venv/lib/python3.10/site-packages/scipy/linalg/_matfuncs_sqrtm.py

"""
Matrix square root for general matrices and for upper triangular matrices.

This module exists to avoid cyclic imports.

"""
__all__ = ['sqrtm']

import numpy as np

from scipy._lib._util import _asarray_validated

# Local imports
from ._misc import norm
from .lapack import ztrsyl, dtrsyl
from ._decomp_schur import schur, rsf2csf



class SqrtmError(np.linalg.LinAlgError):
    pass


from ._matfuncs_sqrtm_triu import within_block_loop  # noqa: E402


def _sqrtm_triu(T, blocksize=64):
    """
    Matrix square root of an upper triangular matrix.

    This is a helper function for `sqrtm` and `logm`.

    Parameters
    ----------
    T : (N, N) array_like upper triangular
        Matrix whose square root to evaluate
    blocksize : int, optional
        If the blocksize is not degenerate with respect to the
        size of the input array, then use a blocked algorithm. (Default: 64)

    Returns
    -------
    sqrtm : (N, N) ndarray
        Value of the sqrt function at `T`

    References
    ----------
    .. [1] Edvin Deadman, Nicholas J. Higham, Rui Ralha (2013)
           "Blocked Schur Algorithms for Computing the Matrix Square Root,
           Lecture Notes in Computer Science, 7782. pp. 171-182.

    """
    T_diag = np.diag(T)
    keep_it_real = np.isrealobj(T) and np.min(T_diag) >= 0

    # Cast to complex as necessary + ensure double precision
    if not keep_it_real:
        T = np.asarray(T, dtype=np.complex128, order="C")
        T_diag = np.asarray(T_diag, dtype=np.complex128)
    else:
        T = np.asarray(T, dtype=np.float64, order="C")
        T_diag = np.asarray(T_diag, dtype=np.float64)

    R = np.diag(np.sqrt(T_diag))

    # Compute the number of blocks to use; use at least one block.
    n, n = T.shape
    nblocks = max(n // blocksize, 1)

    # Compute the smaller of the two sizes of blocks that
    # we will actually use, and compute the number of large blocks.
    bsmall, nlarge = divmod(n, nblocks)
    blarge = bsmall + 1
    nsmall = nblocks - nlarge
    if nsmall * bsmall + nlarge * blarge != n:
        raise Exception('internal inconsistency')

    # Define the index range covered by each block.
    start_stop_pairs = []
    start = 0
    for count, size in ((nsmall, bsmall), (nlarge, blarge)):
        for i in range(count):
            start_stop_pairs.append((start, start + size))
            start += size

    # Within-block interactions (Cythonized)
    try:
        within_block_loop(R, T, start_stop_pairs, nblocks)
    except RuntimeError as e:
        raise SqrtmError(*e.args) from e

    # Between-block interactions (Cython would give no significant speedup)
    for j in range(nblocks):
        jstart, jstop = start_stop_pairs[j]
        for i in range(j-1, -1, -1):
            istart, istop = start_stop_pairs[i]
            S = T[istart:istop, jstart:jstop]
            if j - i > 1:
                S = S - R[istart:istop, istop:jstart].dot(R[istop:jstart,
                                                            jstart:jstop])

            # Invoke LAPACK.
            # For more details, see the solve_sylvester implementation
            # and the fortran dtrsyl and ztrsyl docs.
            Rii = R[istart:istop, istart:istop]
            Rjj = R[jstart:jstop, jstart:jstop]
            if keep_it_real:
                x, scale, info = dtrsyl(Rii, Rjj, S)
            else:
                x, scale, info = ztrsyl(Rii, Rjj, S)
            R[istart:istop, jstart:jstop] = x * scale

    # Return the matrix square root.
    return R


def sqrtm(A, disp=True, blocksize=64):
    """
    Matrix square root.

    Parameters
    ----------
    A : (N, N) array_like
        Matrix whose square root to evaluate
    disp : bool, optional
        Print warning if error in the result is estimated large
        instead of returning estimated error. (Default: True)
    blocksize : integer, optional
        If the blocksize is not degenerate with respect to the
        size of the input array, then use a blocked algorithm. (Default: 64)

    Returns
    -------
    sqrtm : (N, N) ndarray
        Value of the sqrt function at `A`. The dtype is float or complex.
        The precision (data size) is determined based on the precision of
        input `A`. When the dtype is float, the precision is the same as `A`.
        When the dtype is complex, the precision is double that of `A`. The
        precision might be clipped by each dtype precision range.

    errest : float
        (if disp == False)

        Frobenius norm of the estimated error, ||err||_F / ||A||_F

    References
    ----------
    .. [1] Edvin Deadman, Nicholas J. Higham, Rui Ralha (2013)
           "Blocked Schur Algorithms for Computing the Matrix Square Root,
           Lecture Notes in Computer Science, 7782. pp. 171-182.

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.linalg import sqrtm
    >>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
    >>> r = sqrtm(a)
    >>> r
    array([[ 0.75592895,  1.13389342],
           [ 0.37796447,  1.88982237]])
    >>> r.dot(r)
    array([[ 1.,  3.],
           [ 1.,  4.]])

    """
    byte_size = np.asarray(A).dtype.itemsize
    A = _asarray_validated(A, check_finite=True, as_inexact=True)
    if len(A.shape) != 2:
        raise ValueError("Non-matrix input to matrix function.")
    if blocksize < 1:
        raise ValueError("The blocksize should be at least 1.")
    keep_it_real = np.isrealobj(A)
    if keep_it_real:
        T, Z = schur(A)
        d0 = np.diagonal(T)
        d1 = np.diagonal(T, -1)
        eps = np.finfo(T.dtype).eps
        needs_conversion = abs(d1) > eps * (abs(d0[1:]) + abs(d0[:-1]))
        if needs_conversion.any():
            T, Z = rsf2csf(T, Z)
    else:
        T, Z = schur(A, output='complex')
    failflag = False
    try:
        R = _sqrtm_triu(T, blocksize=blocksize)
        ZH = np.conjugate(Z).T
        X = Z.dot(R).dot(ZH)
        if not np.iscomplexobj(X):
            # float byte size range: f2 ~ f16
            X = X.astype(f"f{np.clip(byte_size, 2, 16)}", copy=False)
        else:
            # complex byte size range: c8 ~ c32.
            # c32(complex256) might not be supported in some environments.
            if hasattr(np, 'complex256'):
                X = X.astype(f"c{np.clip(byte_size*2, 8, 32)}", copy=False)
            else:
                X = X.astype(f"c{np.clip(byte_size*2, 8, 16)}", copy=False)
    except SqrtmError:
        failflag = True
        X = np.empty_like(A)
        X.fill(np.nan)

    if disp:
        if failflag:
            print("Failed to find a square root.")
        return X
    else:
        try:
            arg2 = norm(X.dot(X) - A, 'fro')**2 / norm(A, 'fro')
        except ValueError:
            # NaNs in matrix
            arg2 = np.inf

        return X, arg2