a column of ``y``).
+
+        Setting ``vectorized=True`` allows for faster finite difference
+        approximation of the Jacobian by methods 'Radau' and 'BDF', but
+        will result in slower execution for this solver.
+
+    Attributes
+    ----------
+    n : int
+        Number of equations.
+    status : string
+        Current status of the solver: 'running', 'finished' or 'failed'.
+    t_bound : float
+        Boundary time.
+    direction : float
+        Integration direction: +1 or -1.
+    t : float
+        Current time.
+    y : ndarray
+        Current state.
+    t_old : float
+        Previous time. None if no steps were made yet.
+    step_size : float
+        Size of the last successful step. None if no steps were made yet.
+    nfev : int
+        Number evaluations of the system's right-hand side.
+    njev : int
+        Number of evaluations of the Jacobian.
+        Is always 0 for this solver as it does not use the Jacobian.
+    nlu : int
+        Number of LU decompositions. Is always 0 for this solver.
+
+    References
+    ----------
+    .. [1] P. Bogacki, L.F. Shampine, "A 3(2) Pair of Runge-Kutta Formulas",
+           Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989.
+    """
+    order = 3
+    error_estimator_order = 2
+    n_stages = 3
+    C = np.array([0, 1/2, 3/4])
+    A = np.array([
+        [0, 0, 0],
+        [1/2, 0, 0],
+        [0, 3/4, 0]
+    ])
+    B = np.array([2/9, 1/3, 4/9])
+    E = np.array([5/72, -1/12, -1/9, 1/8])
+    P = np.array([[1, -4 / 3, 5 / 9],
+                  [0, 1, -2/3],
+                  [0, 4/3, -8/9],
+                  [0, -1, 1]])
+
+
+class RK45(RungeKutta):
+    """Explicit Runge-Kutta method of order 5(4).
+
+    This uses the Dormand-Prince pair of formulas [1]_. The error is controlled
+    assuming accuracy of the fourth-order method accuracy, but steps are taken
+    using the fifth-order accurate formula (local extrapolation is done).
+    A quartic interpolation polynomial is used for the dense output [2]_.
+
+    Can be applied in the complex domain.
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system. The calling signature is ``fun(t, y)``.
+        Here ``t`` is a scalar, and there are two options for the ndarray ``y``:
+        It can either have shape (n,); then ``fun`` must return array_like with
+        shape (n,). Alternatively it can have shape (n, k); then ``fun``
+        must return an array_like with shape (n, k), i.e., each column
+        corresponds to a single column in ``y``. The choice between the two
+        options is determined by `vectorized` argument (see below).
+    t0 : float
+        Initial time.
+    y0 : array_like, shape (n,)
+        Initial state.
+    t_bound : float
+        Boundary time - the integration won't continue beyond it. It also
+        determines the direction of the integration.
+    first_step : float or None, optional
+        Initial step size. Default is ``None`` which means that the algorithm
+        should choose.
+    max_step : float, optional
+        Maximum allowed step size. Default is np.inf, i.e., the step size is not
+        bounded and determined solely by the solver.
+    rtol, atol : float and array_like, optional
+        Relative and absolute tolerances. The solver keeps the local error
+        estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
+        relative accuracy (number of correct digits), while `atol` controls
+        absolute accuracy (number of correct decimal places). To achieve the
+        desired `rtol`, set `atol` to be smaller than the smallest value that
+        can be expected from ``rtol * abs(y)`` so that `rtol` dominates the
+        allowable error. If `atol` is larger than ``rtol * abs(y)`` the
+        number of correct digits is not guaranteed. Conversely, to achieve the
+        desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller
+        than `atol`. If components of y have different scales, it might be
+        beneficial to set different `atol` values for different components by
+        passing array_like with shape (n,) for `atol`. Default values are
+        1e-3 for `rtol` and 1e-6 for `atol`.
+    vectorized : bool, optional
+        Whether `fun` is implemented in a vectorized fashion. Default is False.
+
+    Attributes
+    ----------
+    n : int
+        Number of equations.
+    status : string
+        Current status of the solver: 'running', 'finished' or 'failed'.
+    t_bound : float
+        Boundary time.
+    direction : float
+        Integration direction: +1 or -1.
+    t : float
+        Current time.
+    y : ndarray
+        Current state.
+    t_old : float
+        Previous time. None if no steps were made yet.
+    step_size : float
+        Size of the last successful step. None if no steps were made yet.
+    nfev : int
+        Number evaluations of the system's right-hand side.
+    njev : int
+        Number of evaluations of the Jacobian.
+        Is always 0 for this solver as it does not use the Jacobian.
+    nlu : int
+        Number of LU decompositions. Is always 0 for this solver.
+
+    References
+    ----------
+    .. [1] J. R. Dormand, P. J. Prince, "A family of embedded Runge-Kutta
+           formulae", Journal of Computational and Applied Mathematics, Vol. 6,
+           No. 1, pp. 19-26, 1980.
+    .. [2] L. W. Shampine, "Some Practical Runge-Kutta Formulas", Mathematics
+           of Computation,, Vol. 46, No. 173, pp. 135-150, 1986.
+    """
+    order = 5
+    error_estimator_order = 4
+    n_stages = 6
+    C = np.array([0, 1/5, 3/10, 4/5, 8/9, 1])
+    A = np.array([
+        [0, 0, 0, 0, 0],
+        [1/5, 0, 0, 0, 0],
+        [3/40, 9/40, 0, 0, 0],
+        [44/45, -56/15, 32/9, 0, 0],
+        [19372/6561, -25360/2187, 64448/6561, -212/729, 0],
+        [9017/3168, -355/33, 46732/5247, 49/176, -5103/18656]
+    ])
+    B = np.array([35/384, 0, 500/1113, 125/192, -2187/6784, 11/84])
+    E = np.array([-71/57600, 0, 71/16695, -71/1920, 17253/339200, -22/525,
+                  1/40])
+    # Corresponds to the optimum value of c_6 from [2]_.
+    P = np.array([
+        [1, -8048581381/2820520608, 8663915743/2820520608,
+         -12715105075/11282082432],
+        [0, 0, 0, 0],
+        [0, 131558114200/32700410799, -68118460800/10900136933,
+         87487479700/32700410799],
+        [0, -1754552775/470086768, 14199869525/1410260304,
+         -10690763975/1880347072],
+        [0, 127303824393/49829197408, -318862633887/49829197408,
+         701980252875 / 199316789632],
+        [0, -282668133/205662961, 2019193451/616988883, -1453857185/822651844],
+        [0, 40617522/29380423, -110615467/29380423, 69997945/29380423]])
+
+
+class DOP853(RungeKutta):
+    """Explicit Runge-Kutta method of order 8.
+
+    This is a Python implementation of "DOP853" algorithm originally written
+    in Fortran [1]_, [2]_. Note that this is not a literal translation, but
+    the algorithmic core and coefficients are the same.
+
+    Can be applied in the complex domain.
+
+    Parameters
+    ----------
+    fun : callable
+        Right-hand side of the system. The calling signature is ``fun(t, y)``.
+        Here, ``t`` is a scalar, and there are two options for the ndarray ``y``:
+        It can either have shape (n,); then ``fun`` must return array_like with
+        shape (n,). Alternatively it can have shape (n, k); then ``fun``
+        must return an array_like with shape (n, k), i.e. each column
+        corresponds to a single column in ``y``. The choice between the two
+        options is determined by `vectorized` argument (see below).
+    t0 : float
+        Initial time.
+    y0 : array_like, shape (n,)
+        Initial state.
+    t_bound : float
+        Boundary time - the integration won't continue beyond it. It also
+        determines the direction of the integration.
+    first_step : float or None, optional
+        Initial step size. Default is ``None`` which means that the algorithm
+        should choose.
+    max_step : float, optional
+        Maximum allowed step size. Default is np.inf, i.e. the step size is not
+        bounded and determined solely by the solver.
+    rtol, atol : float and array_like, optional
+        Relative and absolute tolerances. The solver keeps the local error
+        estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
+        relative accuracy (number of correct digits), while `atol` controls
+        absolute accuracy (number of correct decimal places). To achieve the
+        desired `rtol`, set `atol` to be smaller than the smallest value that
+        can be expected from ``rtol * abs(y)`` so that `rtol` dominates the
+        allowable error. If `atol` is larger than ``rtol * abs(y)`` the
+        number of correct digits is not guaranteed. Conversely, to achieve the
+        desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller
+        than `atol`. If components of y have different scales, it might be
+        beneficial to set different `atol` values for different components by
+        passing array_like with shape (n,) for `atol`. Default values are
+        1e-3 for `rtol` and 1e-6 for `atol`.
+    vectorized : bool, optional
+        Whether `fun` is implemented in a vectorized fashion. Default is False.
+
+    Attributes
+    ----------
+    n : int
+        Number of equations.
+    status : string
+        Current status of the solver: 'running', 'finished' or 'failed'.
+    t_bound : float
+        Boundary time.
+    direction : float
+        Integration direction: +1 or -1.
+    t : float
+        Current time.
+    y : ndarray
+        Current state.
+    t_old : float
+        Previous time. None if no steps were made yet.
+    step_size : float
+        Size of the last successful step. None if no steps were made yet.
+    nfev : int
+        Number evaluations of the system's right-hand side.
+    njev : int
+        Number of evaluations of the Jacobian. Is always 0 for this solver
+        as it does not use the Jacobian.
+    nlu : int
+        Number of LU decompositions. Is always 0 for this solver.
+
+    References
+    ----------
+    .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
+           Equations I: Nonstiff Problems", Sec. II.
+    .. [2] `Page with original Fortran code of DOP853
+            <http://www.unige.ch/~hairer/software.html>`_.
+    """
+    n_stages = dop853_coefficients.N_STAGES
+    order = 8
+    error_estimator_order = 7
+    A = dop853_coefficients.A[:n_stages, :n_stages]
+    B = dop853_coefficients.B
+    C = dop853_coefficients.C[:n_stages]
+    E3 = dop853_coefficients.E3
+    E5 = dop853_coefficients.E5
+    D = dop853_coefficients.D
+
+    A_EXTRA = dop853_coefficients.A[n_stages + 1:]
+    C_EXTRA = dop853_coefficients.C[n_stages + 1:]
+
+    def __init__(self, fun, t0, y0, t_bound, max_step=np.inf,
+                 rtol=1e-3, atol=1e-6, vectorized=False,
+                 first_step=None, **extraneous):
+        super().__init__(fun, t0, y0, t_bound, max_step, rtol, atol,
+                         vectorized, first_step, **extraneous)
+        self.K_extended = np.empty((dop853_coefficients.N_STAGES_EXTENDED,
+                                    self.n), dtype=self.y.dtype)
+        self.K = self.K_extended[:self.n_stages + 1]
+
+    def _estimate_error(self, K, h):  # Left for testing purposes.
+        err5 = np.dot(K.T, self.E5)
+        err3 = np.dot(K.T, self.E3)
+        denom = np.hypot(np.abs(err5), 0.1 * np.abs(err3))
+        correction_factor = np.ones_like(err5)
+        mask = denom > 0
+        correction_factor[mask] = np.abs(err5[mask]) / denom[mask]
+        return h * err5 * correction_factor
+
+    def _estimate_error_norm(self, K, h, scale):
+        err5 = np.dot(K.T, self.E5) / scale
+        err3 = np.dot(K.T, self.E3) / scale
+        err5_norm_2 = np.linalg.norm(err5)**2
+        err3_norm_2 = np.linalg.norm(err3)**2
+        if err5_norm_2 == 0 and err3_norm_2 == 0:
+            return 0.0
+        denom = err5_norm_2 + 0.01 * err3_norm_2
+        return np.abs(h) * err5_norm_2 / np.sqrt(denom * len(scale))
+
+    def _dense_output_impl(self):
+        K = self.K_extended
+        h = self.h_previous
+        for s, (a, c) in enumerate(zip(self.A_EXTRA, self.C_EXTRA),
+                                   start=self.n_stages + 1):
+            dy = np.dot(K[:s].T, a[:s]) * h
+            K[s] = self.fun(self.t_old + c * h, self.y_old + dy)
+
+        F = np.empty((dop853_coefficients.INTERPOLATOR_POWER, self.n),
+                     dtype=self.y_old.dtype)
+
+        f_old = K[0]
+        delta_y = self.y - self.y_old
+
+        F[0] = delta_y
+        F[1] = h * f_old - delta_y
+        F[2] = 2 * delta_y - h * (self.f + f_old)
+        F[3:] = h * np.dot(self.D, K)
+
+        return Dop853DenseOutput(self.t_old, self.t, self.y_old, F)
+
+
+class RkDenseOutput(DenseOutput):
+    def __init__(self, t_old, t, y_old, Q):
+        super().__init__(t_old, t)
+        self.h = t - t_old
+        self.Q = Q
+        self.order = Q.shape[1] - 1
+        self.y_old = y_old
+
+    def _call_impl(self, t):
+        x = (t - self.t_old) / self.h
+        if t.ndim == 0:
+            p = np.tile(x, self.order + 1)
+            p = np.cumprod(p)
+        else:
+            p = np.tile(x, (self.order + 1, 1))
+            p = np.cumprod(p, axis=0)
+        y = self.h * np.dot(self.Q, p)
+        if y.ndim == 2:
+            y += self.y_old[:, None]
+        else:
+            y += self.y_old
+
+        return y
+
+
+class Dop853DenseOutput(DenseOutput):
+    def __init__(self, t_old, t, y_old, F):
+        super().__init__(t_old, t)
+        self.h = t - t_old
+        self.F = F
+        self.y_old = y_old
+
+    def _call_impl(self, t):
+        x = (t - self.t_old) / self.h
+
+        if t.ndim == 0:
+            y = np.zeros_like(self.y_old)
+        else:
+            x = x[:, None]
+            y = np.zeros((len(x), len(self.y_old)), dtype=self.y_old.dtype)
+
+        for i, f in enumerate(reversed(self.F)):
+            y += f
+            if i % 2 == 0:
+                y *= x
+            else:
+                y *= 1 - x
+        y += self.y_old
+
+        return y.T

venv/lib/python3.10/site-packages/scipy/integrate/_ivp/tests/test_ivp.py

@@ -0,0 +1,1135 @@
+from itertools import product
+from numpy.testing import (assert_, assert_allclose, assert_array_less,
+                           assert_equal, assert_no_warnings, suppress_warnings)
+import pytest
+from pytest import raises as assert_raises
+import numpy as np
+from scipy.optimize._numdiff import group_columns
+from scipy.integrate import solve_ivp, RK23, RK45, DOP853, Radau, BDF, LSODA
+from scipy.integrate import OdeSolution
+from scipy.integrate._ivp.common import num_jac
+from scipy.integrate._ivp.base import ConstantDenseOutput
+from scipy.sparse import coo_matrix, csc_matrix
+
+
+def fun_zero(t, y):
+    return np.zeros_like(y)
+
+
+def fun_linear(t, y):
+    return np.array([-y[0] - 5 * y[1], y[0] + y[1]])
+
+
+def jac_linear():
+    return np.array([[-1, -5], [1, 1]])
+
+
+def sol_linear(t):
+    return np.vstack((-5 * np.sin(2 * t),
+                      2 * np.cos(2 * t) + np.sin(2 * t)))
+
+
+def fun_rational(t, y):
+    return np.array([y[1] / t,
+                     y[1] * (y[0] + 2 * y[1] - 1) / (t * (y[0] - 1))])
+
+
+def fun_rational_vectorized(t, y):
+    return np.vstack((y[1] / t,
+                      y[1] * (y[0] + 2 * y[1] - 1) / (t * (y[0] - 1))))
+
+
+def jac_rational(t, y):
+    return np.array([
+        [0, 1 / t],
+        [-2 * y[1] ** 2 / (t * (y[0] - 1) ** 2),
+         (y[0] + 4 * y[1] - 1) / (t * (y[0] - 1))]
+    ])
+
+
+def jac_rational_sparse(t, y):
+    return csc_matrix([
+        [0, 1 / t],
+        [-2 * y[1] ** 2 / (t * (y[0] - 1) ** 2),
+         (y[0] + 4 * y[1] - 1) / (t * (y[0] - 1))]
+    ])
+
+
+def sol_rational(t):
+    return np.asarray((t / (t + 10), 10 * t / (t + 10) ** 2))
+
+
+def fun_medazko(t, y):
+    n = y.shape[0] // 2
+    k = 100
+    c = 4
+
+    phi = 2 if t <= 5 else 0
+    y = np.hstack((phi, 0, y, y[-2]))
+
+    d = 1 / n
+    j = np.arange(n) + 1
+    alpha = 2 * (j * d - 1) ** 3 / c ** 2
+    beta = (j * d - 1) ** 4 / c ** 2
+
+    j_2_p1 = 2 * j + 2
+    j_2_m3 = 2 * j - 2
+    j_2_m1 = 2 * j
+    j_2 = 2 * j + 1
+
+    f = np.empty(2 * n)
+    f[::2] = (alpha * (y[j_2_p1] - y[j_2_m3]) / (2 * d) +
+              beta * (y[j_2_m3] - 2 * y[j_2_m1] + y[j_2_p1]) / d ** 2 -
+              k * y[j_2_m1] * y[j_2])
+    f[1::2] = -k * y[j_2] * y[j_2_m1]
+
+    return f
+
+
+def medazko_sparsity(n):
+    cols = []
+    rows = []
+
+    i = np.arange(n) * 2
+
+    cols.append(i[1:])
+    rows.append(i[1:] - 2)
+
+    cols.append(i)
+    rows.append(i)
+
+    cols.append(i)
+    rows.append(i + 1)
+
+    cols.append(i[:-1])
+    rows.append(i[:-1] + 2)
+
+    i = np.arange(n) * 2 + 1
+
+    cols.append(i)
+    rows.append(i)
+
+    cols.append(i)
+    rows.append(i - 1)
+
+    cols = np.hstack(cols)
+    rows = np.hstack(rows)
+
+    return coo_matrix((np.ones_like(cols), (cols, rows)))
+
+
+def fun_complex(t, y):
+    return -y
+
+
+def jac_complex(t, y):
+    return -np.eye(y.shape[0])
+
+
+def jac_complex_sparse(t, y):
+    return csc_matrix(jac_complex(t, y))
+
+
+def sol_complex(t):
+    y = (0.5 + 1j) * np.exp(-t)
+    return y.reshape((1, -1))
+
+
+def fun_event_dense_output_LSODA(t, y):
+    return y * (t - 2)
+
+
+def jac_event_dense_output_LSODA(t, y):
+    return t - 2
+
+
+def sol_event_dense_output_LSODA(t):
+    return np.exp(t ** 2 / 2 - 2 * t + np.log(0.05) - 6)
+
+
+def compute_error(y, y_true, rtol, atol):
+    e = (y - y_true) / (atol + rtol * np.abs(y_true))
+    return np.linalg.norm(e, axis=0) / np.sqrt(e.shape[0])
+
+
+def test_integration():
+    rtol = 1e-3
+    atol = 1e-6
+    y0 = [1/3, 2/9]
+
+    for vectorized, method, t_span, jac in product(
+            [False, True],
+            ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA'],
+            [[5, 9], [5, 1]],
+            [None, jac_rational, jac_rational_sparse]):
+
+        if vectorized:
+            fun = fun_rational_vectorized
+        else:
+            fun = fun_rational
+
+        with suppress_warnings() as sup:
+            sup.filter(UserWarning,
+                       "The following arguments have no effect for a chosen "
+                       "solver: `jac`")
+            res = solve_ivp(fun, t_span, y0, rtol=rtol,
+                            atol=atol, method=method, dense_output=True,
+                            jac=jac, vectorized=vectorized)
+        assert_equal(res.t[0], t_span[0])
+        assert_(res.t_events is None)
+        assert_(res.y_events is None)
+        assert_(res.success)
+        assert_equal(res.status, 0)
+
+        if method == 'DOP853':
+            # DOP853 spends more functions evaluation because it doesn't
+            # have enough time to develop big enough step size.
+            assert_(res.nfev < 50)
+        else:
+            assert_(res.nfev < 40)
+
+        if method in ['RK23', 'RK45', 'DOP853', 'LSODA']:
+            assert_equal(res.njev, 0)
+            assert_equal(res.nlu, 0)
+        else:
+            assert_(0 < res.njev < 3)
+            assert_(0 < res.nlu < 10)
+
+        y_true = sol_rational(res.t)
+        e = compute_error(res.y, y_true, rtol, atol)
+        assert_(np.all(e < 5))
+
+        tc = np.linspace(*t_span)
+        yc_true = sol_rational(tc)
+        yc = res.sol(tc)
+
+        e = compute_error(yc, yc_true, rtol, atol)
+        assert_(np.all(e < 5))
+
+        tc = (t_span[0] + t_span[-1]) / 2
+        yc_true = sol_rational(tc)
+        yc = res.sol(tc)
+
+        e = compute_error(yc, yc_true, rtol, atol)
+        assert_(np.all(e < 5))
+
+        assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15)
+
+
+def test_integration_complex():
+    rtol = 1e-3
+    atol = 1e-6
+    y0 = [0.5 + 1j]
+    t_span = [0, 1]
+    tc = np.linspace(t_span[0], t_span[1])
+    for method, jac in product(['RK23', 'RK45', 'DOP853', 'BDF'],
+                               [None, jac_complex, jac_complex_sparse]):
+        with suppress_warnings() as sup:
+            sup.filter(UserWarning,
+                       "The following arguments have no effect for a chosen "
+                       "solver: `jac`")
+            res = solve_ivp(fun_complex, t_span, y0, method=method,
+                            dense_output=True, rtol=rtol, atol=atol, jac=jac)
+
+        assert_equal(res.t[0], t_span[0])
+        assert_(res.t_events is None)
+        assert_(res.y_events is None)
+        assert_(res.success)
+        assert_equal(res.status, 0)
+
+        if method == 'DOP853':
+            assert res.nfev < 35
+        else:
+            assert res.nfev < 25
+
+        if method == 'BDF':
+            assert_equal(res.njev, 1)
+            assert res.nlu < 6
+        else:
+            assert res.njev == 0
+            assert res.nlu == 0
+
+        y_true = sol_complex(res.t)
+        e = compute_error(res.y, y_true, rtol, atol)
+        assert np.all(e < 5)
+
+        yc_true = sol_complex(tc)
+        yc = res.sol(tc)
+        e = compute_error(yc, yc_true, rtol, atol)
+
+        assert np.all(e < 5)
+
+
+def test_integration_sparse_difference():
+    n = 200
+    t_span = [0, 20]
+    y0 = np.zeros(2 * n)
+    y0[1::2] = 1
+    sparsity = medazko_sparsity(n)
+
+    for method in ['BDF', 'Radau']:
+        res = solve_ivp(fun_medazko, t_span, y0, method=method,
+                        jac_sparsity=sparsity)
+
+        assert_equal(res.t[0], t_span[0])
+        assert_(res.t_events is None)
+        assert_(res.y_events is None)
+        assert_(res.success)
+        assert_equal(res.status, 0)
+
+        assert_allclose(res.y[78, -1], 0.233994e-3, rtol=1e-2)
+        assert_allclose(res.y[79, -1], 0, atol=1e-3)
+        assert_allclose(res.y[148, -1], 0.359561e-3, rtol=1e-2)
+        assert_allclose(res.y[149, -1], 0, atol=1e-3)
+        assert_allclose(res.y[198, -1], 0.117374129e-3, rtol=1e-2)
+        assert_allclose(res.y[199, -1], 0.6190807e-5, atol=1e-3)
+        assert_allclose(res.y[238, -1], 0, atol=1e-3)
+        assert_allclose(res.y[239, -1], 0.9999997, rtol=1e-2)
+
+
+def test_integration_const_jac():
+    rtol = 1e-3
+    atol = 1e-6
+    y0 = [0, 2]
+    t_span = [0, 2]
+    J = jac_linear()
+    J_sparse = csc_matrix(J)
+
+    for method, jac in product(['Radau', 'BDF'], [J, J_sparse]):
+        res = solve_ivp(fun_linear, t_span, y0, rtol=rtol, atol=atol,
+                        method=method, dense_output=True, jac=jac)
+        assert_equal(res.t[0], t_span[0])
+        assert_(res.t_events is None)
+        assert_(res.y_events is None)
+        assert_(res.success)
+        assert_equal(res.status, 0)
+
+        assert_(res.nfev < 100)
+        assert_equal(res.njev, 0)
+        assert_(0 < res.nlu < 15)
+
+        y_true = sol_linear(res.t)
+        e = compute_error(res.y, y_true, rtol, atol)
+        assert_(np.all(e < 10))
+
+        tc = np.linspace(*t_span)
+        yc_true = sol_linear(tc)
+        yc = res.sol(tc)
+
+        e = compute_error(yc, yc_true, rtol, atol)
+        assert_(np.all(e < 15))
+
+        assert_allclose(res.sol(res.t), res.y, rtol=1e-14, atol=1e-14)
+
+
+@pytest.mark.slow
+@pytest.mark.parametrize('method', ['Radau', 'BDF', 'LSODA'])
+def test_integration_stiff(method):
+    rtol = 1e-6
+    atol = 1e-6
+    y0 = [1e4, 0, 0]
+    tspan = [0, 1e8]
+
+    def fun_robertson(t, state):
+        x, y, z = state
+        return [
+            -0.04 * x + 1e4 * y * z,
+            0.04 * x - 1e4 * y * z - 3e7 * y * y,
+            3e7 * y * y,
+        ]
+
+    res = solve_ivp(fun_robertson, tspan, y0, rtol=rtol,
+                    atol=atol, method=method)
+
+    # If the stiff mode is not activated correctly, these numbers will be much bigger
+    assert res.nfev < 5000
+    assert res.njev < 200
+
+
+def test_events():
+    def event_rational_1(t, y):
+        return y[0] - y[1] ** 0.7
+
+    def event_rational_2(t, y):
+        return y[1] ** 0.6 - y[0]
+
+    def event_rational_3(t, y):
+        return t - 7.4
+
+    event_rational_3.terminal = True
+
+    for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']:
+        res = solve_ivp(fun_rational, [5, 8], [1/3, 2/9], method=method,
+                        events=(event_rational_1, event_rational_2))
+        assert_equal(res.status, 0)
+        assert_equal(res.t_events[0].size, 1)
+        assert_equal(res.t_events[1].size, 1)
+        assert_(5.3 < res.t_events[0][0] < 5.7)
+        assert_(7.3 < res.t_events[1][0] < 7.7)
+
+        assert_equal(res.y_events[0].shape, (1, 2))
+        assert_equal(res.y_events[1].shape, (1, 2))
+        assert np.isclose(
+            event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0)
+        assert np.isclose(
+            event_rational_2(res.t_events[1][0], res.y_events[1][0]), 0)
+
+        event_rational_1.direction = 1
+        event_rational_2.direction = 1
+        res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method,
+                        events=(event_rational_1, event_rational_2))
+        assert_equal(res.status, 0)
+        assert_equal(res.t_events[0].size, 1)
+        assert_equal(res.t_events[1].size, 0)
+        assert_(5.3 < res.t_events[0][0] < 5.7)
+        assert_equal(res.y_events[0].shape, (1, 2))
+        assert_equal(res.y_events[1].shape, (0,))
+        assert np.isclose(
+            event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0)
+
+        event_rational_1.direction = -1
+        event_rational_2.direction = -1
+        res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method,
+                        events=(event_rational_1, event_rational_2))
+        assert_equal(res.status, 0)
+        assert_equal(res.t_events[0].size, 0)
+        assert_equal(res.t_events[1].size, 1)
+        assert_(7.3 < res.t_events[1][0] < 7.7)
+        assert_equal(res.y_events[0].shape, (0,))
+        assert_equal(res.y_events[1].shape, (1, 2))
+        assert np.isclose(
+            event_rational_2(res.t_events[1][0], res.y_events[1][0]), 0)
+
+        event_rational_1.direction = 0
+        event_rational_2.direction = 0
+
+        res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method,
+                        events=(event_rational_1, event_rational_2,
+                                event_rational_3), dense_output=True)
+        assert_equal(res.status, 1)
+        assert_equal(res.t_events[0].size, 1)
+        assert_equal(res.t_events[1].size, 0)
+        assert_equal(res.t_events[2].size, 1)
+        assert_(5.3 < res.t_events[0][0] < 5.7)
+        assert_(7.3 < res.t_events[2][0] < 7.5)
+        assert_equal(res.y_events[0].shape, (1, 2))
+        assert_equal(res.y_events[1].shape, (0,))
+        assert_equal(res.y_events[2].shape, (1, 2))
+        assert np.isclose(
+            event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0)
+        assert np.isclose(
+            event_rational_3(res.t_events[2][0], res.y_events[2][0]), 0)
+
+        res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method,
+                        events=event_rational_1, dense_output=True)
+        assert_equal(res.status, 0)
+        assert_equal(res.t_events[0].size, 1)
+        assert_(5.3 < res.t_events[0][0] < 5.7)
+
+        assert_equal(res.y_events[0].shape, (1, 2))
+        assert np.isclose(
+            event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0)
+
+        # Also test that termination by event doesn't break interpolants.
+        tc = np.linspace(res.t[0], res.t[-1])
+        yc_true = sol_rational(tc)
+        yc = res.sol(tc)
+        e = compute_error(yc, yc_true, 1e-3, 1e-6)
+        assert_(np.all(e < 5))
+
+        # Test that the y_event matches solution
+        assert np.allclose(sol_rational(res.t_events[0][0]), res.y_events[0][0],
+                           rtol=1e-3, atol=1e-6)
+
+    # Test in backward direction.
+    event_rational_1.direction = 0
+    event_rational_2.direction = 0
+    for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']:
+        res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method,
+                        events=(event_rational_1, event_rational_2))
+        assert_equal(res.status, 0)
+        assert_equal(res.t_events[0].size, 1)
+        assert_equal(res.t_events[1].size, 1)
+        assert_(5.3 < res.t_events[0][0] < 5.7)
+        assert_(7.3 < res.t_events[1][0] < 7.7)
+
+        assert_equal(res.y_events[0].shape, (1, 2))
+        assert_equal(res.y_events[1].shape, (1, 2))
+        assert np.isclose(
+            event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0)
+        assert np.isclose(
+            event_rational_2(res.t_events[1][0], res.y_events[1][0]), 0)
+
+        event_rational_1.direction = -1
+        event_rational_2.direction = -1
+        res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method,
+                        events=(event_rational_1, event_rational_2))
+        assert_equal(res.status, 0)
+        assert_equal(res.t_events[0].size, 1)
+        assert_equal(res.t_events[1].size, 0)
+        assert_(5.3 < res.t_events[0][0] < 5.7)
+
+        assert_equal(res.y_events[0].shape, (1, 2))
+        assert_equal(res.y_events[1].shape, (0,))
+        assert np.isclose(
+            event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0)
+
+        event_rational_1.direction = 1
+        event_rational_2.direction = 1
+        res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method,
+                        events=(event_rational_1, event_rational_2))
+        assert_equal(res.status, 0)
+        assert_equal(res.t_events[0].size, 0)
+        assert_equal(res.t_events[1].size, 1)
+        assert_(7.3 < res.t_events[1][0] < 7.7)
+
+        assert_equal(res.y_events[0].shape, (0,))
+        assert_equal(res.y_events[1].shape, (1, 2))
+        assert np.isclose(
+            event_rational_2(res.t_events[1][0], res.y_events[1][0]), 0)
+
+        event_rational_1.direction = 0
+        event_rational_2.direction = 0
+
+        res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method,
+                        events=(event_rational_1, event_rational_2,
+                                event_rational_3), dense_output=True)
+        assert_equal(res.status, 1)
+        assert_equal(res.t_events[0].size, 0)
+        assert_equal(res.t_events[1].size, 1)
+        assert_equal(res.t_events[2].size, 1)
+        assert_(7.3 < res.t_events[1][0] < 7.7)
+        assert_(7.3 < res.t_events[2][0] < 7.5)
+
+        assert_equal(res.y_events[0].shape, (0,))
+        assert_equal(res.y_events[1].shape, (1, 2))
+        assert_equal(res.y_events[2].shape, (1, 2))
+        assert np.isclose(
+            event_rational_2(res.t_events[1][0], res.y_events[1][0]), 0)
+        assert np.isclose(
+            event_rational_3(res.t_events[2][0], res.y_events[2][0]), 0)
+
+        # Also test that termination by event doesn't break interpolants.
+        tc = np.linspace(res.t[-1], res.t[0])
+        yc_true = sol_rational(tc)
+        yc = res.sol(tc)
+        e = compute_error(yc, yc_true, 1e-3, 1e-6)
+        assert_(np.all(e < 5))
+
+        assert np.allclose(sol_rational(res.t_events[1][0]), res.y_events[1][0],
+                           rtol=1e-3, atol=1e-6)
+        assert np.allclose(sol_rational(res.t_events[2][0]), res.y_events[2][0],
+                           rtol=1e-3, atol=1e-6)
+
+
+def _get_harmonic_oscillator():
+    def f(t, y):
+        return [y[1], -y[0]]
+
+    def event(t, y):
+        return y[0]
+
+    return f, event
+
+
+@pytest.mark.parametrize('n_events', [3, 4])
+def test_event_terminal_integer(n_events):
+    f, event = _get_harmonic_oscillator()
+    event.terminal = n_events
+    res = solve_ivp(f, (0, 100), [1, 0], events=event)
+    assert len(res.t_events[0]) == n_events
+    assert len(res.y_events[0]) == n_events
+    assert_allclose(res.y_events[0][:, 0], 0, atol=1e-14)
+
+
+def test_event_terminal_iv():
+    f, event = _get_harmonic_oscillator()
+    args = (f, (0, 100), [1, 0])
+
+    event.terminal = None
+    res = solve_ivp(*args, events=event)
+    event.terminal = 0
+    ref = solve_ivp(*args, events=event)
+    assert_allclose(res.t_events, ref.t_events)
+
+    message = "The `terminal` attribute..."
+    event.terminal = -1
+    with pytest.raises(ValueError, match=message):
+        solve_ivp(*args, events=event)
+    event.terminal = 3.5
+    with pytest.raises(ValueError, match=message):
+        solve_ivp(*args, events=event)
+
+
+def test_max_step():
+    rtol = 1e-3
+    atol = 1e-6
+    y0 = [1/3, 2/9]
+    for method in [RK23, RK45, DOP853, Radau, BDF, LSODA]:
+        for t_span in ([5, 9], [5, 1]):
+            res = solve_ivp(fun_rational, t_span, y0, rtol=rtol,
+                            max_step=0.5, atol=atol, method=method,
+                            dense_output=True)
+            assert_equal(res.t[0], t_span[0])
+            assert_equal(res.t[-1], t_span[-1])
+            assert_(np.all(np.abs(np.diff(res.t)) <= 0.5 + 1e-15))
+            assert_(res.t_events is None)
+            assert_(res.success)
+            assert_equal(res.status, 0)
+
+            y_true = sol_rational(res.t)
+            e = compute_error(res.y, y_true, rtol, atol)
+            assert_(np.all(e < 5))
+
+            tc = np.linspace(*t_span)
+            yc_true = sol_rational(tc)
+            yc = res.sol(tc)
+
+            e = compute_error(yc, yc_true, rtol, atol)
+            assert_(np.all(e < 5))
+
+            assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15)
+
+            assert_raises(ValueError, method, fun_rational, t_span[0], y0,
+                          t_span[1], max_step=-1)
+
+            if method is not LSODA:
+                solver = method(fun_rational, t_span[0], y0, t_span[1],
+                                rtol=rtol, atol=atol, max_step=1e-20)
+                message = solver.step()
+
+                assert_equal(solver.status, 'failed')
+                assert_("step size is less" in message)
+                assert_raises(RuntimeError, solver.step)
+
+
+def test_first_step():
+    rtol = 1e-3
+    atol = 1e-6
+    y0 = [1/3, 2/9]
+    first_step = 0.1
+    for method in [RK23, RK45, DOP853, Radau, BDF, LSODA]:
+        for t_span in ([5, 9], [5, 1]):
+            res = solve_ivp(fun_rational, t_span, y0, rtol=rtol,
+                            max_step=0.5, atol=atol, method=method,
+                            dense_output=True, first_step=first_step)
+
+            assert_equal(res.t[0], t_span[0])
+            assert_equal(res.t[-1], t_span[-1])
+            assert_allclose(first_step, np.abs(res.t[1] - 5))
+            assert_(res.t_events is None)
+            assert_(res.success)
+            assert_equal(res.status, 0)
+
+            y_true = sol_rational(res.t)
+            e = compute_error(res.y, y_true, rtol, atol)
+            assert_(np.all(e < 5))
+
+            tc = np.linspace(*t_span)
+            yc_true = sol_rational(tc)
+            yc = res.sol(tc)
+
+            e = compute_error(yc, yc_true, rtol, atol)
+            assert_(np.all(e < 5))
+
+            assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15)
+
+            assert_raises(ValueError, method, fun_rational, t_span[0], y0,
+                          t_span[1], first_step=-1)
+            assert_raises(ValueError, method, fun_rational, t_span[0], y0,
+                          t_span[1], first_step=5)
+
+
+def test_t_eval():
+    rtol = 1e-3
+    atol = 1e-6
+    y0 = [1/3, 2/9]
+    for t_span in ([5, 9], [5, 1]):
+        t_eval = np.linspace(t_span[0], t_span[1], 10)
+        res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol,
+                        t_eval=t_eval)
+        assert_equal(res.t, t_eval)
+        assert_(res.t_events is None)
+        assert_(res.success)
+        assert_equal(res.status, 0)
+
+        y_true = sol_rational(res.t)
+        e = compute_error(res.y, y_true, rtol, atol)
+        assert_(np.all(e < 5))
+
+    t_eval = [5, 5.01, 7, 8, 8.01, 9]
+    res = solve_ivp(fun_rational, [5, 9], y0, rtol=rtol, atol=atol,
+                    t_eval=t_eval)
+    assert_equal(res.t, t_eval)
+    assert_(res.t_events is None)
+    assert_(res.success)
+    assert_equal(res.status, 0)
+
+    y_true = sol_rational(res.t)
+    e = compute_error(res.y, y_true, rtol, atol)
+    assert_(np.all(e < 5))
+
+    t_eval = [5, 4.99, 3, 1.5, 1.1, 1.01, 1]
+    res = solve_ivp(fun_rational, [5, 1], y0, rtol=rtol, atol=atol,
+                    t_eval=t_eval)
+    assert_equal(res.t, t_eval)
+    assert_(res.t_events is None)
+    assert_(res.success)
+    assert_equal(res.status, 0)
+
+    t_eval = [5.01, 7, 8, 8.01]
+    res = solve_ivp(fun_rational, [5, 9], y0, rtol=rtol, atol=atol,
+                    t_eval=t_eval)
+    assert_equal(res.t, t_eval)
+    assert_(res.t_events is None)
+    assert_(res.success)
+    assert_equal(res.status, 0)
+
+    y_true = sol_rational(res.t)
+    e = compute_error(res.y, y_true, rtol, atol)
+    assert_(np.all(e < 5))
+
+    t_eval = [4.99, 3, 1.5, 1.1, 1.01]
+    res = solve_ivp(fun_rational, [5, 1], y0, rtol=rtol, atol=atol,
+                    t_eval=t_eval)
+    assert_equal(res.t, t_eval)
+    assert_(res.t_events is None)
+    assert_(res.success)
+    assert_equal(res.status, 0)
+
+    t_eval = [4, 6]
+    assert_raises(ValueError, solve_ivp, fun_rational, [5, 9], y0,
+                  rtol=rtol, atol=atol, t_eval=t_eval)
+
+
+def test_t_eval_dense_output():
+    rtol = 1e-3
+    atol = 1e-6
+    y0 = [1/3, 2/9]
+    t_span = [5, 9]
+    t_eval = np.linspace(t_span[0], t_span[1], 10)
+    res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol,
+                    t_eval=t_eval)
+    res_d = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol,
+                      t_eval=t_eval, dense_output=True)
+    assert_equal(res.t, t_eval)
+    assert_(res.t_events is None)
+    assert_(res.success)
+    assert_equal(res.status, 0)
+
+    assert_equal(res.t, res_d.t)
+    assert_equal(res.y, res_d.y)
+    assert_(res_d.t_events is None)
+    assert_(res_d.success)
+    assert_equal(res_d.status, 0)
+
+    # if t and y are equal only test values for one case
+    y_true = sol_rational(res.t)
+    e = compute_error(res.y, y_true, rtol, atol)
+    assert_(np.all(e < 5))
+
+
+def test_t_eval_early_event():
+    def early_event(t, y):
+        return t - 7
+
+    early_event.terminal = True
+
+    rtol = 1e-3
+    atol = 1e-6
+    y0 = [1/3, 2/9]
+    t_span = [5, 9]
+    t_eval = np.linspace(7.5, 9, 16)
+    for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']:
+        with suppress_warnings() as sup:
+            sup.filter(UserWarning,
+                       "The following arguments have no effect for a chosen "
+                       "solver: `jac`")
+            res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol,
+                            method=method, t_eval=t_eval, events=early_event,
+                            jac=jac_rational)
+        assert res.success
+        assert res.message == 'A termination event occurred.'
+        assert res.status == 1
+        assert not res.t and not res.y
+        assert len(res.t_events) == 1
+        assert res.t_events[0].size == 1
+        assert res.t_events[0][0] == 7
+
+
+def test_event_dense_output_LSODA():
+    def event_lsoda(t, y):
+        return y[0] - 2.02e-5
+
+    rtol = 1e-3
+    atol = 1e-6
+    y0 = [0.05]
+    t_span = [-2, 2]
+    first_step = 1e-3
+    res = solve_ivp(
+        fun_event_dense_output_LSODA,
+        t_span,
+        y0,
+        method="LSODA",
+        dense_output=True,
+        events=event_lsoda,
+        first_step=first_step,
+        max_step=1,
+        rtol=rtol,
+        atol=atol,
+        jac=jac_event_dense_output_LSODA,
+    )
+
+    assert_equal(res.t[0], t_span[0])
+    assert_equal(res.t[-1], t_span[-1])
+    assert_allclose(first_step, np.abs(res.t[1] - t_span[0]))
+    assert res.success
+    assert_equal(res.status, 0)
+
+    y_true = sol_event_dense_output_LSODA(res.t)
+    e = compute_error(res.y, y_true, rtol, atol)
+    assert_array_less(e, 5)
+
+    tc = np.linspace(*t_span)
+    yc_true = sol_event_dense_output_LSODA(tc)
+    yc = res.sol(tc)
+    e = compute_error(yc, yc_true, rtol, atol)
+    assert_array_less(e, 5)
+
+    assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15)
+
+
+def test_no_integration():
+    for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']:
+        sol = solve_ivp(lambda t, y: -y, [4, 4], [2, 3],
+                        method=method, dense_output=True)
+        assert_equal(sol.sol(4), [2, 3])
+        assert_equal(sol.sol([4, 5, 6]), [[2, 2, 2], [3, 3, 3]])
+
+
+def test_no_integration_class():
+    for method in [RK23, RK45, DOP853, Radau, BDF, LSODA]:
+        solver = method(lambda t, y: -y, 0.0, [10.0, 0.0], 0.0)
+        solver.step()
+        assert_equal(solver.status, 'finished')
+        sol = solver.dense_output()
+        assert_equal(sol(0.0), [10.0, 0.0])
+        assert_equal(sol([0, 1, 2]), [[10, 10, 10], [0, 0, 0]])
+
+        solver = method(lambda t, y: -y, 0.0, [], np.inf)
+        solver.step()
+        assert_equal(solver.status, 'finished')
+        sol = solver.dense_output()
+        assert_equal(sol(100.0), [])
+        assert_equal(sol([0, 1, 2]), np.empty((0, 3)))
+
+
+def test_empty():
+    def fun(t, y):
+        return np.zeros((0,))
+
+    y0 = np.zeros((0,))
+
+    for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']:
+        sol = assert_no_warnings(solve_ivp, fun, [0, 10], y0,
+                                 method=method, dense_output=True)
+        assert_equal(sol.sol(10), np.zeros((0,)))
+        assert_equal(sol.sol([1, 2, 3]), np.zeros((0, 3)))
+
+    for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']:
+        sol = assert_no_warnings(solve_ivp, fun, [0, np.inf], y0,
+                                 method=method, dense_output=True)
+        assert_equal(sol.sol(10), np.zeros((0,)))
+        assert_equal(sol.sol([1, 2, 3]), np.zeros((0, 3)))
+
+
+def test_ConstantDenseOutput():
+    sol = ConstantDenseOutput(0, 1, np.array([1, 2]))
+    assert_allclose(sol(1.5), [1, 2])
+    assert_allclose(sol([1, 1.5, 2]), [[1, 1, 1], [2, 2, 2]])
+
+    sol = ConstantDenseOutput(0, 1, np.array([]))
+    assert_allclose(sol(1.5), np.empty(0))
+    assert_allclose(sol([1, 1.5, 2]), np.empty((0, 3)))
+
+
+def test_classes():
+    y0 = [1 / 3, 2 / 9]
+    for cls in [RK23, RK45, DOP853, Radau, BDF, LSODA]:
+        solver = cls(fun_rational, 5, y0, np.inf)
+        assert_equal(solver.n, 2)
+        assert_equal(solver.status, 'running')
+        assert_equal(solver.t_bound, np.inf)
+        assert_equal(solver.direction, 1)
+        assert_equal(solver.t, 5)
+        assert_equal(solver.y, y0)
+        assert_(solver.step_size is None)
+        if cls is not LSODA:
+            assert_(solver.nfev > 0)
+            assert_(solver.njev >= 0)
+            assert_equal(solver.nlu, 0)
+        else:
+            assert_equal(solver.nfev, 0)
+            assert_equal(solver.njev, 0)
+            assert_equal(solver.nlu, 0)
+
+        assert_raises(RuntimeError, solver.dense_output)
+
+        message = solver.step()
+        assert_equal(solver.status, 'running')
+        assert_equal(message, None)
+        assert_equal(solver.n, 2)
+        assert_equal(solver.t_bound, np.inf)
+        assert_equal(solver.direction, 1)
+        assert_(solver.t > 5)
+        assert_(not np.all(np.equal(solver.y, y0)))
+        assert_(solver.step_size > 0)
+        assert_(solver.nfev > 0)
+        assert_(solver.njev >= 0)
+        assert_(solver.nlu >= 0)
+        sol = solver.dense_output()
+        assert_allclose(sol(5), y0, rtol=1e-15, atol=0)
+
+
+def test_OdeSolution():
+    ts = np.array([0, 2, 5], dtype=float)
+    s1 = ConstantDenseOutput(ts[0], ts[1], np.array([-1]))
+    s2 = ConstantDenseOutput(ts[1], ts[2], np.array([1]))
+
+    sol = OdeSolution(ts, [s1, s2])
+
+    assert_equal(sol(-1), [-1])
+    assert_equal(sol(1), [-1])
+    assert_equal(sol(2), [-1])
+    assert_equal(sol(3), [1])
+    assert_equal(sol(5), [1])
+    assert_equal(sol(6), [1])
+
+    assert_equal(sol([0, 6, -2, 1.5, 4.5, 2.5, 5, 5.5, 2]),
+                 np.array([[-1, 1, -1, -1, 1, 1, 1, 1, -1]]))
+
+    ts = np.array([10, 4, -3])
+    s1 = ConstantDenseOutput(ts[0], ts[1], np.array([-1]))
+    s2 = ConstantDenseOutput(ts[1], ts[2], np.array([1]))
+
+    sol = OdeSolution(ts, [s1, s2])
+    assert_equal(sol(11), [-1])
+    assert_equal(sol(10), [-1])
+    assert_equal(sol(5), [-1])
+    assert_equal(sol(4), [-1])
+    assert_equal(sol(0), [1])
+    assert_equal(sol(-3), [1])
+    assert_equal(sol(-4), [1])
+
+    assert_equal(sol([12, -5, 10, -3, 6, 1, 4]),
+                 np.array([[-1, 1, -1, 1, -1, 1, -1]]))
+
+    ts = np.array([1, 1])
+    s = ConstantDenseOutput(1, 1, np.array([10]))
+    sol = OdeSolution(ts, [s])
+    assert_equal(sol(0), [10])
+    assert_equal(sol(1), [10])
+    assert_equal(sol(2), [10])
+
+    assert_equal(sol([2, 1, 0]), np.array([[10, 10, 10]]))
+
+
+def test_num_jac():
+    def fun(t, y):
+        return np.vstack([
+            -0.04 * y[0] + 1e4 * y[1] * y[2],
+            0.04 * y[0] - 1e4 * y[1] * y[2] - 3e7 * y[1] ** 2,
+            3e7 * y[1] ** 2
+        ])
+
+    def jac(t, y):
+        return np.array([
+            [-0.04, 1e4 * y[2], 1e4 * y[1]],
+            [0.04, -1e4 * y[2] - 6e7 * y[1], -1e4 * y[1]],
+            [0, 6e7 * y[1], 0]
+        ])
+
+    t = 1
+    y = np.array([1, 0, 0])
+    J_true = jac(t, y)
+    threshold = 1e-5
+    f = fun(t, y).ravel()
+
+    J_num, factor = num_jac(fun, t, y, f, threshold, None)
+    assert_allclose(J_num, J_true, rtol=1e-5, atol=1e-5)
+
+    J_num, factor = num_jac(fun, t, y, f, threshold, factor)
+    assert_allclose(J_num, J_true, rtol=1e-5, atol=1e-5)
+
+
+def test_num_jac_sparse():
+    def fun(t, y):
+        e = y[1:]**3 - y[:-1]**2
+        z = np.zeros(y.shape[1])
+        return np.vstack((z, 3 * e)) + np.vstack((2 * e, z))
+
+    def structure(n):
+        A = np.zeros((n, n), dtype=int)
+        A[0, 0] = 1
+        A[0, 1] = 1
+        for i in range(1, n - 1):
+            A[i, i - 1: i + 2] = 1
+        A[-1, -1] = 1
+        A[-1, -2] = 1
+
+        return A
+
+    np.random.seed(0)
+    n = 20
+    y = np.random.randn(n)
+    A = structure(n)
+    groups = group_columns(A)
+
+    f = fun(0, y[:, None]).ravel()
+
+    # Compare dense and sparse results, assuming that dense implementation
+    # is correct (as it is straightforward).
+    J_num_sparse, factor_sparse = num_jac(fun, 0, y.ravel(), f, 1e-8, None,
+                                          sparsity=(A, groups))
+    J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, None)
+    assert_allclose(J_num_dense, J_num_sparse.toarray(),
+                    rtol=1e-12, atol=1e-14)
+    assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14)
+
+    # Take small factors to trigger their recomputing inside.
+    factor = np.random.uniform(0, 1e-12, size=n)
+    J_num_sparse, factor_sparse = num_jac(fun, 0, y.ravel(), f, 1e-8, factor,
+                                          sparsity=(A, groups))
+    J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, factor)
+
+    assert_allclose(J_num_dense, J_num_sparse.toarray(),
+                    rtol=1e-12, atol=1e-14)
+    assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14)
+
+
+def test_args():
+
+    # sys3 is actually two decoupled systems. (x, y) form a
+    # linear oscillator, while z is a nonlinear first order
+    # system with equilibria at z=0 and z=1. If k > 0, z=1
+    # is stable and z=0 is unstable.
+
+    def sys3(t, w, omega, k, zfinal):
+        x, y, z = w
+        return [-omega*y, omega*x, k*z*(1 - z)]
+
+    def sys3_jac(t, w, omega, k, zfinal):
+        x, y, z = w
+        J = np.array([[0, -omega, 0],
+                      [omega, 0, 0],
+                      [0, 0, k*(1 - 2*z)]])
+        return J
+
+    def sys3_x0decreasing(t, w, omega, k, zfinal):
+        x, y, z = w
+        return x
+
+    def sys3_y0increasing(t, w, omega, k, zfinal):
+        x, y, z = w
+        return y
+
+    def sys3_zfinal(t, w, omega, k, zfinal):
+        x, y, z = w
+        return z - zfinal
+
+    # Set the event flags for the event functions.
+    sys3_x0decreasing.direction = -1
+    sys3_y0increasing.direction = 1
+    sys3_zfinal.terminal = True
+
+    omega = 2
+    k = 4
+
+    tfinal = 5
+    zfinal = 0.99
+    # Find z0 such that when z(0) = z0, z(tfinal) = zfinal.
+    # The condition z(tfinal) = zfinal is the terminal event.
+    z0 = np.exp(-k*tfinal)/((1 - zfinal)/zfinal + np.exp(-k*tfinal))
+
+    w0 = [0, -1, z0]
+
+    # Provide the jac argument and use the Radau method to ensure that the use
+    # of the Jacobian function is exercised.
+    # If event handling is working, the solution will stop at tfinal, not tend.
+    tend = 2*tfinal
+    sol = solve_ivp(sys3, [0, tend], w0,
+                    events=[sys3_x0decreasing, sys3_y0increasing, sys3_zfinal],
+                    dense_output=True, args=(omega, k, zfinal),
+                    method='Radau', jac=sys3_jac,
+                    rtol=1e-10, atol=1e-13)
+
+    # Check that we got the expected events at the expected times.
+    x0events_t = sol.t_events[0]
+    y0events_t = sol.t_events[1]
+    zfinalevents_t = sol.t_events[2]
+    assert_allclose(x0events_t, [0.5*np.pi, 1.5*np.pi])
+    assert_allclose(y0events_t, [0.25*np.pi, 1.25*np.pi])
+    assert_allclose(zfinalevents_t, [tfinal])
+
+    # Check that the solution agrees with the known exact solution.
+    t = np.linspace(0, zfinalevents_t[0], 250)
+    w = sol.sol(t)
+    assert_allclose(w[0], np.sin(omega*t), rtol=1e-9, atol=1e-12)
+    assert_allclose(w[1], -np.cos(omega*t), rtol=1e-9, atol=1e-12)
+    assert_allclose(w[2], 1/(((1 - z0)/z0)*np.exp(-k*t) + 1),
+                    rtol=1e-9, atol=1e-12)
+
+    # Check that the state variables have the expected values at the events.
+    x0events = sol.sol(x0events_t)
+    y0events = sol.sol(y0events_t)
+    zfinalevents = sol.sol(zfinalevents_t)
+    assert_allclose(x0events[0], np.zeros_like(x0events[0]), atol=5e-14)
+    assert_allclose(x0events[1], np.ones_like(x0events[1]))
+    assert_allclose(y0events[0], np.ones_like(y0events[0]))
+    assert_allclose(y0events[1], np.zeros_like(y0events[1]), atol=5e-14)
+    assert_allclose(zfinalevents[2], [zfinal])
+
+
+def test_array_rtol():
+    # solve_ivp had a bug with array_like `rtol`; see gh-15482
+    # check that it's fixed
+    def f(t, y):
+        return y[0], y[1]
+
+    # no warning (or error) when `rtol` is array_like
+    sol = solve_ivp(f, (0, 1), [1., 1.], rtol=[1e-1, 1e-1])
+    err1 = np.abs(np.linalg.norm(sol.y[:, -1] - np.exp(1)))
+
+    # warning when an element of `rtol` is too small
+    with pytest.warns(UserWarning, match="At least one element..."):
+        sol = solve_ivp(f, (0, 1), [1., 1.], rtol=[1e-1, 1e-16])
+        err2 = np.abs(np.linalg.norm(sol.y[:, -1] - np.exp(1)))
+
+    # tighter rtol improves the error
+    assert err2 < err1
+
+@pytest.mark.parametrize('method', ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA'])
+def test_integration_zero_rhs(method):
+    result = solve_ivp(fun_zero, [0, 10], np.ones(3), method=method)
+    assert_(result.success)
+    assert_equal(result.status, 0)
+    assert_allclose(result.y, 1.0, rtol=1e-15)
+
+
+def test_args_single_value():
+    def fun_with_arg(t, y, a):
+        return a*y
+
+    message = "Supplied 'args' cannot be unpacked."
+    with pytest.raises(TypeError, match=message):
+        solve_ivp(fun_with_arg, (0, 0.1), [1], args=-1)
+
+    sol = solve_ivp(fun_with_arg, (0, 0.1), [1], args=(-1,))
+    assert_allclose(sol.y[0, -1], np.exp(-0.1))
+
+@pytest.mark.parametrize("f0_fill", [np.nan, np.inf])
+def test_initial_state_finiteness(f0_fill):
+    # regression test for gh-17846
+    msg = "All components of the initial state `y0` must be finite."
+    with pytest.raises(ValueError, match=msg):
+        solve_ivp(fun_zero, [0, 10], np.full(3, f0_fill))

venv/lib/python3.10/site-packages/scipy/integrate/_ivp/tests/test_rk.py

@@ -0,0 +1,37 @@
+import pytest
+from numpy.testing import assert_allclose, assert_
+import numpy as np
+from scipy.integrate import RK23, RK45, DOP853
+from scipy.integrate._ivp import dop853_coefficients
+
+
+@pytest.mark.parametrize("solver", [RK23, RK45, DOP853])
+def test_coefficient_properties(solver):
+    assert_allclose(np.sum(solver.B), 1, rtol=1e-15)
+    assert_allclose(np.sum(solver.A, axis=1), solver.C, rtol=1e-14)
+
+
+def test_coefficient_properties_dop853():
+    assert_allclose(np.sum(dop853_coefficients.B), 1, rtol=1e-15)
+    assert_allclose(np.sum(dop853_coefficients.A, axis=1),
+                    dop853_coefficients.C,
+                    rtol=1e-14)
+
+
+@pytest.mark.parametrize("solver_class", [RK23, RK45, DOP853])
+def test_error_estimation(solver_class):
+    step = 0.2
+    solver = solver_class(lambda t, y: y, 0, [1], 1, first_step=step)
+    solver.step()
+    error_estimate = solver._estimate_error(solver.K, step)
+    error = solver.y - np.exp([step])
+    assert_(np.abs(error) < np.abs(error_estimate))
+
+
+@pytest.mark.parametrize("solver_class", [RK23, RK45, DOP853])
+def test_error_estimation_complex(solver_class):
+    h = 0.2
+    solver = solver_class(lambda t, y: 1j * y, 0, [1j], 1, first_step=h)
+    solver.step()
+    err_norm = solver._estimate_error_norm(solver.K, h, scale=[1])
+    assert np.isrealobj(err_norm)

venv/lib/python3.10/site-packages/scipy/spatial/__init__.py

@@ -0,0 +1,129 @@
+"""
+=============================================================
+Spatial algorithms and data structures (:mod:`scipy.spatial`)
+=============================================================
+
+.. currentmodule:: scipy.spatial
+
+.. toctree::
+   :hidden:
+
+   spatial.distance
+
+Spatial transformations
+=======================
+
+These are contained in the `scipy.spatial.transform` submodule.
+
+Nearest-neighbor queries
+========================
+.. autosummary::
+   :toctree: generated/
+
+   KDTree      -- class for efficient nearest-neighbor queries
+   cKDTree     -- class for efficient nearest-neighbor queries (faster implementation)
+   Rectangle
+
+Distance metrics
+================
+
+Distance metrics are contained in the :mod:`scipy.spatial.distance` submodule.
+
+Delaunay triangulation, convex hulls, and Voronoi diagrams
+==========================================================
+
+.. autosummary::
+   :toctree: generated/
+
+   Delaunay    -- compute Delaunay triangulation of input points
+   ConvexHull  -- compute a convex hull for input points
+   Voronoi     -- compute a Voronoi diagram hull from input points
+   SphericalVoronoi -- compute a Voronoi diagram from input points on the surface of a sphere
+   HalfspaceIntersection -- compute the intersection points of input halfspaces
+
+Plotting helpers
+================
+
+.. autosummary::
+   :toctree: generated/
+
+   delaunay_plot_2d     -- plot 2-D triangulation
+   convex_hull_plot_2d  -- plot 2-D convex hull
+   voronoi_plot_2d      -- plot 2-D Voronoi diagram
+
+.. seealso:: :ref:`Tutorial <qhulltutorial>`
+
+
+Simplex representation
+======================
+The simplices (triangles, tetrahedra, etc.) appearing in the Delaunay
+tessellation (N-D simplices), convex hull facets, and Voronoi ridges
+(N-1-D simplices) are represented in the following scheme::
+
+    tess = Delaunay(points)
+    hull = ConvexHull(points)
+    voro = Voronoi(points)
+
+    # coordinates of the jth vertex of the ith simplex
+    tess.points[tess.simplices[i, j], :]        # tessellation element
+    hull.points[hull.simplices[i, j], :]        # convex hull facet
+    voro.vertices[voro.ridge_vertices[i, j], :] # ridge between Voronoi cells
+
+For Delaunay triangulations and convex hulls, the neighborhood
+structure of the simplices satisfies the condition:
+``tess.neighbors[i,j]`` is the neighboring simplex of the ith
+simplex, opposite to the ``j``-vertex. It is -1 in case of no neighbor.
+
+Convex hull facets also define a hyperplane equation::
+
+    (hull.equations[i,:-1] * coord).sum() + hull.equations[i,-1] == 0
+
+Similar hyperplane equations for the Delaunay triangulation correspond
+to the convex hull facets on the corresponding N+1-D
+paraboloid.
+
+The Delaunay triangulation objects offer a method for locating the
+simplex containing a given point, and barycentric coordinate
+computations.
+
+Functions
+---------
+
+.. autosummary::
+   :toctree: generated/
+
+   tsearch
+   distance_matrix
+   minkowski_distance
+   minkowski_distance_p
+   procrustes
+   geometric_slerp
+
+Warnings / Errors used in :mod:`scipy.spatial`
+----------------------------------------------
+.. autosummary::
+   :toctree: generated/
+
+   QhullError
+"""  # noqa: E501
+
+from ._kdtree import *
+from ._ckdtree import *
+from ._qhull import *
+from ._spherical_voronoi import SphericalVoronoi
+from ._plotutils import *
+from ._procrustes import procrustes
+from ._geometric_slerp import geometric_slerp
+
+# Deprecated namespaces, to be removed in v2.0.0
+from . import ckdtree, kdtree, qhull
+
+__all__ = [s for s in dir() if not s.startswith('_')]
+
+from . import distance, transform
+
+__all__ += ['distance', 'transform']
+
+from scipy._lib._testutils import PytestTester
+test = PytestTester(__name__)
+del PytestTester

venv/lib/python3.10/site-packages/scipy/spatial/_ckdtree.pyi

@@ -0,0 +1,214 @@
+from __future__ import annotations
+from typing import (
+    Any,
+    Generic,
+    overload,
+    TypeVar,
+)
+
+import numpy as np
+import numpy.typing as npt
+from scipy.sparse import coo_matrix, dok_matrix
+
+from typing import Literal
+
+# TODO: Replace `ndarray` with a 1D float64 array when possible
+_BoxType = TypeVar("_BoxType", None, npt.NDArray[np.float64])
+
+# Copied from `numpy.typing._scalar_like._ScalarLike`
+# TODO: Expand with 0D arrays once we have shape support
+_ArrayLike0D = bool | int | float | complex | str | bytes | np.generic
+
+_WeightType = npt.ArrayLike | tuple[npt.ArrayLike | None, npt.ArrayLike | None]
+
+class cKDTreeNode:
+    @property
+    def data_points(self) -> npt.NDArray[np.float64]: ...
+    @property
+    def indices(self) -> npt.NDArray[np.intp]: ...
+
+    # These are read-only attributes in cython, which behave like properties
+    @property
+    def level(self) -> int: ...
+    @property
+    def split_dim(self) -> int: ...
+    @property
+    def children(self) -> int: ...
+    @property
+    def start_idx(self) -> int: ...
+    @property
+    def end_idx(self) -> int: ...
+    @property
+    def split(self) -> float: ...
+    @property
+    def lesser(self) -> cKDTreeNode | None: ...
+    @property
+    def greater(self) -> cKDTreeNode | None: ...
+
+class cKDTree(Generic[_BoxType]):
+    @property
+    def n(self) -> int: ...
+    @property
+    def m(self) -> int: ...
+    @property
+    def leafsize(self) -> int: ...
+    @property
+    def size(self) -> int: ...
+    @property
+    def tree(self) -> cKDTreeNode: ...
+
+    # These are read-only attributes in cython, which behave like properties
+    @property
+    def data(self) -> npt.NDArray[np.float64]: ...
+    @property
+    def maxes(self) -> npt.NDArray[np.float64]: ...
+    @property
+    def mins(self) -> npt.NDArray[np.float64]: ...
+    @property
+    def indices(self) -> npt.NDArray[np.float64]: ...
+    @property
+    def boxsize(self) -> _BoxType: ...
+
+    # NOTE: In practice `__init__` is used as constructor, not `__new__`.
+    # The latter gives us more flexibility in setting the generic parameter
+    # though.
+    @overload
+    def __new__(  # type: ignore[misc]
+        cls,
+        data: npt.ArrayLike,
+        leafsize: int = ...,
+        compact_nodes: bool = ...,
+        copy_data: bool = ...,
+        balanced_tree: bool = ...,
+        boxsize: None = ...,
+    ) -> cKDTree[None]: ...
+    @overload
+    def __new__(
+        cls,
+        data: npt.ArrayLike,
+        leafsize: int = ...,
+        compact_nodes: bool = ...,
+        copy_data: bool = ...,
+        balanced_tree: bool = ...,
+        boxsize: npt.ArrayLike = ...,
+    ) -> cKDTree[npt.NDArray[np.float64]]: ...
+
+    # TODO: returns a 2-tuple of scalars if `x.ndim == 1` and `k == 1`,
+    # returns a 2-tuple of arrays otherwise
+    def query(
+        self,
+        x: npt.ArrayLike,
+        k: npt.ArrayLike = ...,
+        eps: float = ...,
+        p: float = ...,
+        distance_upper_bound: float = ...,
+        workers: int | None = ...,
+    ) -> tuple[Any, Any]: ...
+
+    # TODO: returns a list scalars if `x.ndim <= 1`,
+    # returns an object array of lists otherwise
+    def query_ball_point(
+        self,
+        x: npt.ArrayLike,
+        r: npt.ArrayLike,
+        p: float,
+        eps: float = ...,
+        workers: int | None = ...,
+        return_sorted: bool | None = ...,
+        return_length: bool = ...
+    ) -> Any: ...
+
+    def query_ball_tree(
+        self,
+        other: cKDTree,
+        r: float,
+        p: float,
+        eps: float = ...,
+    ) -> list[list[int]]: ...
+
+    @overload
+    def query_pairs(  # type: ignore[misc]
+        self,
+        r: float,
+        p: float = ...,
+        eps: float = ...,
+        output_type: Literal["set"] = ...,
+    ) -> set[tuple[int, int]]: ...
+    @overload
+    def query_pairs(
+        self,
+        r: float,
+        p: float = ...,
+        eps: float = ...,
+        output_type: Literal["ndarray"] = ...,
+    ) -> npt.NDArray[np.intp]: ...
+
+    @overload
+    def count_neighbors(  # type: ignore[misc]
+        self,
+        other: cKDTree,
+        r: _ArrayLike0D,
+        p: float = ...,
+        weights: None | tuple[None, None] = ...,
+        cumulative: bool = ...,
+    ) -> int: ...
+    @overload
+    def count_neighbors(  # type: ignore[misc]
+        self,
+        other: cKDTree,
+        r: _ArrayLike0D,
+        p: float = ...,
+        weights: _WeightType = ...,
+        cumulative: bool = ...,
+    ) -> np.float64: ...
+    @overload
+    def count_neighbors(  # type: ignore[misc]
+        self,
+        other: cKDTree,
+        r: npt.ArrayLike,
+        p: float = ...,
+        weights: None | tuple[None, None] = ...,
+        cumulative: bool = ...,
+    ) -> npt.NDArray[np.intp]: ...
+    @overload
+    def count_neighbors(
+        self,
+        other: cKDTree,
+        r: npt.ArrayLike,
+        p: float = ...,
+        weights: _WeightType = ...,
+        cumulative: bool = ...,
+    ) -> npt.NDArray[np.float64]: ...
+
+    @overload
+    def sparse_distance_matrix(  # type: ignore[misc]
+        self,
+        other: cKDTree,
+        max_distance: float,
+        p: float = ...,
+        output_type: Literal["dok_matrix"] = ...,
+    ) -> dok_matrix: ...
+    @overload
+    def sparse_distance_matrix(  # type: ignore[misc]
+        self,
+        other: cKDTree,
+        max_distance: float,
+        p: float = ...,
+        output_type: Literal["coo_matrix"] = ...,
+    ) -> coo_matrix: ...
+    @overload
+    def sparse_distance_matrix(  # type: ignore[misc]
+        self,
+        other: cKDTree,
+        max_distance: float,
+        p: float = ...,
+        output_type: Literal["dict"] = ...,
+    ) -> dict[tuple[int, int], float]: ...
+    @overload
+    def sparse_distance_matrix(
+        self,
+        other: cKDTree,
+        max_distance: float,
+        p: float = ...,
+        output_type: Literal["ndarray"] = ...,
+    ) -> npt.NDArray[np.void]: ...

venv/lib/python3.10/site-packages/scipy/spatial/_geometric_slerp.py

@@ -0,0 +1,240 @@
+from __future__ import annotations
+
+__all__ = ['geometric_slerp']
+
+import warnings
+from typing import TYPE_CHECKING
+
+import numpy as np
+from scipy.spatial.distance import euclidean
+
+if TYPE_CHECKING:
+    import numpy.typing as npt
+
+
+def _geometric_slerp(start, end, t):
+    # create an orthogonal basis using QR decomposition
+    basis = np.vstack([start, end])
+    Q, R = np.linalg.qr(basis.T)
+    signs = 2 * (np.diag(R) >= 0) - 1
+    Q = Q.T * signs.T[:, np.newaxis]
+    R = R.T * signs.T[:, np.newaxis]
+
+    # calculate the angle between `start` and `end`
+    c = np.dot(start, end)
+    s = np.linalg.det(R)
+    omega = np.arctan2(s, c)
+
+    # interpolate
+    start, end = Q
+    s = np.sin(t * omega)
+    c = np.cos(t * omega)
+    return start * c[:, np.newaxis] + end * s[:, np.newaxis]
+
+
+def geometric_slerp(
+    start: npt.ArrayLike,
+    end: npt.ArrayLike,
+    t: npt.ArrayLike,
+    tol: float = 1e-7,
+) -> np.ndarray:
+    """
+    Geometric spherical linear interpolation.
+
+    The interpolation occurs along a unit-radius
+    great circle arc in arbitrary dimensional space.
+
+    Parameters
+    ----------
+    start : (n_dimensions, ) array-like
+        Single n-dimensional input coordinate in a 1-D array-like
+        object. `n` must be greater than 1.
+    end : (n_dimensions, ) array-like
+        Single n-dimensional input coordinate in a 1-D array-like
+        object. `n` must be greater than 1.
+    t : float or (n_points,) 1D array-like
+        A float or 1D array-like of doubles representing interpolation
+        parameters, with values required in the inclusive interval
+        between 0 and 1. A common approach is to generate the array
+        with ``np.linspace(0, 1, n_pts)`` for linearly spaced points.
+        Ascending, descending, and scrambled orders are permitted.
+    tol : float
+        The absolute tolerance for determining if the start and end
+        coordinates are antipodes.
+
+    Returns
+    -------
+    result : (t.size, D)
+        An array of doubles containing the interpolated
+        spherical path and including start and
+        end when 0 and 1 t are used. The
+        interpolated values should correspond to the
+        same sort order provided in the t array. The result
+        may be 1-dimensional if ``t`` is a float.
+
+    Raises
+    ------
+    ValueError
+        If ``start`` and ``end`` are antipodes, not on the
+        unit n-sphere, or for a variety of degenerate conditions.
+
+    See Also
+    --------
+    scipy.spatial.transform.Slerp : 3-D Slerp that works with quaternions
+
+    Notes
+    -----
+    The implementation is based on the mathematical formula provided in [1]_,
+    and the first known presentation of this algorithm, derived from study of
+    4-D geometry, is credited to Glenn Davis in a footnote of the original
+    quaternion Slerp publication by Ken Shoemake [2]_.
+
+    .. versionadded:: 1.5.0
+
+    References
+    ----------
+    .. [1] https://en.wikipedia.org/wiki/Slerp#Geometric_Slerp
+    .. [2] Ken Shoemake (1985) Animating rotation with quaternion curves.
+           ACM SIGGRAPH Computer Graphics, 19(3): 245-254.
+
+    Examples
+    --------
+    Interpolate four linearly-spaced values on the circumference of
+    a circle spanning 90 degrees:
+
+    >>> import numpy as np
+    >>> from scipy.spatial import geometric_slerp
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure()
+    >>> ax = fig.add_subplot(111)
+    >>> start = np.array([1, 0])
+    >>> end = np.array([0, 1])
+    >>> t_vals = np.linspace(0, 1, 4)
+    >>> result = geometric_slerp(start,
+    ...                          end,
+    ...                          t_vals)
+
+    The interpolated results should be at 30 degree intervals
+    recognizable on the unit circle:
+
+    >>> ax.scatter(result[...,0], result[...,1], c='k')
+    >>> circle = plt.Circle((0, 0), 1, color='grey')
+    >>> ax.add_artist(circle)
+    >>> ax.set_aspect('equal')
+    >>> plt.show()
+
+    Attempting to interpolate between antipodes on a circle is
+    ambiguous because there are two possible paths, and on a
+    sphere there are infinite possible paths on the geodesic surface.
+    Nonetheless, one of the ambiguous paths is returned along
+    with a warning:
+
+    >>> opposite_pole = np.array([-1, 0])
+    >>> with np.testing.suppress_warnings() as sup:
+    ...     sup.filter(UserWarning)
+    ...     geometric_slerp(start,
+    ...                     opposite_pole,
+    ...                     t_vals)
+    array([[ 1.00000000e+00,  0.00000000e+00],
+           [ 5.00000000e-01,  8.66025404e-01],
+           [-5.00000000e-01,  8.66025404e-01],
+           [-1.00000000e+00,  1.22464680e-16]])
+
+    Extend the original example to a sphere and plot interpolation
+    points in 3D:
+
+    >>> from mpl_toolkits.mplot3d import proj3d
+    >>> fig = plt.figure()
+    >>> ax = fig.add_subplot(111, projection='3d')
+
+    Plot the unit sphere for reference (optional):
+
+    >>> u = np.linspace(0, 2 * np.pi, 100)
+    >>> v = np.linspace(0, np.pi, 100)
+    >>> x = np.outer(np.cos(u), np.sin(v))
+    >>> y = np.outer(np.sin(u), np.sin(v))
+    >>> z = np.outer(np.ones(np.size(u)), np.cos(v))
+    >>> ax.plot_surface(x, y, z, color='y', alpha=0.1)
+
+    Interpolating over a larger number of points
+    may provide the appearance of a smooth curve on
+    the surface of the sphere, which is also useful
+    for discretized integration calculations on a
+    sphere surface:
+
+    >>> start = np.array([1, 0, 0])
+    >>> end = np.array([0, 0, 1])
+    >>> t_vals = np.linspace(0, 1, 200)
+    >>> result = geometric_slerp(start,
+    ...                          end,
+    ...                          t_vals)
+    >>> ax.plot(result[...,0],
+    ...         result[...,1],
+    ...         result[...,2],
+    ...         c='k')
+    >>> plt.show()
+    """
+
+    start = np.asarray(start, dtype=np.float64)
+    end = np.asarray(end, dtype=np.float64)
+    t = np.asarray(t)
+
+    if t.ndim > 1:
+        raise ValueError("The interpolation parameter "
+                         "value must be one dimensional.")
+
+    if start.ndim != 1 or end.ndim != 1:
+        raise ValueError("Start and end coordinates "
+                         "must be one-dimensional")
+
+    if start.size != end.size:
+        raise ValueError("The dimensions of start and "
+                         "end must match (have same size)")
+
+    if start.size < 2 or end.size < 2:
+        raise ValueError("The start and end coordinates must "
+                         "both be in at least two-dimensional "
+                         "space")
+
+    if np.array_equal(start, end):
+        return np.linspace(start, start, t.size)
+
+    # for points that violate equation for n-sphere
+    for coord in [start, end]:
+        if not np.allclose(np.linalg.norm(coord), 1.0,
+                           rtol=1e-9,
+                           atol=0):
+            raise ValueError("start and end are not"
+                             " on a unit n-sphere")
+
+    if not isinstance(tol, float):
+        raise ValueError("tol must be a float")
+    else:
+        tol = np.fabs(tol)
+
+    coord_dist = euclidean(start, end)
+
+    # diameter of 2 within tolerance means antipodes, which is a problem
+    # for all unit n-spheres (even the 0-sphere would have an ambiguous path)
+    if np.allclose(coord_dist, 2.0, rtol=0, atol=tol):
+        warnings.warn("start and end are antipodes "
+                      "using the specified tolerance; "
+                      "this may cause ambiguous slerp paths",
+                      stacklevel=2)
+
+    t = np.asarray(t, dtype=np.float64)
+
+    if t.size == 0:
+        return np.empty((0, start.size))
+
+    if t.min() < 0 or t.max() > 1:
+        raise ValueError("interpolation parameter must be in [0, 1]")
+
+    if t.ndim == 0:
+        return _geometric_slerp(start,
+                                end,
+                                np.atleast_1d(t)).ravel()
+    else:
+        return _geometric_slerp(start,
+                                end,
+                                t)

venv/lib/python3.10/site-packages/scipy/spatial/_kdtree.py

@@ -0,0 +1,920 @@
+# Copyright Anne M. Archibald 2008
+# Released under the scipy license
+import numpy as np
+from ._ckdtree import cKDTree, cKDTreeNode
+
+__all__ = ['minkowski_distance_p', 'minkowski_distance',
+           'distance_matrix',
+           'Rectangle', 'KDTree']
+
+
+def minkowski_distance_p(x, y, p=2):
+    """Compute the pth power of the L**p distance between two arrays.
+
+    For efficiency, this function computes the L**p distance but does
+    not extract the pth root. If `p` is 1 or infinity, this is equal to
+    the actual L**p distance.
+
+    The last dimensions of `x` and `y` must be the same length.  Any
+    other dimensions must be compatible for broadcasting.
+
+    Parameters
+    ----------
+    x : (..., K) array_like
+        Input array.
+    y : (..., K) array_like
+        Input array.
+    p : float, 1 <= p <= infinity
+        Which Minkowski p-norm to use.
+
+    Returns
+    -------
+    dist : ndarray
+        pth power of the distance between the input arrays.
+
+    Examples
+    --------
+    >>> from scipy.spatial import minkowski_distance_p
+    >>> minkowski_distance_p([[0, 0], [0, 0]], [[1, 1], [0, 1]])
+    array([2, 1])
+
+    """
+    x = np.asarray(x)
+    y = np.asarray(y)
+
+    # Find smallest common datatype with float64 (return type of this
+    # function) - addresses #10262.
+    # Don't just cast to float64 for complex input case.
+    common_datatype = np.promote_types(np.promote_types(x.dtype, y.dtype),
+                                       'float64')
+
+    # Make sure x and y are NumPy arrays of correct datatype.
+    x = x.astype(common_datatype)
+    y = y.astype(common_datatype)
+
+    if p == np.inf:
+        return np.amax(np.abs(y-x), axis=-1)
+    elif p == 1:
+        return np.sum(np.abs(y-x), axis=-1)
+    else:
+        return np.sum(np.abs(y-x)**p, axis=-1)
+
+
+def minkowski_distance(x, y, p=2):
+    """Compute the L**p distance between two arrays.
+
+    The last dimensions of `x` and `y` must be the same length.  Any
+    other dimensions must be compatible for broadcasting.
+
+    Parameters
+    ----------
+    x : (..., K) array_like
+        Input array.
+    y : (..., K) array_like
+        Input array.
+    p : float, 1 <= p <= infinity
+        Which Minkowski p-norm to use.
+
+    Returns
+    -------
+    dist : ndarray
+        Distance between the input arrays.
+
+    Examples
+    --------
+    >>> from scipy.spatial import minkowski_distance
+    >>> minkowski_distance([[0, 0], [0, 0]], [[1, 1], [0, 1]])
+    array([ 1.41421356,  1.        ])
+
+    """
+    x = np.asarray(x)
+    y = np.asarray(y)
+    if p == np.inf or p == 1:
+        return minkowski_distance_p(x, y, p)
+    else:
+        return minkowski_distance_p(x, y, p)**(1./p)
+
+
+class Rectangle:
+    """Hyperrectangle class.
+
+    Represents a Cartesian product of intervals.
+    """
+    def __init__(self, maxes, mins):
+        """Construct a hyperrectangle."""
+        self.maxes = np.maximum(maxes,mins).astype(float)
+        self.mins = np.minimum(maxes,mins).astype(float)
+        self.m, = self.maxes.shape
+
+    def __repr__(self):
+        return "<Rectangle %s>" % list(zip(self.mins, self.maxes))
+
+    def volume(self):
+        """Total volume."""
+        return np.prod(self.maxes-self.mins)
+
+    def split(self, d, split):
+        """Produce two hyperrectangles by splitting.
+
+        In general, if you need to compute maximum and minimum
+        distances to the children, it can be done more efficiently
+        by updating the maximum and minimum distances to the parent.
+
+        Parameters
+        ----------
+        d : int
+            Axis to split hyperrectangle along.
+        split : float
+            Position along axis `d` to split at.
+
+        """
+        mid = np.copy(self.maxes)
+        mid[d] = split
+        less = Rectangle(self.mins, mid)
+        mid = np.copy(self.mins)
+        mid[d] = split
+        greater = Rectangle(mid, self.maxes)
+        return less, greater
+
+    def min_distance_point(self, x, p=2.):
+        """
+        Return the minimum distance between input and points in the
+        hyperrectangle.
+
+        Parameters
+        ----------
+        x : array_like
+            Input.
+        p : float, optional
+            Input.
+
+        """
+        return minkowski_distance(
+            0, np.maximum(0, np.maximum(self.mins-x, x-self.maxes)),
+            p
+        )
+
+    def max_distance_point(self, x, p=2.):
+        """
+        Return the maximum distance between input and points in the hyperrectangle.
+
+        Parameters
+        ----------
+        x : array_like
+            Input array.
+        p : float, optional
+            Input.
+
+        """
+        return minkowski_distance(0, np.maximum(self.maxes-x, x-self.mins), p)
+
+    def min_distance_rectangle(self, other, p=2.):
+        """
+        Compute the minimum distance between points in the two hyperrectangles.
+
+        Parameters
+        ----------
+        other : hyperrectangle
+            Input.
+        p : float
+            Input.
+
+        """
+        return minkowski_distance(
+            0,
+            np.maximum(0, np.maximum(self.mins-other.maxes,
+                                     other.mins-self.maxes)),
+            p
+        )
+
+    def max_distance_rectangle(self, other, p=2.):
+        """
+        Compute the maximum distance between points in the two hyperrectangles.
+
+        Parameters
+        ----------
+        other : hyperrectangle
+            Input.
+        p : float, optional
+            Input.
+
+        """
+        return minkowski_distance(
+            0, np.maximum(self.maxes-other.mins, other.maxes-self.mins), p)
+
+
+class KDTree(cKDTree):
+    """kd-tree for quick nearest-neighbor lookup.
+
+    This class provides an index into a set of k-dimensional points
+    which can be used to rapidly look up the nearest neighbors of any
+    point.
+
+    Parameters
+    ----------
+    data : array_like, shape (n,m)
+        The n data points of dimension m to be indexed. This array is
+        not copied unless this is necessary to produce a contiguous
+        array of doubles, and so modifying this data will result in
+        bogus results. The data are also copied if the kd-tree is built
+        with copy_data=True.
+    leafsize : positive int, optional
+        The number of points at which the algorithm switches over to
+        brute-force.  Default: 10.
+    compact_nodes : bool, optional
+        If True, the kd-tree is built to shrink the hyperrectangles to
+        the actual data range. This usually gives a more compact tree that
+        is robust against degenerated input data and gives faster queries
+        at the expense of longer build time. Default: True.
+    copy_data : bool, optional
+        If True the data is always copied to protect the kd-tree against
+        data corruption. Default: False.
+    balanced_tree : bool, optional
+        If True, the median is used to split the hyperrectangles instead of
+        the midpoint. This usually gives a more compact tree and
+        faster queries at the expense of longer build time. Default: True.
+    boxsize : array_like or scalar, optional
+        Apply a m-d toroidal topology to the KDTree.. The topology is generated
+        by :math:`x_i + n_i L_i` where :math:`n_i` are integers and :math:`L_i`
+        is the boxsize along i-th dimension. The input data shall be wrapped
+        into :math:`[0, L_i)`. A ValueError is raised if any of the data is
+        outside of this bound.
+
+    Notes
+    -----
+    The algorithm used is described in Maneewongvatana and Mount 1999.
+    The general idea is that the kd-tree is a binary tree, each of whose
+    nodes represents an axis-aligned hyperrectangle. Each node specifies
+    an axis and splits the set of points based on whether their coordinate
+    along that axis is greater than or less than a particular value.
+
+    During construction, the axis and splitting point are chosen by the
+    "sliding midpoint" rule, which ensures that the cells do not all
+    become long and thin.
+
+    The tree can be queried for the r closest neighbors of any given point
+    (optionally returning only those within some maximum distance of the
+    point). It can also be queried, with a substantial gain in efficiency,
+    for the r approximate closest neighbors.
+
+    For large dimensions (20 is already large) do not expect this to run
+    significantly faster than brute force. High-dimensional nearest-neighbor
+    queries are a substantial open problem in computer science.
+
+    Attributes
+    ----------
+    data : ndarray, shape (n,m)
+        The n data points of dimension m to be indexed. This array is
+        not copied unless this is necessary to produce a contiguous
+        array of doubles. The data are also copied if the kd-tree is built
+        with `copy_data=True`.
+    leafsize : positive int
+        The number of points at which the algorithm switches over to
+        brute-force.
+    m : int
+        The dimension of a single data-point.
+    n : int
+        The number of data points.
+    maxes : ndarray, shape (m,)
+        The maximum value in each dimension of the n data points.
+    mins : ndarray, shape (m,)
+        The minimum value in each dimension of the n data points.
+    size : int
+        The number of nodes in the tree.
+
+    """
+
+    class node:
+        @staticmethod
+        def _create(ckdtree_node=None):
+            """Create either an inner or leaf node, wrapping a cKDTreeNode instance"""
+            if ckdtree_node is None:
+                return KDTree.node(ckdtree_node)
+            elif ckdtree_node.split_dim == -1:
+                return KDTree.leafnode(ckdtree_node)
+            else:
+                return KDTree.innernode(ckdtree_node)
+
+        def __init__(self, ckdtree_node=None):
+            if ckdtree_node is None:
+                ckdtree_node = cKDTreeNode()
+            self._node = ckdtree_node
+
+        def __lt__(self, other):
+            return id(self) < id(other)
+
+        def __gt__(self, other):
+            return id(self) > id(other)
+
+        def __le__(self, other):
+            return id(self) <= id(other)
+
+        def __ge__(self, other):
+            return id(self) >= id(other)
+
+        def __eq__(self, other):
+            return id(self) == id(other)
+
+    class leafnode(node):
+        @property
+        def idx(self):
+            return self._node.indices
+
+        @property
+        def children(self):
+            return self._node.children
+
+    class innernode(node):
+        def __init__(self, ckdtreenode):
+            assert isinstance(ckdtreenode, cKDTreeNode)
+            super().__init__(ckdtreenode)
+            self.less = KDTree.node._create(ckdtreenode.lesser)
+            self.greater = KDTree.node._create(ckdtreenode.greater)
+
+        @property
+        def split_dim(self):
+            return self._node.split_dim
+
+        @property
+        def split(self):
+            return self._node.split
+
+        @property
+        def children(self):
+            return self._node.children
+
+    @property
+    def tree(self):
+        if not hasattr(self, "_tree"):
+            self._tree = KDTree.node._create(super().tree)
+
+        return self._tree
+
+    def __init__(self, data, leafsize=10, compact_nodes=True, copy_data=False,
+                 balanced_tree=True, boxsize=None):
+        data = np.asarray(data)
+        if data.dtype.kind == 'c':
+            raise TypeError("KDTree does not work with complex data")
+
+        # Note KDTree has different default leafsize from cKDTree
+        super().__init__(data, leafsize, compact_nodes, copy_data,
+                         balanced_tree, boxsize)
+
+    def query(
+            self, x, k=1, eps=0, p=2, distance_upper_bound=np.inf, workers=1):
+        r"""Query the kd-tree for nearest neighbors.
+
+        Parameters
+        ----------
+        x : array_like, last dimension self.m
+            An array of points to query.
+        k : int or Sequence[int], optional
+            Either the number of nearest neighbors to return, or a list of the
+            k-th nearest neighbors to return, starting from 1.
+        eps : nonnegative float, optional
+            Return approximate nearest neighbors; the kth returned value
+            is guaranteed to be no further than (1+eps) times the
+            distance to the real kth nearest neighbor.
+        p : float, 1<=p<=infinity, optional
+            Which Minkowski p-norm to use.
+            1 is the sum-of-absolute-values distance ("Manhattan" distance).
+            2 is the usual Euclidean distance.
+            infinity is the maximum-coordinate-difference distance.
+            A large, finite p may cause a ValueError if overflow can occur.
+        distance_upper_bound : nonnegative float, optional
+            Return only neighbors within this distance. This is used to prune
+            tree searches, so if you are doing a series of nearest-neighbor
+            queries, it may help to supply the distance to the nearest neighbor
+            of the most recent point.
+        workers : int, optional
+            Number of workers to use for parallel processing. If -1 is given
+            all CPU threads are used. Default: 1.
+
+            .. versionadded:: 1.6.0
+
+        Returns
+        -------
+        d : float or array of floats
+            The distances to the nearest neighbors.
+            If ``x`` has shape ``tuple+(self.m,)``, then ``d`` has shape
+            ``tuple+(k,)``.
+            When k == 1, the last dimension of the output is squeezed.
+            Missing neighbors are indicated with infinite distances.
+            Hits are sorted by distance (nearest first).
+
+            .. versionchanged:: 1.9.0
+               Previously if ``k=None``, then `d` was an object array of
+               shape ``tuple``, containing lists of distances. This behavior
+               has been removed, use `query_ball_point` instead.
+
+        i : integer or array of integers
+            The index of each neighbor in ``self.data``.
+            ``i`` is the same shape as d.
+            Missing neighbors are indicated with ``self.n``.
+
+        Examples
+        --------
+
+        >>> import numpy as np
+        >>> from scipy.spatial import KDTree
+        >>> x, y = np.mgrid[0:5, 2:8]
+        >>> tree = KDTree(np.c_[x.ravel(), y.ravel()])
+
+        To query the nearest neighbours and return squeezed result, use
+
+        >>> dd, ii = tree.query([[0, 0], [2.2, 2.9]], k=1)
+        >>> print(dd, ii, sep='\n')
+        [2.         0.2236068]
+        [ 0 13]
+
+        To query the nearest neighbours and return unsqueezed result, use
+
+        >>> dd, ii = tree.query([[0, 0], [2.2, 2.9]], k=[1])
+        >>> print(dd, ii, sep='\n')
+        [[2.        ]
+         [0.2236068]]
+        [[ 0]
+         [13]]
+
+        To query the second nearest neighbours and return unsqueezed result,
+        use
+
+        >>> dd, ii = tree.query([[0, 0], [2.2, 2.9]], k=[2])
+        >>> print(dd, ii, sep='\n')
+        [[2.23606798]
+         [0.80622577]]
+        [[ 6]
+         [19]]
+
+        To query the first and second nearest neighbours, use
+
+        >>> dd, ii = tree.query([[0, 0], [2.2, 2.9]], k=2)
+        >>> print(dd, ii, sep='\n')
+        [[2.         2.23606798]
+         [0.2236068  0.80622577]]
+        [[ 0  6]
+         [13 19]]
+
+        or, be more specific
+
+        >>> dd, ii = tree.query([[0, 0], [2.2, 2.9]], k=[1, 2])
+        >>> print(dd, ii, sep='\n')
+        [[2.         2.23606798]
+         [0.2236068  0.80622577]]
+        [[ 0  6]
+         [13 19]]
+
+        """
+        x = np.asarray(x)
+        if x.dtype.kind == 'c':
+            raise TypeError("KDTree does not work with complex data")
+
+        if k is None:
+            raise ValueError("k must be an integer or a sequence of integers")
+
+        d, i = super().query(x, k, eps, p, distance_upper_bound, workers)
+        if isinstance(i, int):
+            i = np.intp(i)
+        return d, i
+
+    def query_ball_point(self, x, r, p=2., eps=0, workers=1,
+                         return_sorted=None, return_length=False):
+        """Find all points within distance r of point(s) x.
+
+        Parameters
+        ----------
+        x : array_like, shape tuple + (self.m,)
+            The point or points to search for neighbors of.
+        r : array_like, float
+            The radius of points to return, must broadcast to the length of x.
+        p : float, optional
+            Which Minkowski p-norm to use.  Should be in the range [1, inf].
+            A finite large p may cause a ValueError if overflow can occur.
+        eps : nonnegative float, optional
+            Approximate search. Branches of the tree are not explored if their
+            nearest points are further than ``r / (1 + eps)``, and branches are
+            added in bulk if their furthest points are nearer than
+            ``r * (1 + eps)``.
+        workers : int, optional
+            Number of jobs to schedule for parallel processing. If -1 is given
+            all processors are used. Default: 1.
+
+            .. versionadded:: 1.6.0
+        return_sorted : bool, optional
+            Sorts returned indices if True and does not sort them if False. If
+            None, does not sort single point queries, but does sort
+            multi-point queries which was the behavior before this option
+            was added.
+
+            .. versionadded:: 1.6.0
+        return_length : bool, optional
+            Return the number of points inside the radius instead of a list
+            of the indices.
+
+            .. versionadded:: 1.6.0
+
+        Returns
+        -------
+        results : list or array of lists
+            If `x` is a single point, returns a list of the indices of the
+            neighbors of `x`. If `x` is an array of points, returns an object
+            array of shape tuple containing lists of neighbors.
+
+        Notes
+        -----
+        If you have many points whose neighbors you want to find, you may save
+        substantial amounts of time by putting them in a KDTree and using
+        query_ball_tree.
+
+        Examples
+        --------
+        >>> import numpy as np
+        >>> from scipy import spatial
+        >>> x, y = np.mgrid[0:5, 0:5]
+        >>> points = np.c_[x.ravel(), y.ravel()]
+        >>> tree = spatial.KDTree(points)
+        >>> sorted(tree.query_ball_point([2, 0], 1))
+        [5, 10, 11, 15]
+
+        Query multiple points and plot the results:
+
+        >>> import matplotlib.pyplot as plt
+        >>> points = np.asarray(points)
+        >>> plt.plot(points[:,0], points[:,1], '.')
+        >>> for results in tree.query_ball_point(([2, 0], [3, 3]), 1):
+        ...     nearby_points = points[results]
+        ...     plt.plot(nearby_points[:,0], nearby_points[:,1], 'o')
+        >>> plt.margins(0.1, 0.1)
+        >>> plt.show()
+
+        """
+        x = np.asarray(x)
+        if x.dtype.kind == 'c':
+            raise TypeError("KDTree does not work with complex data")
+        return super().query_ball_point(
+            x, r, p, eps, workers, return_sorted, return_length)
+
+    def query_ball_tree(self, other, r, p=2., eps=0):
+        """
+        Find all pairs of points between `self` and `other` whose distance is
+        at most r.
+
+        Parameters
+        ----------
+        other : KDTree instance
+            The tree containing points to search against.
+        r : float
+            The maximum distance, has to be positive.
+        p : float, optional
+            Which Minkowski norm to use.  `p` has to meet the condition
+            ``1 <= p <= infinity``.
+        eps : float, optional
+            Approximate search.  Branches of the tree are not explored
+            if their nearest points are further than ``r/(1+eps)``, and
+            branches are added in bulk if their furthest points are nearer
+            than ``r * (1+eps)``.  `eps` has to be non-negative.
+
+        Returns
+        -------
+        results : list of lists
+            For each element ``self.data[i]`` of this tree, ``results[i]`` is a
+            list of the indices of its neighbors in ``other.data``.
+
+        Examples
+        --------
+        You can search all pairs of points between two kd-trees within a distance:
+
+        >>> import matplotlib.pyplot as plt
+        >>> import numpy as np
+        >>> from scipy.spatial import KDTree
+        >>> rng = np.random.default_rng()
+        >>> points1 = rng.random((15, 2))
+        >>> points2 = rng.random((15, 2))
+        >>> plt.figure(figsize=(6, 6))
+        >>> plt.plot(points1[:, 0], points1[:, 1], "xk", markersize=14)
+        >>> plt.plot(points2[:, 0], points2[:, 1], "og", markersize=14)
+        >>> kd_tree1 = KDTree(points1)
+        >>> kd_tree2 = KDTree(points2)
+        >>> indexes = kd_tree1.query_ball_tree(kd_tree2, r=0.2)
+        >>> for i in range(len(indexes)):
+        ...     for j in indexes[i]:
+        ...         plt.plot([points1[i, 0], points2[j, 0]],
+        ...             [points1[i, 1], points2[j, 1]], "-r")
+        >>> plt.show()
+
+        """
+        return super().query_ball_tree(other, r, p, eps)
+
+    def query_pairs(self, r, p=2., eps=0, output_type='set'):
+        """Find all pairs of points in `self` whose distance is at most r.
+
+        Parameters
+        ----------
+        r : positive float
+            The maximum distance.
+        p : float, optional
+            Which Minkowski norm to use.  `p` has to meet the condition
+            ``1 <= p <= infinity``.
+        eps : float, optional
+            Approximate search.  Branches of the tree are not explored
+            if their nearest points are further than ``r/(1+eps)``, and
+            branches are added in bulk if their furthest points are nearer
+            than ``r * (1+eps)``.  `eps` has to be non-negative.
+        output_type : string, optional
+            Choose the output container, 'set' or 'ndarray'. Default: 'set'
+
+            .. versionadded:: 1.6.0
+
+        Returns
+        -------
+        results : set or ndarray
+            Set of pairs ``(i,j)``, with ``i < j``, for which the corresponding
+            positions are close. If output_type is 'ndarray', an ndarry is
+            returned instead of a set.
+
+        Examples
+        --------
+        You can search all pairs of points in a kd-tree within a distance:
+
+        >>> import matplotlib.pyplot as plt
+        >>> import numpy as np
+        >>> from scipy.spatial import KDTree
+        >>> rng = np.random.default_rng()
+        >>> points = rng.random((20, 2))
+        >>> plt.figure(figsize=(6, 6))
+        >>> plt.plot(points[:, 0], points[:, 1], "xk", markersize=14)
+        >>> kd_tree = KDTree(points)
+        >>> pairs = kd_tree.query_pairs(r=0.2)
+        >>> for (i, j) in pairs:
+        ...     plt.plot([points[i, 0], points[j, 0]],
+        ...             [points[i, 1], points[j, 1]], "-r")
+        >>> plt.show()
+
+        """
+        return super().query_pairs(r, p, eps, output_type)
+
+    def count_neighbors(self, other, r, p=2., weights=None, cumulative=True):
+        """Count how many nearby pairs can be formed.
+
+        Count the number of pairs ``(x1,x2)`` can be formed, with ``x1`` drawn
+        from ``self`` and ``x2`` drawn from ``other``, and where
+        ``distance(x1, x2, p) <= r``.
+
+        Data points on ``self`` and ``other`` are optionally weighted by the
+        ``weights`` argument. (See below)
+
+        This is adapted from the "two-point correlation" algorithm described by
+        Gray and Moore [1]_.  See notes for further discussion.
+
+        Parameters
+        ----------
+        other : KDTree
+            The other tree to draw points from, can be the same tree as self.
+        r : float or one-dimensional array of floats
+            The radius to produce a count for. Multiple radii are searched with
+            a single tree traversal.
+            If the count is non-cumulative(``cumulative=False``), ``r`` defines
+            the edges of the bins, and must be non-decreasing.
+        p : float, optional
+            1<=p<=infinity.
+            Which Minkowski p-norm to use.
+            Default 2.0.
+            A finite large p may cause a ValueError if overflow can occur.
+        weights : tuple, array_like, or None, optional
+            If None, the pair-counting is unweighted.
+            If given as a tuple, weights[0] is the weights of points in
+            ``self``, and weights[1] is the weights of points in ``other``;
+            either can be None to indicate the points are unweighted.
+            If given as an array_like, weights is the weights of points in
+            ``self`` and ``other``. For this to make sense, ``self`` and
+            ``other`` must be the same tree. If ``self`` and ``other`` are two
+            different trees, a ``ValueError`` is raised.
+            Default: None
+
+            .. versionadded:: 1.6.0
+        cumulative : bool, optional
+            Whether the returned counts are cumulative. When cumulative is set
+            to ``False`` the algorithm is optimized to work with a large number
+            of bins (>10) specified by ``r``. When ``cumulative`` is set to
+            True, the algorithm is optimized to work with a small number of
+            ``r``. Default: True
+
+            .. versionadded:: 1.6.0
+
+        Returns
+        -------
+        result : scalar or 1-D array
+            The number of pairs. For unweighted counts, the result is integer.
+            For weighted counts, the result is float.
+            If cumulative is False, ``result[i]`` contains the counts with
+            ``(-inf if i == 0 else r[i-1]) < R <= r[i]``
+
+        Notes
+        -----
+        Pair-counting is the basic operation used to calculate the two point
+        correlation functions from a data set composed of position of objects.
+
+        Two point correlation function measures the clustering of objects and
+        is widely used in cosmology to quantify the large scale structure
+        in our Universe, but it may be useful for data analysis in other fields
+        where self-similar assembly of objects also occur.
+
+        The Landy-Szalay estimator for the two point correlation function of
+        ``D`` measures the clustering signal in ``D``. [2]_
+
+        For example, given the position of two sets of objects,
+
+        - objects ``D`` (data) contains the clustering signal, and
+
+        - objects ``R`` (random) that contains no signal,
+
+        .. math::
+
+             \\xi(r) = \\frac{<D, D> - 2 f <D, R> + f^2<R, R>}{f^2<R, R>},
+
+        where the brackets represents counting pairs between two data sets
+        in a finite bin around ``r`` (distance), corresponding to setting
+        `cumulative=False`, and ``f = float(len(D)) / float(len(R))`` is the
+        ratio between number of objects from data and random.
+
+        The algorithm implemented here is loosely based on the dual-tree
+        algorithm described in [1]_. We switch between two different
+        pair-cumulation scheme depending on the setting of ``cumulative``.
+        The computing time of the method we use when for
+        ``cumulative == False`` does not scale with the total number of bins.
+        The algorithm for ``cumulative == True`` scales linearly with the
+        number of bins, though it is slightly faster when only
+        1 or 2 bins are used. [5]_.
+
+        As an extension to the naive pair-counting,
+        weighted pair-counting counts the product of weights instead
+        of number of pairs.
+        Weighted pair-counting is used to estimate marked correlation functions
+        ([3]_, section 2.2),
+        or to properly calculate the average of data per distance bin
+        (e.g. [4]_, section 2.1 on redshift).
+
+        .. [1] Gray and Moore,
+               "N-body problems in statistical learning",
+               Mining the sky, 2000,
+               https://arxiv.org/abs/astro-ph/0012333
+
+        .. [2] Landy and Szalay,
+               "Bias and variance of angular correlation functions",
+               The Astrophysical Journal, 1993,
+               http://adsabs.harvard.edu/abs/1993ApJ...412...64L
+
+        .. [3] Sheth, Connolly and Skibba,
+               "Marked correlations in galaxy formation models",
+               Arxiv e-print, 2005,
+               https://arxiv.org/abs/astro-ph/0511773
+
+        .. [4] Hawkins, et al.,
+               "The 2dF Galaxy Redshift Survey: correlation functions,
+               peculiar velocities and the matter density of the Universe",
+               Monthly Notices of the Royal Astronomical Society, 2002,
+               http://adsabs.harvard.edu/abs/2003MNRAS.346...78H
+
+        .. [5] https://github.com/scipy/scipy/pull/5647#issuecomment-168474926
+
+        Examples
+        --------
+        You can count neighbors number between two kd-trees within a distance:
+
+        >>> import numpy as np
+        >>> from scipy.spatial import KDTree
+        >>> rng = np.random.default_rng()
+        >>> points1 = rng.random((5, 2))
+        >>> points2 = rng.random((5, 2))
+        >>> kd_tree1 = KDTree(points1)
+        >>> kd_tree2 = KDTree(points2)
+        >>> kd_tree1.count_neighbors(kd_tree2, 0.2)
+        1
+
+        This number is same as the total pair number calculated by
+        `query_ball_tree`:
+
+        >>> indexes = kd_tree1.query_ball_tree(kd_tree2, r=0.2)
+        >>> sum([len(i) for i in indexes])
+        1
+
+        """
+        return super().count_neighbors(other, r, p, weights, cumulative)
+
+    def sparse_distance_matrix(
+            self, other, max_distance, p=2., output_type='dok_matrix'):
+        """Compute a sparse distance matrix.
+
+        Computes a distance matrix between two KDTrees, leaving as zero
+        any distance greater than max_distance.
+
+        Parameters
+        ----------
+        other : KDTree
+
+        max_distance : positive float
+
+        p : float, 1<=p<=infinity
+            Which Minkowski p-norm to use.
+            A finite large p may cause a ValueError if overflow can occur.
+
+        output_type : string, optional
+            Which container to use for output data. Options: 'dok_matrix',
+            'coo_matrix', 'dict', or 'ndarray'. Default: 'dok_matrix'.
+
+            .. versionadded:: 1.6.0
+
+        Returns
+        -------
+        result : dok_matrix, coo_matrix, dict or ndarray
+            Sparse matrix representing the results in "dictionary of keys"
+            format. If a dict is returned the keys are (i,j) tuples of indices.
+            If output_type is 'ndarray' a record array with fields 'i', 'j',
+            and 'v' is returned,
+
+        Examples
+        --------
+        You can compute a sparse distance matrix between two kd-trees:
+
+        >>> import numpy as np
+        >>> from scipy.spatial import KDTree
+        >>> rng = np.random.default_rng()
+        >>> points1 = rng.random((5, 2))
+        >>> points2 = rng.random((5, 2))
+        >>> kd_tree1 = KDTree(points1)
+        >>> kd_tree2 = KDTree(points2)
+        >>> sdm = kd_tree1.sparse_distance_matrix(kd_tree2, 0.3)
+        >>> sdm.toarray()
+        array([[0.        , 0.        , 0.12295571, 0.        , 0.        ],
+           [0.        , 0.        , 0.        , 0.        , 0.        ],
+           [0.28942611, 0.        , 0.        , 0.2333084 , 0.        ],
+           [0.        , 0.        , 0.        , 0.        , 0.        ],
+           [0.24617575, 0.29571802, 0.26836782, 0.        , 0.        ]])
+
+        You can check distances above the `max_distance` are zeros:
+
+        >>> from scipy.spatial import distance_matrix
+        >>> distance_matrix(points1, points2)
+        array([[0.56906522, 0.39923701, 0.12295571, 0.8658745 , 0.79428925],
+           [0.37327919, 0.7225693 , 0.87665969, 0.32580855, 0.75679479],
+           [0.28942611, 0.30088013, 0.6395831 , 0.2333084 , 0.33630734],
+           [0.31994999, 0.72658602, 0.71124834, 0.55396483, 0.90785663],
+           [0.24617575, 0.29571802, 0.26836782, 0.57714465, 0.6473269 ]])
+
+        """
+        return super().sparse_distance_matrix(
+            other, max_distance, p, output_type)
+
+
+def distance_matrix(x, y, p=2, threshold=1000000):
+    """Compute the distance matrix.
+
+    Returns the matrix of all pair-wise distances.
+
+    Parameters
+    ----------
+    x : (M, K) array_like
+        Matrix of M vectors in K dimensions.
+    y : (N, K) array_like
+        Matrix of N vectors in K dimensions.
+    p : float, 1 <= p <= infinity
+        Which Minkowski p-norm to use.
+    threshold : positive int
+        If ``M * N * K`` > `threshold`, algorithm uses a Python loop instead
+        of large temporary arrays.
+
+    Returns
+    -------
+    result : (M, N) ndarray
+        Matrix containing the distance from every vector in `x` to every vector
+        in `y`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance_matrix
+    >>> distance_matrix([[0,0],[0,1]], [[1,0],[1,1]])
+    array([[ 1.        ,  1.41421356],
+           [ 1.41421356,  1.        ]])
+
+    """
+
+    x = np.asarray(x)
+    m, k = x.shape
+    y = np.asarray(y)
+    n, kk = y.shape
+
+    if k != kk:
+        raise ValueError(f"x contains {k}-dimensional vectors but y contains "
+                         f"{kk}-dimensional vectors")
+
+    if m*n*k <= threshold:
+        return minkowski_distance(x[:,np.newaxis,:],y[np.newaxis,:,:],p)
+    else:
+        result = np.empty((m,n),dtype=float)  # FIXME: figure out the best dtype
+        if m < n:
+            for i in range(m):
+                result[i,:] = minkowski_distance(x[i],y,p)
+        else:
+            for j in range(n):
+                result[:,j] = minkowski_distance(x,y[j],p)
+        return result

venv/lib/python3.10/site-packages/scipy/spatial/_plotutils.py

@@ -0,0 +1,270 @@
+import numpy as np
+from scipy._lib.decorator import decorator as _decorator
+
+__all__ = ['delaunay_plot_2d', 'convex_hull_plot_2d', 'voronoi_plot_2d']
+
+
+@_decorator
+def _held_figure(func, obj, ax=None, **kw):
+    import matplotlib.pyplot as plt
+
+    if ax is None:
+        fig = plt.figure()
+        ax = fig.gca()
+        return func(obj, ax=ax, **kw)
+
+    # As of matplotlib 2.0, the "hold" mechanism is deprecated.
+    # When matplotlib 1.x is no longer supported, this check can be removed.
+    was_held = getattr(ax, 'ishold', lambda: True)()
+    if was_held:
+        return func(obj, ax=ax, **kw)
+    try:
+        ax.hold(True)
+        return func(obj, ax=ax, **kw)
+    finally:
+        ax.hold(was_held)
+
+
+def _adjust_bounds(ax, points):
+    margin = 0.1 * np.ptp(points, axis=0)
+    xy_min = points.min(axis=0) - margin
+    xy_max = points.max(axis=0) + margin
+    ax.set_xlim(xy_min[0], xy_max[0])
+    ax.set_ylim(xy_min[1], xy_max[1])
+
+
+@_held_figure
+def delaunay_plot_2d(tri, ax=None):
+    """
+    Plot the given Delaunay triangulation in 2-D
+
+    Parameters
+    ----------
+    tri : scipy.spatial.Delaunay instance
+        Triangulation to plot
+    ax : matplotlib.axes.Axes instance, optional
+        Axes to plot on
+
+    Returns
+    -------
+    fig : matplotlib.figure.Figure instance
+        Figure for the plot
+
+    See Also
+    --------
+    Delaunay
+    matplotlib.pyplot.triplot
+
+    Notes
+    -----
+    Requires Matplotlib.
+
+    Examples
+    --------
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.spatial import Delaunay, delaunay_plot_2d
+
+    The Delaunay triangulation of a set of random points:
+
+    >>> rng = np.random.default_rng()
+    >>> points = rng.random((30, 2))
+    >>> tri = Delaunay(points)
+
+    Plot it:
+
+    >>> _ = delaunay_plot_2d(tri)
+    >>> plt.show()
+
+    """
+    if tri.points.shape[1] != 2:
+        raise ValueError("Delaunay triangulation is not 2-D")
+
+    x, y = tri.points.T
+    ax.plot(x, y, 'o')
+    ax.triplot(x, y, tri.simplices.copy())
+
+    _adjust_bounds(ax, tri.points)
+
+    return ax.figure
+
+
+@_held_figure
+def convex_hull_plot_2d(hull, ax=None):
+    """
+    Plot the given convex hull diagram in 2-D
+
+    Parameters
+    ----------
+    hull : scipy.spatial.ConvexHull instance
+        Convex hull to plot
+    ax : matplotlib.axes.Axes instance, optional
+        Axes to plot on
+
+    Returns
+    -------
+    fig : matplotlib.figure.Figure instance
+        Figure for the plot
+
+    See Also
+    --------
+    ConvexHull
+
+    Notes
+    -----
+    Requires Matplotlib.
+
+
+    Examples
+    --------
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.spatial import ConvexHull, convex_hull_plot_2d
+
+    The convex hull of a random set of points:
+
+    >>> rng = np.random.default_rng()
+    >>> points = rng.random((30, 2))
+    >>> hull = ConvexHull(points)
+
+    Plot it:
+
+    >>> _ = convex_hull_plot_2d(hull)
+    >>> plt.show()
+
+    """
+    from matplotlib.collections import LineCollection
+
+    if hull.points.shape[1] != 2:
+        raise ValueError("Convex hull is not 2-D")
+
+    ax.plot(hull.points[:, 0], hull.points[:, 1], 'o')
+    line_segments = [hull.points[simplex] for simplex in hull.simplices]
+    ax.add_collection(LineCollection(line_segments,
+                                     colors='k',
+                                     linestyle='solid'))
+    _adjust_bounds(ax, hull.points)
+
+    return ax.figure
+
+
+@_held_figure
+def voronoi_plot_2d(vor, ax=None, **kw):
+    """
+    Plot the given Voronoi diagram in 2-D
+
+    Parameters
+    ----------
+    vor : scipy.spatial.Voronoi instance
+        Diagram to plot
+    ax : matplotlib.axes.Axes instance, optional
+        Axes to plot on
+    show_points : bool, optional
+        Add the Voronoi points to the plot.
+    show_vertices : bool, optional
+        Add the Voronoi vertices to the plot.
+    line_colors : string, optional
+        Specifies the line color for polygon boundaries
+    line_width : float, optional
+        Specifies the line width for polygon boundaries
+    line_alpha : float, optional
+        Specifies the line alpha for polygon boundaries
+    point_size : float, optional
+        Specifies the size of points
+
+    Returns
+    -------
+    fig : matplotlib.figure.Figure instance
+        Figure for the plot
+
+    See Also
+    --------
+    Voronoi
+
+    Notes
+    -----
+    Requires Matplotlib.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.spatial import Voronoi, voronoi_plot_2d
+
+    Create a set of points for the example:
+
+    >>> rng = np.random.default_rng()
+    >>> points = rng.random((10,2))
+
+    Generate the Voronoi diagram for the points:
+
+    >>> vor = Voronoi(points)
+
+    Use `voronoi_plot_2d` to plot the diagram:
+
+    >>> fig = voronoi_plot_2d(vor)
+
+    Use `voronoi_plot_2d` to plot the diagram again, with some settings
+    customized:
+
+    >>> fig = voronoi_plot_2d(vor, show_vertices=False, line_colors='orange',
+    ...                       line_width=2, line_alpha=0.6, point_size=2)
+    >>> plt.show()
+
+    """
+    from matplotlib.collections import LineCollection
+
+    if vor.points.shape[1] != 2:
+        raise ValueError("Voronoi diagram is not 2-D")
+
+    if kw.get('show_points', True):
+        point_size = kw.get('point_size', None)
+        ax.plot(vor.points[:, 0], vor.points[:, 1], '.', markersize=point_size)
+    if kw.get('show_vertices', True):
+        ax.plot(vor.vertices[:, 0], vor.vertices[:, 1], 'o')
+
+    line_colors = kw.get('line_colors', 'k')
+    line_width = kw.get('line_width', 1.0)
+    line_alpha = kw.get('line_alpha', 1.0)
+
+    center = vor.points.mean(axis=0)
+    ptp_bound = np.ptp(vor.points, axis=0)
+
+    finite_segments = []
+    infinite_segments = []
+    for pointidx, simplex in zip(vor.ridge_points, vor.ridge_vertices):
+        simplex = np.asarray(simplex)
+        if np.all(simplex >= 0):
+            finite_segments.append(vor.vertices[simplex])
+        else:
+            i = simplex[simplex >= 0][0]  # finite end Voronoi vertex
+
+            t = vor.points[pointidx[1]] - vor.points[pointidx[0]]  # tangent
+            t /= np.linalg.norm(t)
+            n = np.array([-t[1], t[0]])  # normal
+
+            midpoint = vor.points[pointidx].mean(axis=0)
+            direction = np.sign(np.dot(midpoint - center, n)) * n
+            if (vor.furthest_site):
+                direction = -direction
+            aspect_factor = abs(ptp_bound.max() / ptp_bound.min())
+            far_point = vor.vertices[i] + direction * ptp_bound.max() * aspect_factor
+
+            infinite_segments.append([vor.vertices[i], far_point])
+
+    ax.add_collection(LineCollection(finite_segments,
+                                     colors=line_colors,
+                                     lw=line_width,
+                                     alpha=line_alpha,
+                                     linestyle='solid'))
+    ax.add_collection(LineCollection(infinite_segments,
+                                     colors=line_colors,
+                                     lw=line_width,
+                                     alpha=line_alpha,
+                                     linestyle='dashed'))
+
+    _adjust_bounds(ax, vor.points)
+
+    return ax.figure

venv/lib/python3.10/site-packages/scipy/spatial/_procrustes.py

@@ -0,0 +1,132 @@
+"""
+This module provides functions to perform full Procrustes analysis.
+
+This code was originally written by Justin Kucynski and ported over from
+scikit-bio by Yoshiki Vazquez-Baeza.
+"""
+
+import numpy as np
+from scipy.linalg import orthogonal_procrustes
+
+
+__all__ = ['procrustes']
+
+
+def procrustes(data1, data2):
+    r"""Procrustes analysis, a similarity test for two data sets.
+
+    Each input matrix is a set of points or vectors (the rows of the matrix).
+    The dimension of the space is the number of columns of each matrix. Given
+    two identically sized matrices, procrustes standardizes both such that:
+
+    - :math:`tr(AA^{T}) = 1`.
+
+    - Both sets of points are centered around the origin.
+
+    Procrustes ([1]_, [2]_) then applies the optimal transform to the second
+    matrix (including scaling/dilation, rotations, and reflections) to minimize
+    :math:`M^{2}=\sum(data1-data2)^{2}`, or the sum of the squares of the
+    pointwise differences between the two input datasets.
+
+    This function was not designed to handle datasets with different numbers of
+    datapoints (rows).  If two data sets have different dimensionality
+    (different number of columns), simply add columns of zeros to the smaller
+    of the two.
+
+    Parameters
+    ----------
+    data1 : array_like
+        Matrix, n rows represent points in k (columns) space `data1` is the
+        reference data, after it is standardised, the data from `data2` will be
+        transformed to fit the pattern in `data1` (must have >1 unique points).
+    data2 : array_like
+        n rows of data in k space to be fit to `data1`.  Must be the  same
+        shape ``(numrows, numcols)`` as data1 (must have >1 unique points).
+
+    Returns
+    -------
+    mtx1 : array_like
+        A standardized version of `data1`.
+    mtx2 : array_like
+        The orientation of `data2` that best fits `data1`. Centered, but not
+        necessarily :math:`tr(AA^{T}) = 1`.
+    disparity : float
+        :math:`M^{2}` as defined above.
+
+    Raises
+    ------
+    ValueError
+        If the input arrays are not two-dimensional.
+        If the shape of the input arrays is different.
+        If the input arrays have zero columns or zero rows.
+
+    See Also
+    --------
+    scipy.linalg.orthogonal_procrustes
+    scipy.spatial.distance.directed_hausdorff : Another similarity test
+      for two data sets
+
+    Notes
+    -----
+    - The disparity should not depend on the order of the input matrices, but
+      the output matrices will, as only the first output matrix is guaranteed
+      to be scaled such that :math:`tr(AA^{T}) = 1`.
+
+    - Duplicate data points are generally ok, duplicating a data point will
+      increase its effect on the procrustes fit.
+
+    - The disparity scales as the number of points per input matrix.
+
+    References
+    ----------
+    .. [1] Krzanowski, W. J. (2000). "Principles of Multivariate analysis".
+    .. [2] Gower, J. C. (1975). "Generalized procrustes analysis".
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.spatial import procrustes
+
+    The matrix ``b`` is a rotated, shifted, scaled and mirrored version of
+    ``a`` here:
+
+    >>> a = np.array([[1, 3], [1, 2], [1, 1], [2, 1]], 'd')
+    >>> b = np.array([[4, -2], [4, -4], [4, -6], [2, -6]], 'd')
+    >>> mtx1, mtx2, disparity = procrustes(a, b)
+    >>> round(disparity)
+    0.0
+
+    """
+    mtx1 = np.array(data1, dtype=np.float64, copy=True)
+    mtx2 = np.array(data2, dtype=np.float64, copy=True)
+
+    if mtx1.ndim != 2 or mtx2.ndim != 2:
+        raise ValueError("Input matrices must be two-dimensional")
+    if mtx1.shape != mtx2.shape:
+        raise ValueError("Input matrices must be of same shape")
+    if mtx1.size == 0:
+        raise ValueError("Input matrices must be >0 rows and >0 cols")
+
+    # translate all the data to the origin
+    mtx1 -= np.mean(mtx1, 0)
+    mtx2 -= np.mean(mtx2, 0)
+
+    norm1 = np.linalg.norm(mtx1)
+    norm2 = np.linalg.norm(mtx2)
+
+    if norm1 == 0 or norm2 == 0:
+        raise ValueError("Input matrices must contain >1 unique points")
+
+    # change scaling of data (in rows) such that trace(mtx*mtx') = 1
+    mtx1 /= norm1
+    mtx2 /= norm2
+
+    # transform mtx2 to minimize disparity
+    R, s = orthogonal_procrustes(mtx1, mtx2)
+    mtx2 = np.dot(mtx2, R.T) * s
+
+    # measure the dissimilarity between the two datasets
+    disparity = np.sum(np.square(mtx1 - mtx2))
+
+    return mtx1, mtx2, disparity
+

venv/lib/python3.10/site-packages/scipy/spatial/_qhull.pyi

@@ -0,0 +1,213 @@
+'''
+Static type checking stub file for scipy/spatial/qhull.pyx
+'''
+
+
+import numpy as np
+from numpy.typing import ArrayLike, NDArray
+from typing_extensions import final
+
+class QhullError(RuntimeError):
+    ...
+    
+@final
+class _Qhull:
+    # Read-only cython attribute that behaves, more or less, like a property
+    @property
+    def ndim(self) -> int: ...
+    mode_option: bytes
+    options: bytes
+    furthest_site: bool
+
+    def __init__(
+        self,
+        mode_option: bytes,
+        points: NDArray[np.float64],
+        options: None | bytes = ...,
+        required_options: None | bytes = ...,
+        furthest_site: bool = ...,
+        incremental: bool = ...,
+        interior_point: None | NDArray[np.float64] = ...,
+    ) -> None: ...
+    def check_active(self) -> None: ...
+    def close(self) -> None: ...
+    def get_points(self) -> NDArray[np.float64]: ...
+    def add_points(
+        self,
+        points: ArrayLike,
+        interior_point: ArrayLike = ...
+    ) -> None: ...
+    def get_paraboloid_shift_scale(self) -> tuple[float, float]: ...
+    def volume_area(self) -> tuple[float, float]: ...
+    def triangulate(self) -> None: ...
+    def get_simplex_facet_array(self) -> tuple[
+        NDArray[np.intc],
+        NDArray[np.intc],
+        NDArray[np.float64],
+        NDArray[np.intc],
+        NDArray[np.intc],
+    ]: ...
+    def get_hull_points(self) -> NDArray[np.float64]: ...
+    def get_hull_facets(self) -> tuple[
+        list[list[int]],
+        NDArray[np.float64],
+    ]: ...
+    def get_voronoi_diagram(self) -> tuple[
+        NDArray[np.float64],
+        NDArray[np.intc],
+        list[list[int]],
+        list[list[int]],
+        NDArray[np.intp],
+    ]: ...
+    def get_extremes_2d(self) -> NDArray[np.intc]: ...
+
+def _get_barycentric_transforms(
+    points: NDArray[np.float64],
+    simplices: NDArray[np.intc],
+    eps: float
+) -> NDArray[np.float64]: ...
+
+class _QhullUser:
+    ndim: int
+    npoints: int
+    min_bound: NDArray[np.float64]
+    max_bound: NDArray[np.float64]
+
+    def __init__(self, qhull: _Qhull, incremental: bool = ...) -> None: ...
+    def close(self) -> None: ...
+    def _update(self, qhull: _Qhull) -> None: ...
+    def _add_points(
+        self,
+        points: ArrayLike,
+        restart: bool = ...,
+        interior_point: ArrayLike = ...
+    ) -> None: ...
+
+class Delaunay(_QhullUser):
+    furthest_site: bool
+    paraboloid_scale: float
+    paraboloid_shift: float
+    simplices: NDArray[np.intc]
+    neighbors: NDArray[np.intc]
+    equations: NDArray[np.float64]
+    coplanar: NDArray[np.intc]
+    good: NDArray[np.intc]
+    nsimplex: int
+    vertices: NDArray[np.intc]
+
+    def __init__(
+        self,
+        points: ArrayLike,
+        furthest_site: bool = ...,
+        incremental: bool = ...,
+        qhull_options: None | str = ...
+    ) -> None: ...
+    def _update(self, qhull: _Qhull) -> None: ...
+    def add_points(
+        self,
+        points: ArrayLike,
+        restart: bool = ...
+    ) -> None: ...
+    @property
+    def points(self) -> NDArray[np.float64]: ...
+    @property
+    def transform(self) -> NDArray[np.float64]: ...
+    @property
+    def vertex_to_simplex(self) -> NDArray[np.intc]: ...
+    @property
+    def vertex_neighbor_vertices(self) -> tuple[
+        NDArray[np.intc],
+        NDArray[np.intc],
+    ]: ...
+    @property
+    def convex_hull(self) -> NDArray[np.intc]: ...
+    def find_simplex(
+        self,
+        xi: ArrayLike,
+        bruteforce: bool = ...,
+        tol: float = ...
+    ) -> NDArray[np.intc]: ...
+    def plane_distance(self, xi: ArrayLike) -> NDArray[np.float64]: ...
+    def lift_points(self, x: ArrayLike) -> NDArray[np.float64]: ...
+
+def tsearch(tri: Delaunay, xi: ArrayLike) -> NDArray[np.intc]: ...
+def _copy_docstr(dst: object, src: object) -> None: ...
+
+class ConvexHull(_QhullUser):
+    simplices: NDArray[np.intc]
+    neighbors: NDArray[np.intc]
+    equations: NDArray[np.float64]
+    coplanar: NDArray[np.intc]
+    good: None | NDArray[np.bool_]
+    volume: float
+    area: float
+    nsimplex: int
+
+    def __init__(
+        self,
+        points: ArrayLike,
+        incremental: bool = ...,
+        qhull_options: None | str = ...
+    ) -> None: ...
+    def _update(self, qhull: _Qhull) -> None: ...
+    def add_points(self, points: ArrayLike,
+                   restart: bool = ...) -> None: ...
+    @property
+    def points(self) -> NDArray[np.float64]: ...
+    @property
+    def vertices(self) -> NDArray[np.intc]: ...
+
+class Voronoi(_QhullUser):
+    vertices: NDArray[np.float64]
+    ridge_points: NDArray[np.intc]
+    ridge_vertices: list[list[int]]
+    regions: list[list[int]]
+    point_region: NDArray[np.intp]
+    furthest_site: bool
+
+    def __init__(
+        self,
+        points: ArrayLike,
+        furthest_site: bool = ...,
+        incremental: bool = ...,
+        qhull_options: None | str = ...
+    ) -> None: ...
+    def _update(self, qhull: _Qhull) -> None: ...
+    def add_points(
+        self,
+        points: ArrayLike,
+        restart: bool = ...
+    ) -> None: ...
+    @property
+    def points(self) -> NDArray[np.float64]: ...
+    @property
+    def ridge_dict(self) -> dict[tuple[int, int], list[int]]: ...
+
+class HalfspaceIntersection(_QhullUser):
+    interior_point: NDArray[np.float64]
+    dual_facets: list[list[int]]
+    dual_equations: NDArray[np.float64]
+    dual_points: NDArray[np.float64]
+    dual_volume: float
+    dual_area: float
+    intersections: NDArray[np.float64]
+    ndim: int
+    nineq: int
+
+    def __init__(
+        self,
+        halfspaces: ArrayLike,
+        interior_point: ArrayLike,
+        incremental: bool = ...,
+        qhull_options: None | str = ...
+    ) -> None: ...
+    def _update(self, qhull: _Qhull) -> None: ...
+    def add_halfspaces(
+        self,
+        halfspaces: ArrayLike,
+        restart: bool = ...
+    ) -> None: ...
+    @property
+    def halfspaces(self) -> NDArray[np.float64]: ...
+    @property
+    def dual_vertices(self) -> NDArray[np.integer]: ...

venv/lib/python3.10/site-packages/scipy/spatial/_spherical_voronoi.py

@@ -0,0 +1,341 @@
+"""
+Spherical Voronoi Code
+
+.. versionadded:: 0.18.0
+
+"""
+#
+# Copyright (C)  Tyler Reddy, Ross Hemsley, Edd Edmondson,
+#                Nikolai Nowaczyk, Joe Pitt-Francis, 2015.
+#
+# Distributed under the same BSD license as SciPy.
+#
+
+import numpy as np
+import scipy
+from . import _voronoi
+from scipy.spatial import cKDTree
+
+__all__ = ['SphericalVoronoi']
+
+
+def calculate_solid_angles(R):
+    """Calculates the solid angles of plane triangles. Implements the method of
+    Van Oosterom and Strackee [VanOosterom]_ with some modifications. Assumes
+    that input points have unit norm."""
+    # Original method uses a triple product `R1 . (R2 x R3)` for the numerator.
+    # This is equal to the determinant of the matrix [R1 R2 R3], which can be
+    # computed with better stability.
+    numerator = np.linalg.det(R)
+    denominator = 1 + (np.einsum('ij,ij->i', R[:, 0], R[:, 1]) +
+                       np.einsum('ij,ij->i', R[:, 1], R[:, 2]) +
+                       np.einsum('ij,ij->i', R[:, 2], R[:, 0]))
+    return np.abs(2 * np.arctan2(numerator, denominator))
+
+
+class SphericalVoronoi:
+    """ Voronoi diagrams on the surface of a sphere.
+
+    .. versionadded:: 0.18.0
+
+    Parameters
+    ----------
+    points : ndarray of floats, shape (npoints, ndim)
+        Coordinates of points from which to construct a spherical
+        Voronoi diagram.
+    radius : float, optional
+        Radius of the sphere (Default: 1)
+    center : ndarray of floats, shape (ndim,)
+        Center of sphere (Default: origin)
+    threshold : float
+        Threshold for detecting duplicate points and
+        mismatches between points and sphere parameters.
+        (Default: 1e-06)
+
+    Attributes
+    ----------
+    points : double array of shape (npoints, ndim)
+        the points in `ndim` dimensions to generate the Voronoi diagram from
+    radius : double
+        radius of the sphere
+    center : double array of shape (ndim,)
+        center of the sphere
+    vertices : double array of shape (nvertices, ndim)
+        Voronoi vertices corresponding to points
+    regions : list of list of integers of shape (npoints, _ )
+        the n-th entry is a list consisting of the indices
+        of the vertices belonging to the n-th point in points
+
+    Methods
+    -------
+    calculate_areas
+        Calculates the areas of the Voronoi regions. For 2D point sets, the
+        regions are circular arcs. The sum of the areas is `2 * pi * radius`.
+        For 3D point sets, the regions are spherical polygons. The sum of the
+        areas is `4 * pi * radius**2`.
+
+    Raises
+    ------
+    ValueError
+        If there are duplicates in `points`.
+        If the provided `radius` is not consistent with `points`.
+
+    Notes
+    -----
+    The spherical Voronoi diagram algorithm proceeds as follows. The Convex
+    Hull of the input points (generators) is calculated, and is equivalent to
+    their Delaunay triangulation on the surface of the sphere [Caroli]_.
+    The Convex Hull neighbour information is then used to
+    order the Voronoi region vertices around each generator. The latter
+    approach is substantially less sensitive to floating point issues than
+    angle-based methods of Voronoi region vertex sorting.
+
+    Empirical assessment of spherical Voronoi algorithm performance suggests
+    quadratic time complexity (loglinear is optimal, but algorithms are more
+    challenging to implement).
+
+    References
+    ----------
+    .. [Caroli] Caroli et al. Robust and Efficient Delaunay triangulations of
+                points on or close to a sphere. Research Report RR-7004, 2009.
+
+    .. [VanOosterom] Van Oosterom and Strackee. The solid angle of a plane
+                     triangle. IEEE Transactions on Biomedical Engineering,
+                     2, 1983, pp 125--126.
+
+    See Also
+    --------
+    Voronoi : Conventional Voronoi diagrams in N dimensions.
+
+    Examples
+    --------
+    Do some imports and take some points on a cube:
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.spatial import SphericalVoronoi, geometric_slerp
+    >>> from mpl_toolkits.mplot3d import proj3d
+    >>> # set input data
+    >>> points = np.array([[0, 0, 1], [0, 0, -1], [1, 0, 0],
+    ...                    [0, 1, 0], [0, -1, 0], [-1, 0, 0], ])
+
+    Calculate the spherical Voronoi diagram:
+
+    >>> radius = 1
+    >>> center = np.array([0, 0, 0])
+    >>> sv = SphericalVoronoi(points, radius, center)
+
+    Generate plot:
+
+    >>> # sort vertices (optional, helpful for plotting)
+    >>> sv.sort_vertices_of_regions()
+    >>> t_vals = np.linspace(0, 1, 2000)
+    >>> fig = plt.figure()
+    >>> ax = fig.add_subplot(111, projection='3d')
+    >>> # plot the unit sphere for reference (optional)
+    >>> u = np.linspace(0, 2 * np.pi, 100)
+    >>> v = np.linspace(0, np.pi, 100)
+    >>> x = np.outer(np.cos(u), np.sin(v))
+    >>> y = np.outer(np.sin(u), np.sin(v))
+    >>> z = np.outer(np.ones(np.size(u)), np.cos(v))
+    >>> ax.plot_surface(x, y, z, color='y', alpha=0.1)
+    >>> # plot generator points
+    >>> ax.scatter(points[:, 0], points[:, 1], points[:, 2], c='b')
+    >>> # plot Voronoi vertices
+    >>> ax.scatter(sv.vertices[:, 0], sv.vertices[:, 1], sv.vertices[:, 2],
+    ...                    c='g')
+    >>> # indicate Voronoi regions (as Euclidean polygons)
+    >>> for region in sv.regions:
+    ...    n = len(region)
+    ...    for i in range(n):
+    ...        start = sv.vertices[region][i]
+    ...        end = sv.vertices[region][(i + 1) % n]
+    ...        result = geometric_slerp(start, end, t_vals)
+    ...        ax.plot(result[..., 0],
+    ...                result[..., 1],
+    ...                result[..., 2],
+    ...                c='k')
+    >>> ax.azim = 10
+    >>> ax.elev = 40
+    >>> _ = ax.set_xticks([])
+    >>> _ = ax.set_yticks([])
+    >>> _ = ax.set_zticks([])
+    >>> fig.set_size_inches(4, 4)
+    >>> plt.show()
+
+    """
+    def __init__(self, points, radius=1, center=None, threshold=1e-06):
+
+        if radius is None:
+            raise ValueError('`radius` is `None`. '
+                             'Please provide a floating point number '
+                             '(i.e. `radius=1`).')
+
+        self.radius = float(radius)
+        self.points = np.array(points).astype(np.float64)
+        self._dim = self.points.shape[1]
+        if center is None:
+            self.center = np.zeros(self._dim)
+        else:
+            self.center = np.array(center, dtype=float)
+
+        # test degenerate input
+        self._rank = np.linalg.matrix_rank(self.points - self.points[0],
+                                           tol=threshold * self.radius)
+        if self._rank < self._dim:
+            raise ValueError(f"Rank of input points must be at least {self._dim}")
+
+        if cKDTree(self.points).query_pairs(threshold * self.radius):
+            raise ValueError("Duplicate generators present.")
+
+        radii = np.linalg.norm(self.points - self.center, axis=1)
+        max_discrepancy = np.abs(radii - self.radius).max()
+        if max_discrepancy >= threshold * self.radius:
+            raise ValueError("Radius inconsistent with generators.")
+
+        self._calc_vertices_regions()
+
+    def _calc_vertices_regions(self):
+        """
+        Calculates the Voronoi vertices and regions of the generators stored
+        in self.points. The vertices will be stored in self.vertices and the
+        regions in self.regions.
+
+        This algorithm was discussed at PyData London 2015 by
+        Tyler Reddy, Ross Hemsley and Nikolai Nowaczyk
+        """
+        # get Convex Hull
+        conv = scipy.spatial.ConvexHull(self.points)
+        # get circumcenters of Convex Hull triangles from facet equations
+        # for 3D input circumcenters will have shape: (2N-4, 3)
+        self.vertices = self.radius * conv.equations[:, :-1] + self.center
+        self._simplices = conv.simplices
+        # calculate regions from triangulation
+        # for 3D input simplex_indices will have shape: (2N-4,)
+        simplex_indices = np.arange(len(self._simplices))
+        # for 3D input tri_indices will have shape: (6N-12,)
+        tri_indices = np.column_stack([simplex_indices] * self._dim).ravel()
+        # for 3D input point_indices will have shape: (6N-12,)
+        point_indices = self._simplices.ravel()
+        # for 3D input indices will have shape: (6N-12,)
+        indices = np.argsort(point_indices, kind='mergesort')
+        # for 3D input flattened_groups will have shape: (6N-12,)
+        flattened_groups = tri_indices[indices].astype(np.intp)
+        # intervals will have shape: (N+1,)
+        intervals = np.cumsum(np.bincount(point_indices + 1))
+        # split flattened groups to get nested list of unsorted regions
+        groups = [list(flattened_groups[intervals[i]:intervals[i + 1]])
+                  for i in range(len(intervals) - 1)]
+        self.regions = groups
+
+    def sort_vertices_of_regions(self):
+        """Sort indices of the vertices to be (counter-)clockwise ordered.
+
+        Raises
+        ------
+        TypeError
+            If the points are not three-dimensional.
+
+        Notes
+        -----
+        For each region in regions, it sorts the indices of the Voronoi
+        vertices such that the resulting points are in a clockwise or
+        counterclockwise order around the generator point.
+
+        This is done as follows: Recall that the n-th region in regions
+        surrounds the n-th generator in points and that the k-th
+        Voronoi vertex in vertices is the circumcenter of the k-th triangle
+        in self._simplices.  For each region n, we choose the first triangle
+        (=Voronoi vertex) in self._simplices and a vertex of that triangle
+        not equal to the center n. These determine a unique neighbor of that
+        triangle, which is then chosen as the second triangle. The second
+        triangle will have a unique vertex not equal to the current vertex or
+        the center. This determines a unique neighbor of the second triangle,
+        which is then chosen as the third triangle and so forth. We proceed
+        through all the triangles (=Voronoi vertices) belonging to the
+        generator in points and obtain a sorted version of the vertices
+        of its surrounding region.
+        """
+        if self._dim != 3:
+            raise TypeError("Only supported for three-dimensional point sets")
+        _voronoi.sort_vertices_of_regions(self._simplices, self.regions)
+
+    def _calculate_areas_3d(self):
+        self.sort_vertices_of_regions()
+        sizes = [len(region) for region in self.regions]
+        csizes = np.cumsum(sizes)
+        num_regions = csizes[-1]
+
+        # We create a set of triangles consisting of one point and two Voronoi
+        # vertices. The vertices of each triangle are adjacent in the sorted
+        # regions list.
+        point_indices = [i for i, size in enumerate(sizes)
+                         for j in range(size)]
+
+        nbrs1 = np.array([r for region in self.regions for r in region])
+
+        # The calculation of nbrs2 is a vectorized version of:
+        # np.array([r for region in self.regions for r in np.roll(region, 1)])
+        nbrs2 = np.roll(nbrs1, 1)
+        indices = np.roll(csizes, 1)
+        indices[0] = 0
+        nbrs2[indices] = nbrs1[csizes - 1]
+
+        # Normalize points and vertices.
+        pnormalized = (self.points - self.center) / self.radius
+        vnormalized = (self.vertices - self.center) / self.radius
+
+        # Create the complete set of triangles and calculate their solid angles
+        triangles = np.hstack([pnormalized[point_indices],
+                               vnormalized[nbrs1],
+                               vnormalized[nbrs2]
+                               ]).reshape((num_regions, 3, 3))
+        triangle_solid_angles = calculate_solid_angles(triangles)
+
+        # Sum the solid angles of the triangles in each region
+        solid_angles = np.cumsum(triangle_solid_angles)[csizes - 1]
+        solid_angles[1:] -= solid_angles[:-1]
+
+        # Get polygon areas using A = omega * r**2
+        return solid_angles * self.radius**2
+
+    def _calculate_areas_2d(self):
+        # Find start and end points of arcs
+        arcs = self.points[self._simplices] - self.center
+
+        # Calculate the angle subtended by arcs
+        d = np.sum((arcs[:, 1] - arcs[:, 0]) ** 2, axis=1)
+        theta = np.arccos(1 - (d / (2 * (self.radius ** 2))))
+
+        # Get areas using A = r * theta
+        areas = self.radius * theta
+
+        # Correct arcs which go the wrong way (single-hemisphere inputs)
+        signs = np.sign(np.einsum('ij,ij->i', arcs[:, 0],
+                                              self.vertices - self.center))
+        indices = np.where(signs < 0)
+        areas[indices] = 2 * np.pi * self.radius - areas[indices]
+        return areas
+
+    def calculate_areas(self):
+        """Calculates the areas of the Voronoi regions.
+
+        For 2D point sets, the regions are circular arcs. The sum of the areas
+        is `2 * pi * radius`.
+
+        For 3D point sets, the regions are spherical polygons. The sum of the
+        areas is `4 * pi * radius**2`.
+
+        .. versionadded:: 1.5.0
+
+        Returns
+        -------
+        areas : double array of shape (npoints,)
+            The areas of the Voronoi regions.
+        """
+        if self._dim == 2:
+            return self._calculate_areas_2d()
+        elif self._dim == 3:
+            return self._calculate_areas_3d()
+        else:
+            raise TypeError("Only supported for 2D and 3D point sets")

venv/lib/python3.10/site-packages/scipy/spatial/_voronoi.pyi

@@ -0,0 +1,4 @@
+
+import numpy as np
+
+def sort_vertices_of_regions(simplices: np.ndarray, regions: list[list[int]]) -> None: ...  # noqa: E501

venv/lib/python3.10/site-packages/scipy/spatial/ckdtree.py

@@ -0,0 +1,27 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.spatial` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+
+__all__ = [  # noqa: F822
+    'cKDTree',
+    'cKDTreeNode',
+    'coo_entries',
+    'operator',
+    'ordered_pairs',
+    'os',
+    'scipy',
+    'threading',
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="spatial", module="ckdtree",
+                                   private_modules=["_ckdtree"], all=__all__,
+                                   attribute=name)

venv/lib/python3.10/site-packages/scipy/spatial/distance.py

@@ -0,0 +1,2993 @@
+"""
+Distance computations (:mod:`scipy.spatial.distance`)
+=====================================================
+
+.. sectionauthor:: Damian Eads
+
+Function reference
+------------------
+
+Distance matrix computation from a collection of raw observation vectors
+stored in a rectangular array.
+
+.. autosummary::
+   :toctree: generated/
+
+   pdist   -- pairwise distances between observation vectors.
+   cdist   -- distances between two collections of observation vectors
+   squareform -- convert distance matrix to a condensed one and vice versa
+   directed_hausdorff -- directed Hausdorff distance between arrays
+
+Predicates for checking the validity of distance matrices, both
+condensed and redundant. Also contained in this module are functions
+for computing the number of observations in a distance matrix.
+
+.. autosummary::
+   :toctree: generated/
+
+   is_valid_dm -- checks for a valid distance matrix
+   is_valid_y  -- checks for a valid condensed distance matrix
+   num_obs_dm  -- # of observations in a distance matrix
+   num_obs_y   -- # of observations in a condensed distance matrix
+
+Distance functions between two numeric vectors ``u`` and ``v``. Computing
+distances over a large collection of vectors is inefficient for these
+functions. Use ``pdist`` for this purpose.
+
+.. autosummary::
+   :toctree: generated/
+
+   braycurtis       -- the Bray-Curtis distance.
+   canberra         -- the Canberra distance.
+   chebyshev        -- the Chebyshev distance.
+   cityblock        -- the Manhattan distance.
+   correlation      -- the Correlation distance.
+   cosine           -- the Cosine distance.
+   euclidean        -- the Euclidean distance.
+   jensenshannon    -- the Jensen-Shannon distance.
+   mahalanobis      -- the Mahalanobis distance.
+   minkowski        -- the Minkowski distance.
+   seuclidean       -- the normalized Euclidean distance.
+   sqeuclidean      -- the squared Euclidean distance.
+
+Distance functions between two boolean vectors (representing sets) ``u`` and
+``v``.  As in the case of numerical vectors, ``pdist`` is more efficient for
+computing the distances between all pairs.
+
+.. autosummary::
+   :toctree: generated/
+
+   dice             -- the Dice dissimilarity.
+   hamming          -- the Hamming distance.
+   jaccard          -- the Jaccard distance.
+   kulczynski1      -- the Kulczynski 1 distance.
+   rogerstanimoto   -- the Rogers-Tanimoto dissimilarity.
+   russellrao       -- the Russell-Rao dissimilarity.
+   sokalmichener    -- the Sokal-Michener dissimilarity.
+   sokalsneath      -- the Sokal-Sneath dissimilarity.
+   yule             -- the Yule dissimilarity.
+
+:func:`hamming` also operates over discrete numerical vectors.
+"""
+
+# Copyright (C) Damian Eads, 2007-2008. New BSD License.
+
+__all__ = [
+    'braycurtis',
+    'canberra',
+    'cdist',
+    'chebyshev',
+    'cityblock',
+    'correlation',
+    'cosine',
+    'dice',
+    'directed_hausdorff',
+    'euclidean',
+    'hamming',
+    'is_valid_dm',
+    'is_valid_y',
+    'jaccard',
+    'jensenshannon',
+    'kulczynski1',
+    'mahalanobis',
+    'minkowski',
+    'num_obs_dm',
+    'num_obs_y',
+    'pdist',
+    'rogerstanimoto',
+    'russellrao',
+    'seuclidean',
+    'sokalmichener',
+    'sokalsneath',
+    'sqeuclidean',
+    'squareform',
+    'yule'
+]
+
+
+import math
+import warnings
+import numpy as np
+import dataclasses
+
+from typing import Optional, Callable
+
+from functools import partial
+from scipy._lib._util import _asarray_validated
+
+from . import _distance_wrap
+from . import _hausdorff
+from ..linalg import norm
+from ..special import rel_entr
+
+from . import _distance_pybind
+
+
+def _copy_array_if_base_present(a):
+    """Copy the array if its base points to a parent array."""
+    if a.base is not None:
+        return a.copy()
+    return a
+
+
+def _correlation_cdist_wrap(XA, XB, dm, **kwargs):
+    XA = XA - XA.mean(axis=1, keepdims=True)
+    XB = XB - XB.mean(axis=1, keepdims=True)
+    _distance_wrap.cdist_cosine_double_wrap(XA, XB, dm, **kwargs)
+
+
+def _correlation_pdist_wrap(X, dm, **kwargs):
+    X2 = X - X.mean(axis=1, keepdims=True)
+    _distance_wrap.pdist_cosine_double_wrap(X2, dm, **kwargs)
+
+
+def _convert_to_type(X, out_type):
+    return np.ascontiguousarray(X, dtype=out_type)
+
+
+def _nbool_correspond_all(u, v, w=None):
+    if u.dtype == v.dtype == bool and w is None:
+        not_u = ~u
+        not_v = ~v
+        nff = (not_u & not_v).sum()
+        nft = (not_u & v).sum()
+        ntf = (u & not_v).sum()
+        ntt = (u & v).sum()
+    else:
+        dtype = np.result_type(int, u.dtype, v.dtype)
+        u = u.astype(dtype)
+        v = v.astype(dtype)
+        not_u = 1.0 - u
+        not_v = 1.0 - v
+        if w is not None:
+            not_u = w * not_u
+            u = w * u
+        nff = (not_u * not_v).sum()
+        nft = (not_u * v).sum()
+        ntf = (u * not_v).sum()
+        ntt = (u * v).sum()
+    return (nff, nft, ntf, ntt)
+
+
+def _nbool_correspond_ft_tf(u, v, w=None):
+    if u.dtype == v.dtype == bool and w is None:
+        not_u = ~u
+        not_v = ~v
+        nft = (not_u & v).sum()
+        ntf = (u & not_v).sum()
+    else:
+        dtype = np.result_type(int, u.dtype, v.dtype)
+        u = u.astype(dtype)
+        v = v.astype(dtype)
+        not_u = 1.0 - u
+        not_v = 1.0 - v
+        if w is not None:
+            not_u = w * not_u
+            u = w * u
+        nft = (not_u * v).sum()
+        ntf = (u * not_v).sum()
+    return (nft, ntf)
+
+
+def _validate_cdist_input(XA, XB, mA, mB, n, metric_info, **kwargs):
+    # get supported types
+    types = metric_info.types
+    # choose best type
+    typ = types[types.index(XA.dtype)] if XA.dtype in types else types[0]
+    # validate data
+    XA = _convert_to_type(XA, out_type=typ)
+    XB = _convert_to_type(XB, out_type=typ)
+
+    # validate kwargs
+    _validate_kwargs = metric_info.validator
+    if _validate_kwargs:
+        kwargs = _validate_kwargs((XA, XB), mA + mB, n, **kwargs)
+    return XA, XB, typ, kwargs
+
+
+def _validate_weight_with_size(X, m, n, **kwargs):
+    w = kwargs.pop('w', None)
+    if w is None:
+        return kwargs
+
+    if w.ndim != 1 or w.shape[0] != n:
+        raise ValueError("Weights must have same size as input vector. "
+                         f"{w.shape[0]} vs. {n}")
+
+    kwargs['w'] = _validate_weights(w)
+    return kwargs
+
+
+def _validate_hamming_kwargs(X, m, n, **kwargs):
+    w = kwargs.get('w', np.ones((n,), dtype='double'))
+
+    if w.ndim != 1 or w.shape[0] != n:
+        raise ValueError(
+            "Weights must have same size as input vector. %d vs. %d" % (w.shape[0], n)
+        )
+
+    kwargs['w'] = _validate_weights(w)
+    return kwargs
+
+
+def _validate_mahalanobis_kwargs(X, m, n, **kwargs):
+    VI = kwargs.pop('VI', None)
+    if VI is None:
+        if m <= n:
+            # There are fewer observations than the dimension of
+            # the observations.
+            raise ValueError("The number of observations (%d) is too "
+                             "small; the covariance matrix is "
+                             "singular. For observations with %d "
+                             "dimensions, at least %d observations "
+                             "are required." % (m, n, n + 1))
+        if isinstance(X, tuple):
+            X = np.vstack(X)
+        CV = np.atleast_2d(np.cov(X.astype(np.float64, copy=False).T))
+        VI = np.linalg.inv(CV).T.copy()
+    kwargs["VI"] = _convert_to_double(VI)
+    return kwargs
+
+
+def _validate_minkowski_kwargs(X, m, n, **kwargs):
+    kwargs = _validate_weight_with_size(X, m, n, **kwargs)
+    if 'p' not in kwargs:
+        kwargs['p'] = 2.
+    else:
+        if kwargs['p'] <= 0:
+            raise ValueError("p must be greater than 0")
+
+    return kwargs
+
+
+def _validate_pdist_input(X, m, n, metric_info, **kwargs):
+    # get supported types
+    types = metric_info.types
+    # choose best type
+    typ = types[types.index(X.dtype)] if X.dtype in types else types[0]
+    # validate data
+    X = _convert_to_type(X, out_type=typ)
+
+    # validate kwargs
+    _validate_kwargs = metric_info.validator
+    if _validate_kwargs:
+        kwargs = _validate_kwargs(X, m, n, **kwargs)
+    return X, typ, kwargs
+
+
+def _validate_seuclidean_kwargs(X, m, n, **kwargs):
+    V = kwargs.pop('V', None)
+    if V is None:
+        if isinstance(X, tuple):
+            X = np.vstack(X)
+        V = np.var(X.astype(np.float64, copy=False), axis=0, ddof=1)
+    else:
+        V = np.asarray(V, order='c')
+        if len(V.shape) != 1:
+            raise ValueError('Variance vector V must '
+                             'be one-dimensional.')
+        if V.shape[0] != n:
+            raise ValueError('Variance vector V must be of the same '
+                             'dimension as the vectors on which the distances '
+                             'are computed.')
+    kwargs['V'] = _convert_to_double(V)
+    return kwargs
+
+
+def _validate_vector(u, dtype=None):
+    # XXX Is order='c' really necessary?
+    u = np.asarray(u, dtype=dtype, order='c')
+    if u.ndim == 1:
+        return u
+    raise ValueError("Input vector should be 1-D.")
+
+
+def _validate_weights(w, dtype=np.float64):
+    w = _validate_vector(w, dtype=dtype)
+    if np.any(w < 0):
+        raise ValueError("Input weights should be all non-negative")
+    return w
+
+
+def directed_hausdorff(u, v, seed=0):
+    """
+    Compute the directed Hausdorff distance between two 2-D arrays.
+
+    Distances between pairs are calculated using a Euclidean metric.
+
+    Parameters
+    ----------
+    u : (M,N) array_like
+        Input array with M points in N dimensions.
+    v : (O,N) array_like
+        Input array with O points in N dimensions.
+    seed : int or None, optional
+        Local `numpy.random.RandomState` seed. Default is 0, a random
+        shuffling of u and v that guarantees reproducibility.
+
+    Returns
+    -------
+    d : double
+        The directed Hausdorff distance between arrays `u` and `v`,
+
+    index_1 : int
+        index of point contributing to Hausdorff pair in `u`
+
+    index_2 : int
+        index of point contributing to Hausdorff pair in `v`
+
+    Raises
+    ------
+    ValueError
+        An exception is thrown if `u` and `v` do not have
+        the same number of columns.
+
+    See Also
+    --------
+    scipy.spatial.procrustes : Another similarity test for two data sets
+
+    Notes
+    -----
+    Uses the early break technique and the random sampling approach
+    described by [1]_. Although worst-case performance is ``O(m * o)``
+    (as with the brute force algorithm), this is unlikely in practice
+    as the input data would have to require the algorithm to explore
+    every single point interaction, and after the algorithm shuffles
+    the input points at that. The best case performance is O(m), which
+    is satisfied by selecting an inner loop distance that is less than
+    cmax and leads to an early break as often as possible. The authors
+    have formally shown that the average runtime is closer to O(m).
+
+    .. versionadded:: 0.19.0
+
+    References
+    ----------
+    .. [1] A. A. Taha and A. Hanbury, "An efficient algorithm for
+           calculating the exact Hausdorff distance." IEEE Transactions On
+           Pattern Analysis And Machine Intelligence, vol. 37 pp. 2153-63,
+           2015.
+
+    Examples
+    --------
+    Find the directed Hausdorff distance between two 2-D arrays of
+    coordinates:
+
+    >>> from scipy.spatial.distance import directed_hausdorff
+    >>> import numpy as np
+    >>> u = np.array([(1.0, 0.0),
+    ...               (0.0, 1.0),
+    ...               (-1.0, 0.0),
+    ...               (0.0, -1.0)])
+    >>> v = np.array([(2.0, 0.0),
+    ...               (0.0, 2.0),
+    ...               (-2.0, 0.0),
+    ...               (0.0, -4.0)])
+
+    >>> directed_hausdorff(u, v)[0]
+    2.23606797749979
+    >>> directed_hausdorff(v, u)[0]
+    3.0
+
+    Find the general (symmetric) Hausdorff distance between two 2-D
+    arrays of coordinates:
+
+    >>> max(directed_hausdorff(u, v)[0], directed_hausdorff(v, u)[0])
+    3.0
+
+    Find the indices of the points that generate the Hausdorff distance
+    (the Hausdorff pair):
+
+    >>> directed_hausdorff(v, u)[1:]
+    (3, 3)
+
+    """
+    u = np.asarray(u, dtype=np.float64, order='c')
+    v = np.asarray(v, dtype=np.float64, order='c')
+    if u.shape[1] != v.shape[1]:
+        raise ValueError('u and v need to have the same '
+                         'number of columns')
+    result = _hausdorff.directed_hausdorff(u, v, seed)
+    return result
+
+
+def minkowski(u, v, p=2, w=None):
+    """
+    Compute the Minkowski distance between two 1-D arrays.
+
+    The Minkowski distance between 1-D arrays `u` and `v`,
+    is defined as
+
+    .. math::
+
+       {\\|u-v\\|}_p = (\\sum{|u_i - v_i|^p})^{1/p}.
+
+
+       \\left(\\sum{w_i(|(u_i - v_i)|^p)}\\right)^{1/p}.
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input array.
+    v : (N,) array_like
+        Input array.
+    p : scalar
+        The order of the norm of the difference :math:`{\\|u-v\\|}_p`. Note
+        that for :math:`0 < p < 1`, the triangle inequality only holds with
+        an additional multiplicative factor, i.e. it is only a quasi-metric.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    minkowski : double
+        The Minkowski distance between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.minkowski([1, 0, 0], [0, 1, 0], 1)
+    2.0
+    >>> distance.minkowski([1, 0, 0], [0, 1, 0], 2)
+    1.4142135623730951
+    >>> distance.minkowski([1, 0, 0], [0, 1, 0], 3)
+    1.2599210498948732
+    >>> distance.minkowski([1, 1, 0], [0, 1, 0], 1)
+    1.0
+    >>> distance.minkowski([1, 1, 0], [0, 1, 0], 2)
+    1.0
+    >>> distance.minkowski([1, 1, 0], [0, 1, 0], 3)
+    1.0
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    if p <= 0:
+        raise ValueError("p must be greater than 0")
+    u_v = u - v
+    if w is not None:
+        w = _validate_weights(w)
+        if p == 1:
+            root_w = w
+        elif p == 2:
+            # better precision and speed
+            root_w = np.sqrt(w)
+        elif p == np.inf:
+            root_w = (w != 0)
+        else:
+            root_w = np.power(w, 1/p)
+        u_v = root_w * u_v
+    dist = norm(u_v, ord=p)
+    return dist
+
+
+def euclidean(u, v, w=None):
+    """
+    Computes the Euclidean distance between two 1-D arrays.
+
+    The Euclidean distance between 1-D arrays `u` and `v`, is defined as
+
+    .. math::
+
+       {\\|u-v\\|}_2
+
+       \\left(\\sum{(w_i |(u_i - v_i)|^2)}\\right)^{1/2}
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input array.
+    v : (N,) array_like
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    euclidean : double
+        The Euclidean distance between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.euclidean([1, 0, 0], [0, 1, 0])
+    1.4142135623730951
+    >>> distance.euclidean([1, 1, 0], [0, 1, 0])
+    1.0
+
+    """
+    return minkowski(u, v, p=2, w=w)
+
+
+def sqeuclidean(u, v, w=None):
+    """
+    Compute the squared Euclidean distance between two 1-D arrays.
+
+    The squared Euclidean distance between `u` and `v` is defined as
+
+    .. math::
+
+       \\sum_i{w_i |u_i - v_i|^2}
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input array.
+    v : (N,) array_like
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    sqeuclidean : double
+        The squared Euclidean distance between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.sqeuclidean([1, 0, 0], [0, 1, 0])
+    2.0
+    >>> distance.sqeuclidean([1, 1, 0], [0, 1, 0])
+    1.0
+
+    """
+    # Preserve float dtypes, but convert everything else to np.float64
+    # for stability.
+    utype, vtype = None, None
+    if not (hasattr(u, "dtype") and np.issubdtype(u.dtype, np.inexact)):
+        utype = np.float64
+    if not (hasattr(v, "dtype") and np.issubdtype(v.dtype, np.inexact)):
+        vtype = np.float64
+
+    u = _validate_vector(u, dtype=utype)
+    v = _validate_vector(v, dtype=vtype)
+    u_v = u - v
+    u_v_w = u_v  # only want weights applied once
+    if w is not None:
+        w = _validate_weights(w)
+        u_v_w = w * u_v
+    return np.dot(u_v, u_v_w)
+
+
+def correlation(u, v, w=None, centered=True):
+    """
+    Compute the correlation distance between two 1-D arrays.
+
+    The correlation distance between `u` and `v`, is
+    defined as
+
+    .. math::
+
+        1 - \\frac{(u - \\bar{u}) \\cdot (v - \\bar{v})}
+                  {{\\|(u - \\bar{u})\\|}_2 {\\|(v - \\bar{v})\\|}_2}
+
+    where :math:`\\bar{u}` is the mean of the elements of `u`
+    and :math:`x \\cdot y` is the dot product of :math:`x` and :math:`y`.
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input array.
+    v : (N,) array_like
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+    centered : bool, optional
+        If True, `u` and `v` will be centered. Default is True.
+
+    Returns
+    -------
+    correlation : double
+        The correlation distance between 1-D array `u` and `v`.
+
+    Examples
+    --------
+    Find the correlation between two arrays.
+
+    >>> from scipy.spatial.distance import correlation
+    >>> correlation([1, 0, 1], [1, 1, 0])
+    1.5
+
+    Using a weighting array, the correlation can be calculated as:
+
+    >>> correlation([1, 0, 1], [1, 1, 0], w=[0.9, 0.1, 0.1])
+    1.1
+
+    If centering is not needed, the correlation can be calculated as:
+
+    >>> correlation([1, 0, 1], [1, 1, 0], centered=False)
+    0.5
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    if w is not None:
+        w = _validate_weights(w)
+        w = w / w.sum()
+    if centered:
+        if w is not None:
+            umu = np.dot(u, w)
+            vmu = np.dot(v, w)
+        else:
+            umu = np.mean(u)
+            vmu = np.mean(v)
+        u = u - umu
+        v = v - vmu
+    if w is not None:
+        vw = v * w
+        uw = u * w
+    else:
+        vw, uw = v, u
+    uv = np.dot(u, vw)
+    uu = np.dot(u, uw)
+    vv = np.dot(v, vw)
+    dist = 1.0 - uv / math.sqrt(uu * vv)
+    # Clip the result to avoid rounding error
+    return np.clip(dist, 0.0, 2.0)
+
+
+def cosine(u, v, w=None):
+    """
+    Compute the Cosine distance between 1-D arrays.
+
+    The Cosine distance between `u` and `v`, is defined as
+
+    .. math::
+
+        1 - \\frac{u \\cdot v}
+                  {\\|u\\|_2 \\|v\\|_2}.
+
+    where :math:`u \\cdot v` is the dot product of :math:`u` and
+    :math:`v`.
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input array.
+    v : (N,) array_like
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    cosine : double
+        The Cosine distance between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.cosine([1, 0, 0], [0, 1, 0])
+    1.0
+    >>> distance.cosine([100, 0, 0], [0, 1, 0])
+    1.0
+    >>> distance.cosine([1, 1, 0], [0, 1, 0])
+    0.29289321881345254
+
+    """
+    # cosine distance is also referred to as 'uncentered correlation',
+    #   or 'reflective correlation'
+    return correlation(u, v, w=w, centered=False)
+
+
+def hamming(u, v, w=None):
+    """
+    Compute the Hamming distance between two 1-D arrays.
+
+    The Hamming distance between 1-D arrays `u` and `v`, is simply the
+    proportion of disagreeing components in `u` and `v`. If `u` and `v` are
+    boolean vectors, the Hamming distance is
+
+    .. math::
+
+       \\frac{c_{01} + c_{10}}{n}
+
+    where :math:`c_{ij}` is the number of occurrences of
+    :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for
+    :math:`k < n`.
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input array.
+    v : (N,) array_like
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    hamming : double
+        The Hamming distance between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.hamming([1, 0, 0], [0, 1, 0])
+    0.66666666666666663
+    >>> distance.hamming([1, 0, 0], [1, 1, 0])
+    0.33333333333333331
+    >>> distance.hamming([1, 0, 0], [2, 0, 0])
+    0.33333333333333331
+    >>> distance.hamming([1, 0, 0], [3, 0, 0])
+    0.33333333333333331
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    if u.shape != v.shape:
+        raise ValueError('The 1d arrays must have equal lengths.')
+    u_ne_v = u != v
+    if w is not None:
+        w = _validate_weights(w)
+        if w.shape != u.shape:
+            raise ValueError("'w' should have the same length as 'u' and 'v'.")
+        w = w / w.sum()
+        return np.dot(u_ne_v, w)
+    return np.mean(u_ne_v)
+
+
+def jaccard(u, v, w=None):
+    """
+    Compute the Jaccard-Needham dissimilarity between two boolean 1-D arrays.
+
+    The Jaccard-Needham dissimilarity between 1-D boolean arrays `u` and `v`,
+    is defined as
+
+    .. math::
+
+       \\frac{c_{TF} + c_{FT}}
+            {c_{TT} + c_{FT} + c_{TF}}
+
+    where :math:`c_{ij}` is the number of occurrences of
+    :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for
+    :math:`k < n`.
+
+    Parameters
+    ----------
+    u : (N,) array_like, bool
+        Input array.
+    v : (N,) array_like, bool
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    jaccard : double
+        The Jaccard distance between vectors `u` and `v`.
+
+    Notes
+    -----
+    When both `u` and `v` lead to a `0/0` division i.e. there is no overlap
+    between the items in the vectors the returned distance is 0. See the
+    Wikipedia page on the Jaccard index [1]_, and this paper [2]_.
+
+    .. versionchanged:: 1.2.0
+        Previously, when `u` and `v` lead to a `0/0` division, the function
+        would return NaN. This was changed to return 0 instead.
+
+    References
+    ----------
+    .. [1] https://en.wikipedia.org/wiki/Jaccard_index
+    .. [2] S. Kosub, "A note on the triangle inequality for the Jaccard
+       distance", 2016, :arxiv:`1612.02696`
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.jaccard([1, 0, 0], [0, 1, 0])
+    1.0
+    >>> distance.jaccard([1, 0, 0], [1, 1, 0])
+    0.5
+    >>> distance.jaccard([1, 0, 0], [1, 2, 0])
+    0.5
+    >>> distance.jaccard([1, 0, 0], [1, 1, 1])
+    0.66666666666666663
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+
+    nonzero = np.bitwise_or(u != 0, v != 0)
+    unequal_nonzero = np.bitwise_and((u != v), nonzero)
+    if w is not None:
+        w = _validate_weights(w)
+        nonzero = w * nonzero
+        unequal_nonzero = w * unequal_nonzero
+    a = np.float64(unequal_nonzero.sum())
+    b = np.float64(nonzero.sum())
+    return (a / b) if b != 0 else 0
+
+
+def kulczynski1(u, v, *, w=None):
+    """
+    Compute the Kulczynski 1 dissimilarity between two boolean 1-D arrays.
+
+    The Kulczynski 1 dissimilarity between two boolean 1-D arrays `u` and `v`
+    of length ``n``, is defined as
+
+    .. math::
+
+         \\frac{c_{11}}
+              {c_{01} + c_{10}}
+
+    where :math:`c_{ij}` is the number of occurrences of
+    :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for
+    :math:`k \\in {0, 1, ..., n-1}`.
+
+    Parameters
+    ----------
+    u : (N,) array_like, bool
+        Input array.
+    v : (N,) array_like, bool
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    kulczynski1 : float
+        The Kulczynski 1 distance between vectors `u` and `v`.
+
+    Notes
+    -----
+    This measure has a minimum value of 0 and no upper limit.
+    It is un-defined when there are no non-matches.
+
+    .. versionadded:: 1.8.0
+
+    References
+    ----------
+    .. [1] Kulczynski S. et al. Bulletin
+           International de l'Academie Polonaise des Sciences
+           et des Lettres, Classe des Sciences Mathematiques
+           et Naturelles, Serie B (Sciences Naturelles). 1927;
+           Supplement II: 57-203.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.kulczynski1([1, 0, 0], [0, 1, 0])
+    0.0
+    >>> distance.kulczynski1([True, False, False], [True, True, False])
+    1.0
+    >>> distance.kulczynski1([True, False, False], [True])
+    0.5
+    >>> distance.kulczynski1([1, 0, 0], [3, 1, 0])
+    -3.0
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    if w is not None:
+        w = _validate_weights(w)
+    (_, nft, ntf, ntt) = _nbool_correspond_all(u, v, w=w)
+
+    return ntt / (ntf + nft)
+
+
+def seuclidean(u, v, V):
+    """
+    Return the standardized Euclidean distance between two 1-D arrays.
+
+    The standardized Euclidean distance between two n-vectors `u` and `v` is
+
+    .. math::
+
+       \\sqrt{\\sum\\limits_i \\frac{1}{V_i} \\left(u_i-v_i \\right)^2}
+
+    ``V`` is the variance vector; ``V[I]`` is the variance computed over all the i-th
+    components of the points. If not passed, it is automatically computed.
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input array.
+    v : (N,) array_like
+        Input array.
+    V : (N,) array_like
+        `V` is an 1-D array of component variances. It is usually computed
+        among a larger collection vectors.
+
+    Returns
+    -------
+    seuclidean : double
+        The standardized Euclidean distance between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.seuclidean([1, 0, 0], [0, 1, 0], [0.1, 0.1, 0.1])
+    4.4721359549995796
+    >>> distance.seuclidean([1, 0, 0], [0, 1, 0], [1, 0.1, 0.1])
+    3.3166247903553998
+    >>> distance.seuclidean([1, 0, 0], [0, 1, 0], [10, 0.1, 0.1])
+    3.1780497164141406
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    V = _validate_vector(V, dtype=np.float64)
+    if V.shape[0] != u.shape[0] or u.shape[0] != v.shape[0]:
+        raise TypeError('V must be a 1-D array of the same dimension '
+                        'as u and v.')
+    return euclidean(u, v, w=1/V)
+
+
+def cityblock(u, v, w=None):
+    """
+    Compute the City Block (Manhattan) distance.
+
+    Computes the Manhattan distance between two 1-D arrays `u` and `v`,
+    which is defined as
+
+    .. math::
+
+       \\sum_i {\\left| u_i - v_i \\right|}.
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input array.
+    v : (N,) array_like
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    cityblock : double
+        The City Block (Manhattan) distance between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.cityblock([1, 0, 0], [0, 1, 0])
+    2
+    >>> distance.cityblock([1, 0, 0], [0, 2, 0])
+    3
+    >>> distance.cityblock([1, 0, 0], [1, 1, 0])
+    1
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    l1_diff = abs(u - v)
+    if w is not None:
+        w = _validate_weights(w)
+        l1_diff = w * l1_diff
+    return l1_diff.sum()
+
+
+def mahalanobis(u, v, VI):
+    """
+    Compute the Mahalanobis distance between two 1-D arrays.
+
+    The Mahalanobis distance between 1-D arrays `u` and `v`, is defined as
+
+    .. math::
+
+       \\sqrt{ (u-v) V^{-1} (u-v)^T }
+
+    where ``V`` is the covariance matrix.  Note that the argument `VI`
+    is the inverse of ``V``.
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input array.
+    v : (N,) array_like
+        Input array.
+    VI : array_like
+        The inverse of the covariance matrix.
+
+    Returns
+    -------
+    mahalanobis : double
+        The Mahalanobis distance between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> iv = [[1, 0.5, 0.5], [0.5, 1, 0.5], [0.5, 0.5, 1]]
+    >>> distance.mahalanobis([1, 0, 0], [0, 1, 0], iv)
+    1.0
+    >>> distance.mahalanobis([0, 2, 0], [0, 1, 0], iv)
+    1.0
+    >>> distance.mahalanobis([2, 0, 0], [0, 1, 0], iv)
+    1.7320508075688772
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    VI = np.atleast_2d(VI)
+    delta = u - v
+    m = np.dot(np.dot(delta, VI), delta)
+    return np.sqrt(m)
+
+
+def chebyshev(u, v, w=None):
+    """
+    Compute the Chebyshev distance.
+
+    Computes the Chebyshev distance between two 1-D arrays `u` and `v`,
+    which is defined as
+
+    .. math::
+
+       \\max_i {|u_i-v_i|}.
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input vector.
+    v : (N,) array_like
+        Input vector.
+    w : (N,) array_like, optional
+        Unused, as 'max' is a weightless operation. Here for API consistency.
+
+    Returns
+    -------
+    chebyshev : double
+        The Chebyshev distance between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.chebyshev([1, 0, 0], [0, 1, 0])
+    1
+    >>> distance.chebyshev([1, 1, 0], [0, 1, 0])
+    1
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    if w is not None:
+        w = _validate_weights(w)
+        has_weight = w > 0
+        if has_weight.sum() < w.size:
+            u = u[has_weight]
+            v = v[has_weight]
+    return max(abs(u - v))
+
+
+def braycurtis(u, v, w=None):
+    """
+    Compute the Bray-Curtis distance between two 1-D arrays.
+
+    Bray-Curtis distance is defined as
+
+    .. math::
+
+       \\sum{|u_i-v_i|} / \\sum{|u_i+v_i|}
+
+    The Bray-Curtis distance is in the range [0, 1] if all coordinates are
+    positive, and is undefined if the inputs are of length zero.
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input array.
+    v : (N,) array_like
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    braycurtis : double
+        The Bray-Curtis distance between 1-D arrays `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.braycurtis([1, 0, 0], [0, 1, 0])
+    1.0
+    >>> distance.braycurtis([1, 1, 0], [0, 1, 0])
+    0.33333333333333331
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v, dtype=np.float64)
+    l1_diff = abs(u - v)
+    l1_sum = abs(u + v)
+    if w is not None:
+        w = _validate_weights(w)
+        l1_diff = w * l1_diff
+        l1_sum = w * l1_sum
+    return l1_diff.sum() / l1_sum.sum()
+
+
+def canberra(u, v, w=None):
+    """
+    Compute the Canberra distance between two 1-D arrays.
+
+    The Canberra distance is defined as
+
+    .. math::
+
+         d(u,v) = \\sum_i \\frac{|u_i-v_i|}
+                              {|u_i|+|v_i|}.
+
+    Parameters
+    ----------
+    u : (N,) array_like
+        Input array.
+    v : (N,) array_like
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    canberra : double
+        The Canberra distance between vectors `u` and `v`.
+
+    Notes
+    -----
+    When `u[i]` and `v[i]` are 0 for given i, then the fraction 0/0 = 0 is
+    used in the calculation.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.canberra([1, 0, 0], [0, 1, 0])
+    2.0
+    >>> distance.canberra([1, 1, 0], [0, 1, 0])
+    1.0
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v, dtype=np.float64)
+    if w is not None:
+        w = _validate_weights(w)
+    with np.errstate(invalid='ignore'):
+        abs_uv = abs(u - v)
+        abs_u = abs(u)
+        abs_v = abs(v)
+        d = abs_uv / (abs_u + abs_v)
+        if w is not None:
+            d = w * d
+        d = np.nansum(d)
+    return d
+
+
+def jensenshannon(p, q, base=None, *, axis=0, keepdims=False):
+    """
+    Compute the Jensen-Shannon distance (metric) between
+    two probability arrays. This is the square root
+    of the Jensen-Shannon divergence.
+
+    The Jensen-Shannon distance between two probability
+    vectors `p` and `q` is defined as,
+
+    .. math::
+
+       \\sqrt{\\frac{D(p \\parallel m) + D(q \\parallel m)}{2}}
+
+    where :math:`m` is the pointwise mean of :math:`p` and :math:`q`
+    and :math:`D` is the Kullback-Leibler divergence.
+
+    This routine will normalize `p` and `q` if they don't sum to 1.0.
+
+    Parameters
+    ----------
+    p : (N,) array_like
+        left probability vector
+    q : (N,) array_like
+        right probability vector
+    base : double, optional
+        the base of the logarithm used to compute the output
+        if not given, then the routine uses the default base of
+        scipy.stats.entropy.
+    axis : int, optional
+        Axis along which the Jensen-Shannon distances are computed. The default
+        is 0.
+
+        .. versionadded:: 1.7.0
+    keepdims : bool, optional
+        If this is set to `True`, the reduced axes are left in the
+        result as dimensions with size one. With this option,
+        the result will broadcast correctly against the input array.
+        Default is False.
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    js : double or ndarray
+        The Jensen-Shannon distances between `p` and `q` along the `axis`.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.2.0
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> import numpy as np
+    >>> distance.jensenshannon([1.0, 0.0, 0.0], [0.0, 1.0, 0.0], 2.0)
+    1.0
+    >>> distance.jensenshannon([1.0, 0.0], [0.5, 0.5])
+    0.46450140402245893
+    >>> distance.jensenshannon([1.0, 0.0, 0.0], [1.0, 0.0, 0.0])
+    0.0
+    >>> a = np.array([[1, 2, 3, 4],
+    ...               [5, 6, 7, 8],
+    ...               [9, 10, 11, 12]])
+    >>> b = np.array([[13, 14, 15, 16],
+    ...               [17, 18, 19, 20],
+    ...               [21, 22, 23, 24]])
+    >>> distance.jensenshannon(a, b, axis=0)
+    array([0.1954288, 0.1447697, 0.1138377, 0.0927636])
+    >>> distance.jensenshannon(a, b, axis=1)
+    array([0.1402339, 0.0399106, 0.0201815])
+
+    """
+    p = np.asarray(p)
+    q = np.asarray(q)
+    p = p / np.sum(p, axis=axis, keepdims=True)
+    q = q / np.sum(q, axis=axis, keepdims=True)
+    m = (p + q) / 2.0
+    left = rel_entr(p, m)
+    right = rel_entr(q, m)
+    left_sum = np.sum(left, axis=axis, keepdims=keepdims)
+    right_sum = np.sum(right, axis=axis, keepdims=keepdims)
+    js = left_sum + right_sum
+    if base is not None:
+        js /= np.log(base)
+    return np.sqrt(js / 2.0)
+
+
+def yule(u, v, w=None):
+    """
+    Compute the Yule dissimilarity between two boolean 1-D arrays.
+
+    The Yule dissimilarity is defined as
+
+    .. math::
+
+         \\frac{R}{c_{TT} * c_{FF} + \\frac{R}{2}}
+
+    where :math:`c_{ij}` is the number of occurrences of
+    :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for
+    :math:`k < n` and :math:`R = 2.0 * c_{TF} * c_{FT}`.
+
+    Parameters
+    ----------
+    u : (N,) array_like, bool
+        Input array.
+    v : (N,) array_like, bool
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    yule : double
+        The Yule dissimilarity between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.yule([1, 0, 0], [0, 1, 0])
+    2.0
+    >>> distance.yule([1, 1, 0], [0, 1, 0])
+    0.0
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    if w is not None:
+        w = _validate_weights(w)
+    (nff, nft, ntf, ntt) = _nbool_correspond_all(u, v, w=w)
+    half_R = ntf * nft
+    if half_R == 0:
+        return 0.0
+    else:
+        return float(2.0 * half_R / (ntt * nff + half_R))
+
+
+def dice(u, v, w=None):
+    """
+    Compute the Dice dissimilarity between two boolean 1-D arrays.
+
+    The Dice dissimilarity between `u` and `v`, is
+
+    .. math::
+
+         \\frac{c_{TF} + c_{FT}}
+              {2c_{TT} + c_{FT} + c_{TF}}
+
+    where :math:`c_{ij}` is the number of occurrences of
+    :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for
+    :math:`k < n`.
+
+    Parameters
+    ----------
+    u : (N,) array_like, bool
+        Input 1-D array.
+    v : (N,) array_like, bool
+        Input 1-D array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    dice : double
+        The Dice dissimilarity between 1-D arrays `u` and `v`.
+
+    Notes
+    -----
+    This function computes the Dice dissimilarity index. To compute the
+    Dice similarity index, convert one to the other with similarity =
+    1 - dissimilarity.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.dice([1, 0, 0], [0, 1, 0])
+    1.0
+    >>> distance.dice([1, 0, 0], [1, 1, 0])
+    0.3333333333333333
+    >>> distance.dice([1, 0, 0], [2, 0, 0])
+    -0.3333333333333333
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    if w is not None:
+        w = _validate_weights(w)
+    if u.dtype == v.dtype == bool and w is None:
+        ntt = (u & v).sum()
+    else:
+        dtype = np.result_type(int, u.dtype, v.dtype)
+        u = u.astype(dtype)
+        v = v.astype(dtype)
+        if w is None:
+            ntt = (u * v).sum()
+        else:
+            ntt = (u * v * w).sum()
+    (nft, ntf) = _nbool_correspond_ft_tf(u, v, w=w)
+    return float((ntf + nft) / np.array(2.0 * ntt + ntf + nft))
+
+
+def rogerstanimoto(u, v, w=None):
+    """
+    Compute the Rogers-Tanimoto dissimilarity between two boolean 1-D arrays.
+
+    The Rogers-Tanimoto dissimilarity between two boolean 1-D arrays
+    `u` and `v`, is defined as
+
+    .. math::
+       \\frac{R}
+            {c_{TT} + c_{FF} + R}
+
+    where :math:`c_{ij}` is the number of occurrences of
+    :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for
+    :math:`k < n` and :math:`R = 2(c_{TF} + c_{FT})`.
+
+    Parameters
+    ----------
+    u : (N,) array_like, bool
+        Input array.
+    v : (N,) array_like, bool
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    rogerstanimoto : double
+        The Rogers-Tanimoto dissimilarity between vectors
+        `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.rogerstanimoto([1, 0, 0], [0, 1, 0])
+    0.8
+    >>> distance.rogerstanimoto([1, 0, 0], [1, 1, 0])
+    0.5
+    >>> distance.rogerstanimoto([1, 0, 0], [2, 0, 0])
+    -1.0
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    if w is not None:
+        w = _validate_weights(w)
+    (nff, nft, ntf, ntt) = _nbool_correspond_all(u, v, w=w)
+    return float(2.0 * (ntf + nft)) / float(ntt + nff + (2.0 * (ntf + nft)))
+
+
+def russellrao(u, v, w=None):
+    """
+    Compute the Russell-Rao dissimilarity between two boolean 1-D arrays.
+
+    The Russell-Rao dissimilarity between two boolean 1-D arrays, `u` and
+    `v`, is defined as
+
+    .. math::
+
+      \\frac{n - c_{TT}}
+           {n}
+
+    where :math:`c_{ij}` is the number of occurrences of
+    :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for
+    :math:`k < n`.
+
+    Parameters
+    ----------
+    u : (N,) array_like, bool
+        Input array.
+    v : (N,) array_like, bool
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    russellrao : double
+        The Russell-Rao dissimilarity between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.russellrao([1, 0, 0], [0, 1, 0])
+    1.0
+    >>> distance.russellrao([1, 0, 0], [1, 1, 0])
+    0.6666666666666666
+    >>> distance.russellrao([1, 0, 0], [2, 0, 0])
+    0.3333333333333333
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    if u.dtype == v.dtype == bool and w is None:
+        ntt = (u & v).sum()
+        n = float(len(u))
+    elif w is None:
+        ntt = (u * v).sum()
+        n = float(len(u))
+    else:
+        w = _validate_weights(w)
+        ntt = (u * v * w).sum()
+        n = w.sum()
+    return float(n - ntt) / n
+
+
+def sokalmichener(u, v, w=None):
+    """
+    Compute the Sokal-Michener dissimilarity between two boolean 1-D arrays.
+
+    The Sokal-Michener dissimilarity between boolean 1-D arrays `u` and `v`,
+    is defined as
+
+    .. math::
+
+       \\frac{R}
+            {S + R}
+
+    where :math:`c_{ij}` is the number of occurrences of
+    :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for
+    :math:`k < n`, :math:`R = 2 * (c_{TF} + c_{FT})` and
+    :math:`S = c_{FF} + c_{TT}`.
+
+    Parameters
+    ----------
+    u : (N,) array_like, bool
+        Input array.
+    v : (N,) array_like, bool
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    sokalmichener : double
+        The Sokal-Michener dissimilarity between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.sokalmichener([1, 0, 0], [0, 1, 0])
+    0.8
+    >>> distance.sokalmichener([1, 0, 0], [1, 1, 0])
+    0.5
+    >>> distance.sokalmichener([1, 0, 0], [2, 0, 0])
+    -1.0
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    if w is not None:
+        w = _validate_weights(w)
+    nff, nft, ntf, ntt = _nbool_correspond_all(u, v, w=w)
+    return float(2.0 * (ntf + nft)) / float(ntt + nff + 2.0 * (ntf + nft))
+
+
+def sokalsneath(u, v, w=None):
+    """
+    Compute the Sokal-Sneath dissimilarity between two boolean 1-D arrays.
+
+    The Sokal-Sneath dissimilarity between `u` and `v`,
+
+    .. math::
+
+       \\frac{R}
+            {c_{TT} + R}
+
+    where :math:`c_{ij}` is the number of occurrences of
+    :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for
+    :math:`k < n` and :math:`R = 2(c_{TF} + c_{FT})`.
+
+    Parameters
+    ----------
+    u : (N,) array_like, bool
+        Input array.
+    v : (N,) array_like, bool
+        Input array.
+    w : (N,) array_like, optional
+        The weights for each value in `u` and `v`. Default is None,
+        which gives each value a weight of 1.0
+
+    Returns
+    -------
+    sokalsneath : double
+        The Sokal-Sneath dissimilarity between vectors `u` and `v`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance
+    >>> distance.sokalsneath([1, 0, 0], [0, 1, 0])
+    1.0
+    >>> distance.sokalsneath([1, 0, 0], [1, 1, 0])
+    0.66666666666666663
+    >>> distance.sokalsneath([1, 0, 0], [2, 1, 0])
+    0.0
+    >>> distance.sokalsneath([1, 0, 0], [3, 1, 0])
+    -2.0
+
+    """
+    u = _validate_vector(u)
+    v = _validate_vector(v)
+    if u.dtype == v.dtype == bool and w is None:
+        ntt = (u & v).sum()
+    elif w is None:
+        ntt = (u * v).sum()
+    else:
+        w = _validate_weights(w)
+        ntt = (u * v * w).sum()
+    (nft, ntf) = _nbool_correspond_ft_tf(u, v, w=w)
+    denom = np.array(ntt + 2.0 * (ntf + nft))
+    if not denom.any():
+        raise ValueError('Sokal-Sneath dissimilarity is not defined for '
+                         'vectors that are entirely false.')
+    return float(2.0 * (ntf + nft)) / denom
+
+
+_convert_to_double = partial(_convert_to_type, out_type=np.float64)
+_convert_to_bool = partial(_convert_to_type, out_type=bool)
+
+# adding python-only wrappers to _distance_wrap module
+_distance_wrap.pdist_correlation_double_wrap = _correlation_pdist_wrap
+_distance_wrap.cdist_correlation_double_wrap = _correlation_cdist_wrap
+
+
+@dataclasses.dataclass(frozen=True)
+class CDistMetricWrapper:
+    metric_name: str
+
+    def __call__(self, XA, XB, *, out=None, **kwargs):
+        XA = np.ascontiguousarray(XA)
+        XB = np.ascontiguousarray(XB)
+        mA, n = XA.shape
+        mB, _ = XB.shape
+        metric_name = self.metric_name
+        metric_info = _METRICS[metric_name]
+        XA, XB, typ, kwargs = _validate_cdist_input(
+            XA, XB, mA, mB, n, metric_info, **kwargs)
+
+        w = kwargs.pop('w', None)
+        if w is not None:
+            metric = metric_info.dist_func
+            return _cdist_callable(
+                XA, XB, metric=metric, out=out, w=w, **kwargs)
+
+        dm = _prepare_out_argument(out, np.float64, (mA, mB))
+        # get cdist wrapper
+        cdist_fn = getattr(_distance_wrap, f'cdist_{metric_name}_{typ}_wrap')
+        cdist_fn(XA, XB, dm, **kwargs)
+        return dm
+
+
+@dataclasses.dataclass(frozen=True)
+class PDistMetricWrapper:
+    metric_name: str
+
+    def __call__(self, X, *, out=None, **kwargs):
+        X = np.ascontiguousarray(X)
+        m, n = X.shape
+        metric_name = self.metric_name
+        metric_info = _METRICS[metric_name]
+        X, typ, kwargs = _validate_pdist_input(
+            X, m, n, metric_info, **kwargs)
+        out_size = (m * (m - 1)) // 2
+        w = kwargs.pop('w', None)
+        if w is not None:
+            metric = metric_info.dist_func
+            return _pdist_callable(
+                X, metric=metric, out=out, w=w, **kwargs)
+
+        dm = _prepare_out_argument(out, np.float64, (out_size,))
+        # get pdist wrapper
+        pdist_fn = getattr(_distance_wrap, f'pdist_{metric_name}_{typ}_wrap')
+        pdist_fn(X, dm, **kwargs)
+        return dm
+
+
+@dataclasses.dataclass(frozen=True)
+class MetricInfo:
+    # Name of python distance function
+    canonical_name: str
+    # All aliases, including canonical_name
+    aka: set[str]
+    # unvectorized distance function
+    dist_func: Callable
+    # Optimized cdist function
+    cdist_func: Callable
+    # Optimized pdist function
+    pdist_func: Callable
+    # function that checks kwargs and computes default values:
+    # f(X, m, n, **kwargs)
+    validator: Optional[Callable] = None
+    # list of supported types:
+    # X (pdist) and XA (cdist) are used to choose the type. if there is no
+    # match the first type is used. Default double
+    types: list[str] = dataclasses.field(default_factory=lambda: ['double'])
+    # true if out array must be C-contiguous
+    requires_contiguous_out: bool = True
+
+
+# Registry of implemented metrics:
+_METRIC_INFOS = [
+    MetricInfo(
+        canonical_name='braycurtis',
+        aka={'braycurtis'},
+        dist_func=braycurtis,
+        cdist_func=_distance_pybind.cdist_braycurtis,
+        pdist_func=_distance_pybind.pdist_braycurtis,
+    ),
+    MetricInfo(
+        canonical_name='canberra',
+        aka={'canberra'},
+        dist_func=canberra,
+        cdist_func=_distance_pybind.cdist_canberra,
+        pdist_func=_distance_pybind.pdist_canberra,
+    ),
+    MetricInfo(
+        canonical_name='chebyshev',
+        aka={'chebychev', 'chebyshev', 'cheby', 'cheb', 'ch'},
+        dist_func=chebyshev,
+        cdist_func=_distance_pybind.cdist_chebyshev,
+        pdist_func=_distance_pybind.pdist_chebyshev,
+    ),
+    MetricInfo(
+        canonical_name='cityblock',
+        aka={'cityblock', 'cblock', 'cb', 'c'},
+        dist_func=cityblock,
+        cdist_func=_distance_pybind.cdist_cityblock,
+        pdist_func=_distance_pybind.pdist_cityblock,
+    ),
+    MetricInfo(
+        canonical_name='correlation',
+        aka={'correlation', 'co'},
+        dist_func=correlation,
+        cdist_func=CDistMetricWrapper('correlation'),
+        pdist_func=PDistMetricWrapper('correlation'),
+    ),
+    MetricInfo(
+        canonical_name='cosine',
+        aka={'cosine', 'cos'},
+        dist_func=cosine,
+        cdist_func=CDistMetricWrapper('cosine'),
+        pdist_func=PDistMetricWrapper('cosine'),
+    ),
+    MetricInfo(
+        canonical_name='dice',
+        aka={'dice'},
+        types=['bool'],
+        dist_func=dice,
+        cdist_func=_distance_pybind.cdist_dice,
+        pdist_func=_distance_pybind.pdist_dice,
+    ),
+    MetricInfo(
+        canonical_name='euclidean',
+        aka={'euclidean', 'euclid', 'eu', 'e'},
+        dist_func=euclidean,
+        cdist_func=_distance_pybind.cdist_euclidean,
+        pdist_func=_distance_pybind.pdist_euclidean,
+    ),
+    MetricInfo(
+        canonical_name='hamming',
+        aka={'matching', 'hamming', 'hamm', 'ha', 'h'},
+        types=['double', 'bool'],
+        validator=_validate_hamming_kwargs,
+        dist_func=hamming,
+        cdist_func=_distance_pybind.cdist_hamming,
+        pdist_func=_distance_pybind.pdist_hamming,
+    ),
+    MetricInfo(
+        canonical_name='jaccard',
+        aka={'jaccard', 'jacc', 'ja', 'j'},
+        types=['double', 'bool'],
+        dist_func=jaccard,
+        cdist_func=_distance_pybind.cdist_jaccard,
+        pdist_func=_distance_pybind.pdist_jaccard,
+    ),
+    MetricInfo(
+        canonical_name='jensenshannon',
+        aka={'jensenshannon', 'js'},
+        dist_func=jensenshannon,
+        cdist_func=CDistMetricWrapper('jensenshannon'),
+        pdist_func=PDistMetricWrapper('jensenshannon'),
+    ),
+    MetricInfo(
+        canonical_name='kulczynski1',
+        aka={'kulczynski1'},
+        types=['bool'],
+        dist_func=kulczynski1,
+        cdist_func=_distance_pybind.cdist_kulczynski1,
+        pdist_func=_distance_pybind.pdist_kulczynski1,
+    ),
+    MetricInfo(
+        canonical_name='mahalanobis',
+        aka={'mahalanobis', 'mahal', 'mah'},
+        validator=_validate_mahalanobis_kwargs,
+        dist_func=mahalanobis,
+        cdist_func=CDistMetricWrapper('mahalanobis'),
+        pdist_func=PDistMetricWrapper('mahalanobis'),
+    ),
+    MetricInfo(
+        canonical_name='minkowski',
+        aka={'minkowski', 'mi', 'm', 'pnorm'},
+        validator=_validate_minkowski_kwargs,
+        dist_func=minkowski,
+        cdist_func=_distance_pybind.cdist_minkowski,
+        pdist_func=_distance_pybind.pdist_minkowski,
+    ),
+    MetricInfo(
+        canonical_name='rogerstanimoto',
+        aka={'rogerstanimoto'},
+        types=['bool'],
+        dist_func=rogerstanimoto,
+        cdist_func=_distance_pybind.cdist_rogerstanimoto,
+        pdist_func=_distance_pybind.pdist_rogerstanimoto,
+    ),
+    MetricInfo(
+        canonical_name='russellrao',
+        aka={'russellrao'},
+        types=['bool'],
+        dist_func=russellrao,
+        cdist_func=_distance_pybind.cdist_russellrao,
+        pdist_func=_distance_pybind.pdist_russellrao,
+    ),
+    MetricInfo(
+        canonical_name='seuclidean',
+        aka={'seuclidean', 'se', 's'},
+        validator=_validate_seuclidean_kwargs,
+        dist_func=seuclidean,
+        cdist_func=CDistMetricWrapper('seuclidean'),
+        pdist_func=PDistMetricWrapper('seuclidean'),
+    ),
+    MetricInfo(
+        canonical_name='sokalmichener',
+        aka={'sokalmichener'},
+        types=['bool'],
+        dist_func=sokalmichener,
+        cdist_func=_distance_pybind.cdist_sokalmichener,
+        pdist_func=_distance_pybind.pdist_sokalmichener,
+    ),
+    MetricInfo(
+        canonical_name='sokalsneath',
+        aka={'sokalsneath'},
+        types=['bool'],
+        dist_func=sokalsneath,
+        cdist_func=_distance_pybind.cdist_sokalsneath,
+        pdist_func=_distance_pybind.pdist_sokalsneath,
+    ),
+    MetricInfo(
+        canonical_name='sqeuclidean',
+        aka={'sqeuclidean', 'sqe', 'sqeuclid'},
+        dist_func=sqeuclidean,
+        cdist_func=_distance_pybind.cdist_sqeuclidean,
+        pdist_func=_distance_pybind.pdist_sqeuclidean,
+    ),
+    MetricInfo(
+        canonical_name='yule',
+        aka={'yule'},
+        types=['bool'],
+        dist_func=yule,
+        cdist_func=_distance_pybind.cdist_yule,
+        pdist_func=_distance_pybind.pdist_yule,
+    ),
+]
+
+_METRICS = {info.canonical_name: info for info in _METRIC_INFOS}
+_METRIC_ALIAS = {alias: info
+                     for info in _METRIC_INFOS
+                     for alias in info.aka}
+
+_METRICS_NAMES = list(_METRICS.keys())
+
+_TEST_METRICS = {'test_' + info.canonical_name: info for info in _METRIC_INFOS}
+
+
+def pdist(X, metric='euclidean', *, out=None, **kwargs):
+    """
+    Pairwise distances between observations in n-dimensional space.
+
+    See Notes for common calling conventions.
+
+    Parameters
+    ----------
+    X : array_like
+        An m by n array of m original observations in an
+        n-dimensional space.
+    metric : str or function, optional
+        The distance metric to use. The distance function can
+        be 'braycurtis', 'canberra', 'chebyshev', 'cityblock',
+        'correlation', 'cosine', 'dice', 'euclidean', 'hamming',
+        'jaccard', 'jensenshannon', 'kulczynski1',
+        'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto',
+        'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath',
+        'sqeuclidean', 'yule'.
+    out : ndarray, optional
+        The output array.
+        If not None, condensed distance matrix Y is stored in this array.
+    **kwargs : dict, optional
+        Extra arguments to `metric`: refer to each metric documentation for a
+        list of all possible arguments.
+
+        Some possible arguments:
+
+        p : scalar
+        The p-norm to apply for Minkowski, weighted and unweighted.
+        Default: 2.
+
+        w : ndarray
+        The weight vector for metrics that support weights (e.g., Minkowski).
+
+        V : ndarray
+        The variance vector for standardized Euclidean.
+        Default: var(X, axis=0, ddof=1)
+
+        VI : ndarray
+        The inverse of the covariance matrix for Mahalanobis.
+        Default: inv(cov(X.T)).T
+
+    Returns
+    -------
+    Y : ndarray
+        Returns a condensed distance matrix Y. For each :math:`i` and :math:`j`
+        (where :math:`i<j<m`),where m is the number of original observations.
+        The metric ``dist(u=X[i], v=X[j])`` is computed and stored in entry ``m
+        * i + j - ((i + 2) * (i + 1)) // 2``.
+
+    See Also
+    --------
+    squareform : converts between condensed distance matrices and
+                 square distance matrices.
+
+    Notes
+    -----
+    See ``squareform`` for information on how to calculate the index of
+    this entry or to convert the condensed distance matrix to a
+    redundant square matrix.
+
+    The following are common calling conventions.
+
+    1. ``Y = pdist(X, 'euclidean')``
+
+       Computes the distance between m points using Euclidean distance
+       (2-norm) as the distance metric between the points. The points
+       are arranged as m n-dimensional row vectors in the matrix X.
+
+    2. ``Y = pdist(X, 'minkowski', p=2.)``
+
+       Computes the distances using the Minkowski distance
+       :math:`\\|u-v\\|_p` (:math:`p`-norm) where :math:`p > 0` (note
+       that this is only a quasi-metric if :math:`0 < p < 1`).
+
+    3. ``Y = pdist(X, 'cityblock')``
+
+       Computes the city block or Manhattan distance between the
+       points.
+
+    4. ``Y = pdist(X, 'seuclidean', V=None)``
+
+       Computes the standardized Euclidean distance. The standardized
+       Euclidean distance between two n-vectors ``u`` and ``v`` is
+
+       .. math::
+
+          \\sqrt{\\sum {(u_i-v_i)^2 / V[x_i]}}
+
+
+       V is the variance vector; V[i] is the variance computed over all
+       the i'th components of the points.  If not passed, it is
+       automatically computed.
+
+    5. ``Y = pdist(X, 'sqeuclidean')``
+
+       Computes the squared Euclidean distance :math:`\\|u-v\\|_2^2` between
+       the vectors.
+
+    6. ``Y = pdist(X, 'cosine')``
+
+       Computes the cosine distance between vectors u and v,
+
+       .. math::
+
+          1 - \\frac{u \\cdot v}
+                   {{\\|u\\|}_2 {\\|v\\|}_2}
+
+       where :math:`\\|*\\|_2` is the 2-norm of its argument ``*``, and
+       :math:`u \\cdot v` is the dot product of ``u`` and ``v``.
+
+    7. ``Y = pdist(X, 'correlation')``
+
+       Computes the correlation distance between vectors u and v. This is
+
+       .. math::
+
+          1 - \\frac{(u - \\bar{u}) \\cdot (v - \\bar{v})}
+                   {{\\|(u - \\bar{u})\\|}_2 {\\|(v - \\bar{v})\\|}_2}
+
+       where :math:`\\bar{v}` is the mean of the elements of vector v,
+       and :math:`x \\cdot y` is the dot product of :math:`x` and :math:`y`.
+
+    8. ``Y = pdist(X, 'hamming')``
+
+       Computes the normalized Hamming distance, or the proportion of
+       those vector elements between two n-vectors ``u`` and ``v``
+       which disagree. To save memory, the matrix ``X`` can be of type
+       boolean.
+
+    9. ``Y = pdist(X, 'jaccard')``
+
+       Computes the Jaccard distance between the points. Given two
+       vectors, ``u`` and ``v``, the Jaccard distance is the
+       proportion of those elements ``u[i]`` and ``v[i]`` that
+       disagree.
+
+    10. ``Y = pdist(X, 'jensenshannon')``
+
+        Computes the Jensen-Shannon distance between two probability arrays.
+        Given two probability vectors, :math:`p` and :math:`q`, the
+        Jensen-Shannon distance is
+
+        .. math::
+
+           \\sqrt{\\frac{D(p \\parallel m) + D(q \\parallel m)}{2}}
+
+        where :math:`m` is the pointwise mean of :math:`p` and :math:`q`
+        and :math:`D` is the Kullback-Leibler divergence.
+
+    11. ``Y = pdist(X, 'chebyshev')``
+
+        Computes the Chebyshev distance between the points. The
+        Chebyshev distance between two n-vectors ``u`` and ``v`` is the
+        maximum norm-1 distance between their respective elements. More
+        precisely, the distance is given by
+
+        .. math::
+
+           d(u,v) = \\max_i {|u_i-v_i|}
+
+    12. ``Y = pdist(X, 'canberra')``
+
+        Computes the Canberra distance between the points. The
+        Canberra distance between two points ``u`` and ``v`` is
+
+        .. math::
+
+          d(u,v) = \\sum_i \\frac{|u_i-v_i|}
+                               {|u_i|+|v_i|}
+
+
+    13. ``Y = pdist(X, 'braycurtis')``
+
+        Computes the Bray-Curtis distance between the points. The
+        Bray-Curtis distance between two points ``u`` and ``v`` is
+
+
+        .. math::
+
+             d(u,v) = \\frac{\\sum_i {|u_i-v_i|}}
+                            {\\sum_i {|u_i+v_i|}}
+
+    14. ``Y = pdist(X, 'mahalanobis', VI=None)``
+
+        Computes the Mahalanobis distance between the points. The
+        Mahalanobis distance between two points ``u`` and ``v`` is
+        :math:`\\sqrt{(u-v)(1/V)(u-v)^T}` where :math:`(1/V)` (the ``VI``
+        variable) is the inverse covariance. If ``VI`` is not None,
+        ``VI`` will be used as the inverse covariance matrix.
+
+    15. ``Y = pdist(X, 'yule')``
+
+        Computes the Yule distance between each pair of boolean
+        vectors. (see yule function documentation)
+
+    16. ``Y = pdist(X, 'matching')``
+
+        Synonym for 'hamming'.
+
+    17. ``Y = pdist(X, 'dice')``
+
+        Computes the Dice distance between each pair of boolean
+        vectors. (see dice function documentation)
+
+    18. ``Y = pdist(X, 'kulczynski1')``
+
+        Computes the kulczynski1 distance between each pair of
+        boolean vectors. (see kulczynski1 function documentation)
+
+    19. ``Y = pdist(X, 'rogerstanimoto')``
+
+        Computes the Rogers-Tanimoto distance between each pair of
+        boolean vectors. (see rogerstanimoto function documentation)
+
+    20. ``Y = pdist(X, 'russellrao')``
+
+        Computes the Russell-Rao distance between each pair of
+        boolean vectors. (see russellrao function documentation)
+
+    21. ``Y = pdist(X, 'sokalmichener')``
+
+        Computes the Sokal-Michener distance between each pair of
+        boolean vectors. (see sokalmichener function documentation)
+
+    22. ``Y = pdist(X, 'sokalsneath')``
+
+        Computes the Sokal-Sneath distance between each pair of
+        boolean vectors. (see sokalsneath function documentation)
+
+    23. ``Y = pdist(X, 'kulczynski1')``
+
+        Computes the Kulczynski 1 distance between each pair of
+        boolean vectors. (see kulczynski1 function documentation)
+
+    24. ``Y = pdist(X, f)``
+
+        Computes the distance between all pairs of vectors in X
+        using the user supplied 2-arity function f. For example,
+        Euclidean distance between the vectors could be computed
+        as follows::
+
+          dm = pdist(X, lambda u, v: np.sqrt(((u-v)**2).sum()))
+
+        Note that you should avoid passing a reference to one of
+        the distance functions defined in this library. For example,::
+
+          dm = pdist(X, sokalsneath)
+
+        would calculate the pair-wise distances between the vectors in
+        X using the Python function sokalsneath. This would result in
+        sokalsneath being called :math:`{n \\choose 2}` times, which
+        is inefficient. Instead, the optimized C version is more
+        efficient, and we call it using the following syntax.::
+
+          dm = pdist(X, 'sokalsneath')
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.spatial.distance import pdist
+
+    ``x`` is an array of five points in three-dimensional space.
+
+    >>> x = np.array([[2, 0, 2], [2, 2, 3], [-2, 4, 5], [0, 1, 9], [2, 2, 4]])
+
+    ``pdist(x)`` with no additional arguments computes the 10 pairwise
+    Euclidean distances:
+
+    >>> pdist(x)
+    array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949,
+           6.40312424, 1.        , 5.38516481, 4.58257569, 5.47722558])
+
+    The following computes the pairwise Minkowski distances with ``p = 3.5``:
+
+    >>> pdist(x, metric='minkowski', p=3.5)
+    array([2.04898923, 5.1154929 , 7.02700737, 2.43802731, 4.19042714,
+           6.03956994, 1.        , 4.45128103, 4.10636143, 5.0619695 ])
+
+    The pairwise city block or Manhattan distances:
+
+    >>> pdist(x, metric='cityblock')
+    array([ 3., 11., 10.,  4.,  8.,  9.,  1.,  9.,  7.,  8.])
+
+    """
+    # You can also call this as:
+    #     Y = pdist(X, 'test_abc')
+    # where 'abc' is the metric being tested.  This computes the distance
+    # between all pairs of vectors in X using the distance metric 'abc' but
+    # with a more succinct, verifiable, but less efficient implementation.
+
+    X = _asarray_validated(X, sparse_ok=False, objects_ok=True, mask_ok=True,
+                           check_finite=False)
+
+    s = X.shape
+    if len(s) != 2:
+        raise ValueError('A 2-dimensional array must be passed.')
+
+    m, n = s
+
+    if callable(metric):
+        mstr = getattr(metric, '__name__', 'UnknownCustomMetric')
+        metric_info = _METRIC_ALIAS.get(mstr, None)
+
+        if metric_info is not None:
+            X, typ, kwargs = _validate_pdist_input(
+                X, m, n, metric_info, **kwargs)
+
+        return _pdist_callable(X, metric=metric, out=out, **kwargs)
+    elif isinstance(metric, str):
+        mstr = metric.lower()
+        metric_info = _METRIC_ALIAS.get(mstr, None)
+
+        if metric_info is not None:
+            pdist_fn = metric_info.pdist_func
+            return pdist_fn(X, out=out, **kwargs)
+        elif mstr.startswith("test_"):
+            metric_info = _TEST_METRICS.get(mstr, None)
+            if metric_info is None:
+                raise ValueError(f'Unknown "Test" Distance Metric: {mstr[5:]}')
+            X, typ, kwargs = _validate_pdist_input(
+                X, m, n, metric_info, **kwargs)
+            return _pdist_callable(
+                X, metric=metric_info.dist_func, out=out, **kwargs)
+        else:
+            raise ValueError('Unknown Distance Metric: %s' % mstr)
+    else:
+        raise TypeError('2nd argument metric must be a string identifier '
+                        'or a function.')
+
+
+def squareform(X, force="no", checks=True):
+    """
+    Convert a vector-form distance vector to a square-form distance
+    matrix, and vice-versa.
+
+    Parameters
+    ----------
+    X : array_like
+        Either a condensed or redundant distance matrix.
+    force : str, optional
+        As with MATLAB(TM), if force is equal to ``'tovector'`` or
+        ``'tomatrix'``, the input will be treated as a distance matrix or
+        distance vector respectively.
+    checks : bool, optional
+        If set to False, no checks will be made for matrix
+        symmetry nor zero diagonals. This is useful if it is known that
+        ``X - X.T1`` is small and ``diag(X)`` is close to zero.
+        These values are ignored any way so they do not disrupt the
+        squareform transformation.
+
+    Returns
+    -------
+    Y : ndarray
+        If a condensed distance matrix is passed, a redundant one is
+        returned, or if a redundant one is passed, a condensed distance
+        matrix is returned.
+
+    Notes
+    -----
+    1. ``v = squareform(X)``
+
+       Given a square n-by-n symmetric distance matrix ``X``,
+       ``v = squareform(X)`` returns a ``n * (n-1) / 2``
+       (i.e. binomial coefficient n choose 2) sized vector `v`
+       where :math:`v[{n \\choose 2} - {n-i \\choose 2} + (j-i-1)]`
+       is the distance between distinct points ``i`` and ``j``.
+       If ``X`` is non-square or asymmetric, an error is raised.
+
+    2. ``X = squareform(v)``
+
+       Given a ``n * (n-1) / 2`` sized vector ``v``
+       for some integer ``n >= 1`` encoding distances as described,
+       ``X = squareform(v)`` returns a n-by-n distance matrix ``X``.
+       The ``X[i, j]`` and ``X[j, i]`` values are set to
+       :math:`v[{n \\choose 2} - {n-i \\choose 2} + (j-i-1)]`
+       and all diagonal elements are zero.
+
+    In SciPy 0.19.0, ``squareform`` stopped casting all input types to
+    float64, and started returning arrays of the same dtype as the input.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.spatial.distance import pdist, squareform
+
+    ``x`` is an array of five points in three-dimensional space.
+
+    >>> x = np.array([[2, 0, 2], [2, 2, 3], [-2, 4, 5], [0, 1, 9], [2, 2, 4]])
+
+    ``pdist(x)`` computes the Euclidean distances between each pair of
+    points in ``x``.  The distances are returned in a one-dimensional
+    array with length ``5*(5 - 1)/2 = 10``.
+
+    >>> distvec = pdist(x)
+    >>> distvec
+    array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949,
+           6.40312424, 1.        , 5.38516481, 4.58257569, 5.47722558])
+
+    ``squareform(distvec)`` returns the 5x5 distance matrix.
+
+    >>> m = squareform(distvec)
+    >>> m
+    array([[0.        , 2.23606798, 6.40312424, 7.34846923, 2.82842712],
+           [2.23606798, 0.        , 4.89897949, 6.40312424, 1.        ],
+           [6.40312424, 4.89897949, 0.        , 5.38516481, 4.58257569],
+           [7.34846923, 6.40312424, 5.38516481, 0.        , 5.47722558],
+           [2.82842712, 1.        , 4.58257569, 5.47722558, 0.        ]])
+
+    When given a square distance matrix ``m``, ``squareform(m)`` returns
+    the one-dimensional condensed distance vector associated with the
+    matrix.  In this case, we recover ``distvec``.
+
+    >>> squareform(m)
+    array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949,
+           6.40312424, 1.        , 5.38516481, 4.58257569, 5.47722558])
+    """
+    X = np.ascontiguousarray(X)
+
+    s = X.shape
+
+    if force.lower() == 'tomatrix':
+        if len(s) != 1:
+            raise ValueError("Forcing 'tomatrix' but input X is not a "
+                             "distance vector.")
+    elif force.lower() == 'tovector':
+        if len(s) != 2:
+            raise ValueError("Forcing 'tovector' but input X is not a "
+                             "distance matrix.")
+
+    # X = squareform(v)
+    if len(s) == 1:
+        if s[0] == 0:
+            return np.zeros((1, 1), dtype=X.dtype)
+
+        # Grab the closest value to the square root of the number
+        # of elements times 2 to see if the number of elements
+        # is indeed a binomial coefficient.
+        d = int(np.ceil(np.sqrt(s[0] * 2)))
+
+        # Check that v is of valid dimensions.
+        if d * (d - 1) != s[0] * 2:
+            raise ValueError('Incompatible vector size. It must be a binomial '
+                             'coefficient n choose 2 for some integer n >= 2.')
+
+        # Allocate memory for the distance matrix.
+        M = np.zeros((d, d), dtype=X.dtype)
+
+        # Since the C code does not support striding using strides.
+        # The dimensions are used instead.
+        X = _copy_array_if_base_present(X)
+
+        # Fill in the values of the distance matrix.
+        _distance_wrap.to_squareform_from_vector_wrap(M, X)
+
+        # Return the distance matrix.
+        return M
+    elif len(s) == 2:
+        if s[0] != s[1]:
+            raise ValueError('The matrix argument must be square.')
+        if checks:
+            is_valid_dm(X, throw=True, name='X')
+
+        # One-side of the dimensions is set here.
+        d = s[0]
+
+        if d <= 1:
+            return np.array([], dtype=X.dtype)
+
+        # Create a vector.
+        v = np.zeros((d * (d - 1)) // 2, dtype=X.dtype)
+
+        # Since the C code does not support striding using strides.
+        # The dimensions are used instead.
+        X = _copy_array_if_base_present(X)
+
+        # Convert the vector to squareform.
+        _distance_wrap.to_vector_from_squareform_wrap(X, v)
+        return v
+    else:
+        raise ValueError(('The first argument must be one or two dimensional '
+                          'array. A %d-dimensional array is not '
+                          'permitted') % len(s))
+
+
+def is_valid_dm(D, tol=0.0, throw=False, name="D", warning=False):
+    """
+    Return True if input array is a valid distance matrix.
+
+    Distance matrices must be 2-dimensional numpy arrays.
+    They must have a zero-diagonal, and they must be symmetric.
+
+    Parameters
+    ----------
+    D : array_like
+        The candidate object to test for validity.
+    tol : float, optional
+        The distance matrix should be symmetric. `tol` is the maximum
+        difference between entries ``ij`` and ``ji`` for the distance
+        metric to be considered symmetric.
+    throw : bool, optional
+        An exception is thrown if the distance matrix passed is not valid.
+    name : str, optional
+        The name of the variable to checked. This is useful if
+        throw is set to True so the offending variable can be identified
+        in the exception message when an exception is thrown.
+    warning : bool, optional
+        Instead of throwing an exception, a warning message is
+        raised.
+
+    Returns
+    -------
+    valid : bool
+        True if the variable `D` passed is a valid distance matrix.
+
+    Notes
+    -----
+    Small numerical differences in `D` and `D.T` and non-zeroness of
+    the diagonal are ignored if they are within the tolerance specified
+    by `tol`.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.spatial.distance import is_valid_dm
+
+    This matrix is a valid distance matrix.
+
+    >>> d = np.array([[0.0, 1.1, 1.2, 1.3],
+    ...               [1.1, 0.0, 1.0, 1.4],
+    ...               [1.2, 1.0, 0.0, 1.5],
+    ...               [1.3, 1.4, 1.5, 0.0]])
+    >>> is_valid_dm(d)
+    True
+
+    In the following examples, the input is not a valid distance matrix.
+
+    Not square:
+
+    >>> is_valid_dm([[0, 2, 2], [2, 0, 2]])
+    False
+
+    Nonzero diagonal element:
+
+    >>> is_valid_dm([[0, 1, 1], [1, 2, 3], [1, 3, 0]])
+    False
+
+    Not symmetric:
+
+    >>> is_valid_dm([[0, 1, 3], [2, 0, 1], [3, 1, 0]])
+    False
+
+    """
+    D = np.asarray(D, order='c')
+    valid = True
+    try:
+        s = D.shape
+        if len(D.shape) != 2:
+            if name:
+                raise ValueError(('Distance matrix \'%s\' must have shape=2 '
+                                  '(i.e. be two-dimensional).') % name)
+            else:
+                raise ValueError('Distance matrix must have shape=2 (i.e. '
+                                 'be two-dimensional).')
+        if tol == 0.0:
+            if not (D == D.T).all():
+                if name:
+                    raise ValueError(('Distance matrix \'%s\' must be '
+                                     'symmetric.') % name)
+                else:
+                    raise ValueError('Distance matrix must be symmetric.')
+            if not (D[range(0, s[0]), range(0, s[0])] == 0).all():
+                if name:
+                    raise ValueError(('Distance matrix \'%s\' diagonal must '
+                                      'be zero.') % name)
+                else:
+                    raise ValueError('Distance matrix diagonal must be zero.')
+        else:
+            if not (D - D.T <= tol).all():
+                if name:
+                    raise ValueError(f'Distance matrix \'{name}\' must be '
+                                     f'symmetric within tolerance {tol:5.5f}.')
+                else:
+                    raise ValueError('Distance matrix must be symmetric within '
+                                     'tolerance %5.5f.' % tol)
+            if not (D[range(0, s[0]), range(0, s[0])] <= tol).all():
+                if name:
+                    raise ValueError(f'Distance matrix \'{name}\' diagonal must be '
+                                     f'close to zero within tolerance {tol:5.5f}.')
+                else:
+                    raise ValueError(('Distance matrix \'{}\' diagonal must be close '
+                                      'to zero within tolerance {:5.5f}.').format(*tol))
+    except Exception as e:
+        if throw:
+            raise
+        if warning:
+            warnings.warn(str(e), stacklevel=2)
+        valid = False
+    return valid
+
+
+def is_valid_y(y, warning=False, throw=False, name=None):
+    """
+    Return True if the input array is a valid condensed distance matrix.
+
+    Condensed distance matrices must be 1-dimensional numpy arrays.
+    Their length must be a binomial coefficient :math:`{n \\choose 2}`
+    for some positive integer n.
+
+    Parameters
+    ----------
+    y : array_like
+        The condensed distance matrix.
+    warning : bool, optional
+        Invokes a warning if the variable passed is not a valid
+        condensed distance matrix. The warning message explains why
+        the distance matrix is not valid.  `name` is used when
+        referencing the offending variable.
+    throw : bool, optional
+        Throws an exception if the variable passed is not a valid
+        condensed distance matrix.
+    name : bool, optional
+        Used when referencing the offending variable in the
+        warning or exception message.
+
+    Returns
+    -------
+    bool
+        True if the input array is a valid condensed distance matrix,
+        False otherwise.
+
+    Examples
+    --------
+    >>> from scipy.spatial.distance import is_valid_y
+
+    This vector is a valid condensed distance matrix.  The length is 6,
+    which corresponds to ``n = 4``, since ``4*(4 - 1)/2`` is 6.
+
+    >>> v = [1.0, 1.2, 1.0, 0.5, 1.3, 0.9]
+    >>> is_valid_y(v)
+    True
+
+    An input vector with length, say, 7, is not a valid condensed distance
+    matrix.
+
+    >>> is_valid_y([1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7])
+    False
+
+    """
+    y = np.asarray(y, order='c')
+    valid = True
+    try:
+        if len(y.shape) != 1:
+            if name:
+                raise ValueError(('Condensed distance matrix \'%s\' must '
+                                  'have shape=1 (i.e. be one-dimensional).')
+                                 % name)
+            else:
+                raise ValueError('Condensed distance matrix must have shape=1 '
+                                 '(i.e. be one-dimensional).')
+        n = y.shape[0]
+        d = int(np.ceil(np.sqrt(n * 2)))
+        if (d * (d - 1) / 2) != n:
+            if name:
+                raise ValueError(('Length n of condensed distance matrix '
+                                  '\'%s\' must be a binomial coefficient, i.e.'
+                                  'there must be a k such that '
+                                  '(k \\choose 2)=n)!') % name)
+            else:
+                raise ValueError('Length n of condensed distance matrix must '
+                                 'be a binomial coefficient, i.e. there must '
+                                 'be a k such that (k \\choose 2)=n)!')
+    except Exception as e:
+        if throw:
+            raise
+        if warning:
+            warnings.warn(str(e), stacklevel=2)
+        valid = False
+    return valid
+
+
+def num_obs_dm(d):
+    """
+    Return the number of original observations that correspond to a
+    square, redundant distance matrix.
+
+    Parameters
+    ----------
+    d : array_like
+        The target distance matrix.
+
+    Returns
+    -------
+    num_obs_dm : int
+        The number of observations in the redundant distance matrix.
+
+    Examples
+    --------
+    Find the number of original observations corresponding
+    to a square redundant distance matrix d.
+    
+    >>> from scipy.spatial.distance import num_obs_dm
+    >>> d = [[0, 100, 200], [100, 0, 150], [200, 150, 0]]
+    >>> num_obs_dm(d)
+    3
+    """
+    d = np.asarray(d, order='c')
+    is_valid_dm(d, tol=np.inf, throw=True, name='d')
+    return d.shape[0]
+
+
+def num_obs_y(Y):
+    """
+    Return the number of original observations that correspond to a
+    condensed distance matrix.
+
+    Parameters
+    ----------
+    Y : array_like
+        Condensed distance matrix.
+
+    Returns
+    -------
+    n : int
+        The number of observations in the condensed distance matrix `Y`.
+
+    Examples
+    --------
+    Find the number of original observations corresponding to a
+    condensed distance matrix Y.
+    
+    >>> from scipy.spatial.distance import num_obs_y
+    >>> Y = [1, 2, 3.5, 7, 10, 4]
+    >>> num_obs_y(Y)
+    4
+    """
+    Y = np.asarray(Y, order='c')
+    is_valid_y(Y, throw=True, name='Y')
+    k = Y.shape[0]
+    if k == 0:
+        raise ValueError("The number of observations cannot be determined on "
+                         "an empty distance matrix.")
+    d = int(np.ceil(np.sqrt(k * 2)))
+    if (d * (d - 1) / 2) != k:
+        raise ValueError("Invalid condensed distance matrix passed. Must be "
+                         "some k where k=(n choose 2) for some n >= 2.")
+    return d
+
+
+def _prepare_out_argument(out, dtype, expected_shape):
+    if out is None:
+        return np.empty(expected_shape, dtype=dtype)
+
+    if out.shape != expected_shape:
+        raise ValueError("Output array has incorrect shape.")
+    if not out.flags.c_contiguous:
+        raise ValueError("Output array must be C-contiguous.")
+    if out.dtype != np.float64:
+        raise ValueError("Output array must be double type.")
+    return out
+
+
+def _pdist_callable(X, *, out, metric, **kwargs):
+    n = X.shape[0]
+    out_size = (n * (n - 1)) // 2
+    dm = _prepare_out_argument(out, np.float64, (out_size,))
+    k = 0
+    for i in range(X.shape[0] - 1):
+        for j in range(i + 1, X.shape[0]):
+            dm[k] = metric(X[i], X[j], **kwargs)
+            k += 1
+    return dm
+
+
+def _cdist_callable(XA, XB, *, out, metric, **kwargs):
+    mA = XA.shape[0]
+    mB = XB.shape[0]
+    dm = _prepare_out_argument(out, np.float64, (mA, mB))
+    for i in range(mA):
+        for j in range(mB):
+            dm[i, j] = metric(XA[i], XB[j], **kwargs)
+    return dm
+
+
+def cdist(XA, XB, metric='euclidean', *, out=None, **kwargs):
+    """
+    Compute distance between each pair of the two collections of inputs.
+
+    See Notes for common calling conventions.
+
+    Parameters
+    ----------
+    XA : array_like
+        An :math:`m_A` by :math:`n` array of :math:`m_A`
+        original observations in an :math:`n`-dimensional space.
+        Inputs are converted to float type.
+    XB : array_like
+        An :math:`m_B` by :math:`n` array of :math:`m_B`
+        original observations in an :math:`n`-dimensional space.
+        Inputs are converted to float type.
+    metric : str or callable, optional
+        The distance metric to use. If a string, the distance function can be
+        'braycurtis', 'canberra', 'chebyshev', 'cityblock', 'correlation',
+        'cosine', 'dice', 'euclidean', 'hamming', 'jaccard', 'jensenshannon',
+        'kulczynski1', 'mahalanobis', 'matching', 'minkowski',
+        'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener',
+        'sokalsneath', 'sqeuclidean', 'yule'.
+    **kwargs : dict, optional
+        Extra arguments to `metric`: refer to each metric documentation for a
+        list of all possible arguments.
+
+        Some possible arguments:
+
+        p : scalar
+        The p-norm to apply for Minkowski, weighted and unweighted.
+        Default: 2.
+
+        w : array_like
+        The weight vector for metrics that support weights (e.g., Minkowski).
+
+        V : array_like
+        The variance vector for standardized Euclidean.
+        Default: var(vstack([XA, XB]), axis=0, ddof=1)
+
+        VI : array_like
+        The inverse of the covariance matrix for Mahalanobis.
+        Default: inv(cov(vstack([XA, XB].T))).T
+
+        out : ndarray
+        The output array
+        If not None, the distance matrix Y is stored in this array.
+
+    Returns
+    -------
+    Y : ndarray
+        A :math:`m_A` by :math:`m_B` distance matrix is returned.
+        For each :math:`i` and :math:`j`, the metric
+        ``dist(u=XA[i], v=XB[j])`` is computed and stored in the
+        :math:`ij` th entry.
+
+    Raises
+    ------
+    ValueError
+        An exception is thrown if `XA` and `XB` do not have
+        the same number of columns.
+
+    Notes
+    -----
+    The following are common calling conventions:
+
+    1. ``Y = cdist(XA, XB, 'euclidean')``
+
+       Computes the distance between :math:`m` points using
+       Euclidean distance (2-norm) as the distance metric between the
+       points. The points are arranged as :math:`m`
+       :math:`n`-dimensional row vectors in the matrix X.
+
+    2. ``Y = cdist(XA, XB, 'minkowski', p=2.)``
+
+       Computes the distances using the Minkowski distance
+       :math:`\\|u-v\\|_p` (:math:`p`-norm) where :math:`p > 0` (note
+       that this is only a quasi-metric if :math:`0 < p < 1`).
+
+    3. ``Y = cdist(XA, XB, 'cityblock')``
+
+       Computes the city block or Manhattan distance between the
+       points.
+
+    4. ``Y = cdist(XA, XB, 'seuclidean', V=None)``
+
+       Computes the standardized Euclidean distance. The standardized
+       Euclidean distance between two n-vectors ``u`` and ``v`` is
+
+       .. math::
+
+          \\sqrt{\\sum {(u_i-v_i)^2 / V[x_i]}}.
+
+       V is the variance vector; V[i] is the variance computed over all
+       the i'th components of the points. If not passed, it is
+       automatically computed.
+
+    5. ``Y = cdist(XA, XB, 'sqeuclidean')``
+
+       Computes the squared Euclidean distance :math:`\\|u-v\\|_2^2` between
+       the vectors.
+
+    6. ``Y = cdist(XA, XB, 'cosine')``
+
+       Computes the cosine distance between vectors u and v,
+
+       .. math::
+
+          1 - \\frac{u \\cdot v}
+                   {{\\|u\\|}_2 {\\|v\\|}_2}
+
+       where :math:`\\|*\\|_2` is the 2-norm of its argument ``*``, and
+       :math:`u \\cdot v` is the dot product of :math:`u` and :math:`v`.
+
+    7. ``Y = cdist(XA, XB, 'correlation')``
+
+       Computes the correlation distance between vectors u and v. This is
+
+       .. math::
+
+          1 - \\frac{(u - \\bar{u}) \\cdot (v - \\bar{v})}
+                   {{\\|(u - \\bar{u})\\|}_2 {\\|(v - \\bar{v})\\|}_2}
+
+       where :math:`\\bar{v}` is the mean of the elements of vector v,
+       and :math:`x \\cdot y` is the dot product of :math:`x` and :math:`y`.
+
+
+    8. ``Y = cdist(XA, XB, 'hamming')``
+
+       Computes the normalized Hamming distance, or the proportion of
+       those vector elements between two n-vectors ``u`` and ``v``
+       which disagree. To save memory, the matrix ``X`` can be of type
+       boolean.
+
+    9. ``Y = cdist(XA, XB, 'jaccard')``
+
+       Computes the Jaccard distance between the points. Given two
+       vectors, ``u`` and ``v``, the Jaccard distance is the
+       proportion of those elements ``u[i]`` and ``v[i]`` that
+       disagree where at least one of them is non-zero.
+
+    10. ``Y = cdist(XA, XB, 'jensenshannon')``
+
+        Computes the Jensen-Shannon distance between two probability arrays.
+        Given two probability vectors, :math:`p` and :math:`q`, the
+        Jensen-Shannon distance is
+
+        .. math::
+
+           \\sqrt{\\frac{D(p \\parallel m) + D(q \\parallel m)}{2}}
+
+        where :math:`m` is the pointwise mean of :math:`p` and :math:`q`
+        and :math:`D` is the Kullback-Leibler divergence.
+
+    11. ``Y = cdist(XA, XB, 'chebyshev')``
+
+        Computes the Chebyshev distance between the points. The
+        Chebyshev distance between two n-vectors ``u`` and ``v`` is the
+        maximum norm-1 distance between their respective elements. More
+        precisely, the distance is given by
+
+        .. math::
+
+           d(u,v) = \\max_i {|u_i-v_i|}.
+
+    12. ``Y = cdist(XA, XB, 'canberra')``
+
+        Computes the Canberra distance between the points. The
+        Canberra distance between two points ``u`` and ``v`` is
+
+        .. math::
+
+          d(u,v) = \\sum_i \\frac{|u_i-v_i|}
+                               {|u_i|+|v_i|}.
+
+    13. ``Y = cdist(XA, XB, 'braycurtis')``
+
+        Computes the Bray-Curtis distance between the points. The
+        Bray-Curtis distance between two points ``u`` and ``v`` is
+
+
+        .. math::
+
+             d(u,v) = \\frac{\\sum_i (|u_i-v_i|)}
+                           {\\sum_i (|u_i+v_i|)}
+
+    14. ``Y = cdist(XA, XB, 'mahalanobis', VI=None)``
+
+        Computes the Mahalanobis distance between the points. The
+        Mahalanobis distance between two points ``u`` and ``v`` is
+        :math:`\\sqrt{(u-v)(1/V)(u-v)^T}` where :math:`(1/V)` (the ``VI``
+        variable) is the inverse covariance. If ``VI`` is not None,
+        ``VI`` will be used as the inverse covariance matrix.
+
+    15. ``Y = cdist(XA, XB, 'yule')``
+
+        Computes the Yule distance between the boolean
+        vectors. (see `yule` function documentation)
+
+    16. ``Y = cdist(XA, XB, 'matching')``
+
+        Synonym for 'hamming'.
+
+    17. ``Y = cdist(XA, XB, 'dice')``
+
+        Computes the Dice distance between the boolean vectors. (see
+        `dice` function documentation)
+
+    18. ``Y = cdist(XA, XB, 'kulczynski1')``
+
+        Computes the kulczynski distance between the boolean
+        vectors. (see `kulczynski1` function documentation)
+
+    19. ``Y = cdist(XA, XB, 'rogerstanimoto')``
+
+        Computes the Rogers-Tanimoto distance between the boolean
+        vectors. (see `rogerstanimoto` function documentation)
+
+    20. ``Y = cdist(XA, XB, 'russellrao')``
+
+        Computes the Russell-Rao distance between the boolean
+        vectors. (see `russellrao` function documentation)
+
+    21. ``Y = cdist(XA, XB, 'sokalmichener')``
+
+        Computes the Sokal-Michener distance between the boolean
+        vectors. (see `sokalmichener` function documentation)
+
+    22. ``Y = cdist(XA, XB, 'sokalsneath')``
+
+        Computes the Sokal-Sneath distance between the vectors. (see
+        `sokalsneath` function documentation)
+
+    23. ``Y = cdist(XA, XB, f)``
+
+        Computes the distance between all pairs of vectors in X
+        using the user supplied 2-arity function f. For example,
+        Euclidean distance between the vectors could be computed
+        as follows::
+
+          dm = cdist(XA, XB, lambda u, v: np.sqrt(((u-v)**2).sum()))
+
+        Note that you should avoid passing a reference to one of
+        the distance functions defined in this library. For example,::
+
+          dm = cdist(XA, XB, sokalsneath)
+
+        would calculate the pair-wise distances between the vectors in
+        X using the Python function `sokalsneath`. This would result in
+        sokalsneath being called :math:`{n \\choose 2}` times, which
+        is inefficient. Instead, the optimized C version is more
+        efficient, and we call it using the following syntax::
+
+          dm = cdist(XA, XB, 'sokalsneath')
+
+    Examples
+    --------
+    Find the Euclidean distances between four 2-D coordinates:
+
+    >>> from scipy.spatial import distance
+    >>> import numpy as np
+    >>> coords = [(35.0456, -85.2672),
+    ...           (35.1174, -89.9711),
+    ...           (35.9728, -83.9422),
+    ...           (36.1667, -86.7833)]
+    >>> distance.cdist(coords, coords, 'euclidean')
+    array([[ 0.    ,  4.7044,  1.6172,  1.8856],
+           [ 4.7044,  0.    ,  6.0893,  3.3561],
+           [ 1.6172,  6.0893,  0.    ,  2.8477],
+           [ 1.8856,  3.3561,  2.8477,  0.    ]])
+
+
+    Find the Manhattan distance from a 3-D point to the corners of the unit
+    cube:
+
+    >>> a = np.array([[0, 0, 0],
+    ...               [0, 0, 1],
+    ...               [0, 1, 0],
+    ...               [0, 1, 1],
+    ...               [1, 0, 0],
+    ...               [1, 0, 1],
+    ...               [1, 1, 0],
+    ...               [1, 1, 1]])
+    >>> b = np.array([[ 0.1,  0.2,  0.4]])
+    >>> distance.cdist(a, b, 'cityblock')
+    array([[ 0.7],
+           [ 0.9],
+           [ 1.3],
+           [ 1.5],
+           [ 1.5],
+           [ 1.7],
+           [ 2.1],
+           [ 2.3]])
+
+    """
+    # You can also call this as:
+    #     Y = cdist(XA, XB, 'test_abc')
+    # where 'abc' is the metric being tested.  This computes the distance
+    # between all pairs of vectors in XA and XB using the distance metric 'abc'
+    # but with a more succinct, verifiable, but less efficient implementation.
+
+    XA = np.asarray(XA)
+    XB = np.asarray(XB)
+
+    s = XA.shape
+    sB = XB.shape
+
+    if len(s) != 2:
+        raise ValueError('XA must be a 2-dimensional array.')
+    if len(sB) != 2:
+        raise ValueError('XB must be a 2-dimensional array.')
+    if s[1] != sB[1]:
+        raise ValueError('XA and XB must have the same number of columns '
+                         '(i.e. feature dimension.)')
+
+    mA = s[0]
+    mB = sB[0]
+    n = s[1]
+
+    if callable(metric):
+        mstr = getattr(metric, '__name__', 'Unknown')
+        metric_info = _METRIC_ALIAS.get(mstr, None)
+        if metric_info is not None:
+            XA, XB, typ, kwargs = _validate_cdist_input(
+                XA, XB, mA, mB, n, metric_info, **kwargs)
+        return _cdist_callable(XA, XB, metric=metric, out=out, **kwargs)
+    elif isinstance(metric, str):
+        mstr = metric.lower()
+        metric_info = _METRIC_ALIAS.get(mstr, None)
+        if metric_info is not None:
+            cdist_fn = metric_info.cdist_func
+            return cdist_fn(XA, XB, out=out, **kwargs)
+        elif mstr.startswith("test_"):
+            metric_info = _TEST_METRICS.get(mstr, None)
+            if metric_info is None:
+                raise ValueError(f'Unknown "Test" Distance Metric: {mstr[5:]}')
+            XA, XB, typ, kwargs = _validate_cdist_input(
+                XA, XB, mA, mB, n, metric_info, **kwargs)
+            return _cdist_callable(
+                XA, XB, metric=metric_info.dist_func, out=out, **kwargs)
+        else:
+            raise ValueError('Unknown Distance Metric: %s' % mstr)
+    else:
+        raise TypeError('2nd argument metric must be a string identifier '
+                        'or a function.')

venv/lib/python3.10/site-packages/scipy/spatial/distance.pyi

@@ -0,0 +1,211 @@
+from __future__ import annotations
+from typing import (overload, Any, SupportsFloat, Literal, Protocol, SupportsIndex)
+
+import numpy as np
+from numpy.typing import ArrayLike, NDArray
+
+# Anything that can be parsed by `np.float64.__init__` and is thus
+# compatible with `ndarray.__setitem__` (for a float64 array)
+_FloatValue = None | str | bytes | SupportsFloat | SupportsIndex
+
+class _MetricCallback1(Protocol):
+    def __call__(
+        self, __XA: NDArray[Any], __XB: NDArray[Any]
+    ) -> _FloatValue: ...
+
+class _MetricCallback2(Protocol):
+    def __call__(
+        self, __XA: NDArray[Any], __XB: NDArray[Any], **kwargs: Any
+    ) -> _FloatValue: ...
+
+# TODO: Use a single protocol with a parameter specification variable
+# once available (PEP 612)
+_MetricCallback = _MetricCallback1 | _MetricCallback2
+
+_MetricKind = Literal[
+    'braycurtis',
+    'canberra',
+    'chebychev', 'chebyshev', 'cheby', 'cheb', 'ch',
+    'cityblock', 'cblock', 'cb', 'c',
+    'correlation', 'co',
+    'cosine', 'cos',
+    'dice',
+    'euclidean', 'euclid', 'eu', 'e',
+    'hamming', 'hamm', 'ha', 'h',
+    'minkowski', 'mi', 'm', 'pnorm',
+    'jaccard', 'jacc', 'ja', 'j',
+    'jensenshannon', 'js',
+    'kulczynski1',
+    'mahalanobis', 'mahal', 'mah',
+    'rogerstanimoto',
+    'russellrao',
+    'seuclidean', 'se', 's',
+    'sokalmichener',
+    'sokalsneath',
+    'sqeuclidean', 'sqe', 'sqeuclid',
+    'yule',
+]
+
+# Function annotations
+
+def braycurtis(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> np.float64: ...
+
+def canberra(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> np.float64: ...
+
+# TODO: Add `metric`-specific overloads
+# Returns a float64 or float128 array, depending on the input dtype
+@overload
+def cdist(
+    XA: ArrayLike,
+    XB: ArrayLike,
+    metric: _MetricKind = ...,
+    *,
+    out: None | NDArray[np.floating[Any]] = ...,
+    p: float = ...,
+    w: ArrayLike | None = ...,
+    V: ArrayLike | None = ...,
+    VI: ArrayLike | None = ...,
+) -> NDArray[np.floating[Any]]: ...
+@overload
+def cdist(
+    XA: ArrayLike,
+    XB: ArrayLike,
+    metric: _MetricCallback,
+    *,
+    out: None | NDArray[np.floating[Any]] = ...,
+    **kwargs: Any,
+) -> NDArray[np.floating[Any]]: ...
+
+# TODO: Wait for dtype support; the return type is
+# dependent on the input arrays dtype
+def chebyshev(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> Any: ...
+
+# TODO: Wait for dtype support; the return type is
+# dependent on the input arrays dtype
+def cityblock(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> Any: ...
+
+def correlation(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ..., centered: bool = ...
+) -> np.float64: ...
+
+def cosine(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> np.float64: ...
+
+def dice(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> float: ...
+
+def directed_hausdorff(
+    u: ArrayLike, v: ArrayLike, seed: int | None = ...
+) -> tuple[float, int, int]: ...
+
+def euclidean(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> float: ...
+
+def hamming(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> np.float64: ...
+
+def is_valid_dm(
+    D: ArrayLike,
+    tol: float = ...,
+    throw: bool = ...,
+    name: str | None = ...,
+    warning: bool = ...,
+) -> bool: ...
+
+def is_valid_y(
+    y: ArrayLike,
+    warning: bool = ...,
+    throw: bool = ...,
+    name: str | None = ...,
+) -> bool: ...
+
+def jaccard(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> np.float64: ...
+
+def jensenshannon(
+    p: ArrayLike, q: ArrayLike, base: float | None = ...
+) -> np.float64: ...
+
+def kulczynski1(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> np.float64: ...
+
+def mahalanobis(
+    u: ArrayLike, v: ArrayLike, VI: ArrayLike
+) -> np.float64: ...
+
+def minkowski(
+    u: ArrayLike, v: ArrayLike, p: float = ..., w: ArrayLike | None = ...
+) -> float: ...
+
+def num_obs_dm(d: ArrayLike) -> int: ...
+
+def num_obs_y(Y: ArrayLike) -> int: ...
+
+# TODO: Add `metric`-specific overloads
+@overload
+def pdist(
+    X: ArrayLike,
+    metric: _MetricKind = ...,
+    *,
+    out: None | NDArray[np.floating[Any]] = ...,
+    p: float = ...,
+    w: ArrayLike | None = ...,
+    V: ArrayLike | None = ...,
+    VI: ArrayLike | None = ...,
+) -> NDArray[np.floating[Any]]: ...
+@overload
+def pdist(
+    X: ArrayLike,
+    metric: _MetricCallback,
+    *,
+    out: None | NDArray[np.floating[Any]] = ...,
+    **kwargs: Any,
+) -> NDArray[np.floating[Any]]: ...
+
+def seuclidean(
+    u: ArrayLike, v: ArrayLike, V: ArrayLike
+) -> float: ...
+
+def sokalmichener(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> float: ...
+
+def sokalsneath(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> np.float64: ...
+
+def sqeuclidean(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> np.float64: ...
+
+def squareform(
+    X: ArrayLike,
+    force: Literal["no", "tomatrix", "tovector"] = ...,
+    checks: bool = ...,
+) -> NDArray[Any]: ...
+
+def rogerstanimoto(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> float: ...
+
+def russellrao(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> float: ...
+
+def yule(
+    u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ...
+) -> float: ...

venv/lib/python3.10/site-packages/scipy/spatial/kdtree.py

@@ -0,0 +1,26 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.spatial` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+
+__all__ = [  # noqa: F822
+    'KDTree',
+    'Rectangle',
+    'cKDTree',
+    'cKDTreeNode',
+    'distance_matrix',
+    'minkowski_distance',
+    'minkowski_distance_p',
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="spatial", module="kdtree",
+                                   private_modules=["_kdtree"], all=__all__,
+                                   attribute=name)

venv/lib/python3.10/site-packages/scipy/spatial/qhull.py

@@ -0,0 +1,25 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.spatial` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+
+__all__ = [  # noqa: F822
+    'ConvexHull',
+    'Delaunay',
+    'HalfspaceIntersection',
+    'QhullError',
+    'Voronoi',
+    'tsearch',
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="spatial", module="qhull",
+                                   private_modules=["_qhull"], all=__all__,
+                                   attribute=name)

venv/lib/python3.10/site-packages/scipy/spatial/qhull_src/COPYING.txt

@@ -0,0 +1,38 @@
+                    Qhull, Copyright (c) 1993-2019
+                    
+                            C.B. Barber
+                           Arlington, MA 
+                          
+                               and
+
+       The National Science and Technology Research Center for
+        Computation and Visualization of Geometric Structures
+                        (The Geometry Center)
+                       University of Minnesota
+
+                       email: [email protected]
+
+This software includes Qhull from C.B. Barber and The Geometry Center.  
+Qhull is copyrighted as noted above.  Qhull is free software and may 
+be obtained via http from www.qhull.org.  It may be freely copied, modified, 
+and redistributed under the following conditions:
+
+1. All copyright notices must remain intact in all files.
+
+2. A copy of this text file must be distributed along with any copies 
+   of Qhull that you redistribute; this includes copies that you have 
+   modified, or copies of programs or other software products that 
+   include Qhull.
+
+3. If you modify Qhull, you must include a notice giving the
+   name of the person performing the modification, the date of
+   modification, and the reason for such modification.
+
+4. When distributing modified versions of Qhull, or other software 
+   products that include Qhull, you must provide notice that the original 
+   source code may be obtained as noted above.
+
+5. There is no warranty or other guarantee of fitness for Qhull, it is 
+   provided solely "as is".  Bug reports or fixes may be sent to 
+   [email protected]; the authors may or may not act on them as 
+   they desire.