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

import numpy as np
from scipy.linalg import svd


__all__ = ['polar']


def polar(a, side="right"):
    """
    Compute the polar decomposition.

    Returns the factors of the polar decomposition [1]_ `u` and `p` such
    that ``a = up`` (if `side` is "right") or ``a = pu`` (if `side` is
    "left"), where `p` is positive semidefinite. Depending on the shape
    of `a`, either the rows or columns of `u` are orthonormal. When `a`
    is a square array, `u` is a square unitary array. When `a` is not
    square, the "canonical polar decomposition" [2]_ is computed.

    Parameters
    ----------
    a : (m, n) array_like
        The array to be factored.
    side : {'left', 'right'}, optional
        Determines whether a right or left polar decomposition is computed.
        If `side` is "right", then ``a = up``.  If `side` is "left",  then
        ``a = pu``.  The default is "right".

    Returns
    -------
    u : (m, n) ndarray
        If `a` is square, then `u` is unitary. If m > n, then the columns
        of `a` are orthonormal, and if m < n, then the rows of `u` are
        orthonormal.
    p : ndarray
        `p` is Hermitian positive semidefinite. If `a` is nonsingular, `p`
        is positive definite. The shape of `p` is (n, n) or (m, m), depending
        on whether `side` is "right" or "left", respectively.

    References
    ----------
    .. [1] R. A. Horn and C. R. Johnson, "Matrix Analysis", Cambridge
           University Press, 1985.
    .. [2] N. J. Higham, "Functions of Matrices: Theory and Computation",
           SIAM, 2008.

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.linalg import polar
    >>> a = np.array([[1, -1], [2, 4]])
    >>> u, p = polar(a)
    >>> u
    array([[ 0.85749293, -0.51449576],
           [ 0.51449576,  0.85749293]])
    >>> p
    array([[ 1.88648444,  1.2004901 ],
           [ 1.2004901 ,  3.94446746]])

    A non-square example, with m < n:

    >>> b = np.array([[0.5, 1, 2], [1.5, 3, 4]])
    >>> u, p = polar(b)
    >>> u
    array([[-0.21196618, -0.42393237,  0.88054056],
           [ 0.39378971,  0.78757942,  0.4739708 ]])
    >>> p
    array([[ 0.48470147,  0.96940295,  1.15122648],
           [ 0.96940295,  1.9388059 ,  2.30245295],
           [ 1.15122648,  2.30245295,  3.65696431]])
    >>> u.dot(p)   # Verify the decomposition.
    array([[ 0.5,  1. ,  2. ],
           [ 1.5,  3. ,  4. ]])
    >>> u.dot(u.T)   # The rows of u are orthonormal.
    array([[  1.00000000e+00,  -2.07353665e-17],
           [ -2.07353665e-17,   1.00000000e+00]])

    Another non-square example, with m > n:

    >>> c = b.T
    >>> u, p = polar(c)
    >>> u
    array([[-0.21196618,  0.39378971],
           [-0.42393237,  0.78757942],
           [ 0.88054056,  0.4739708 ]])
    >>> p
    array([[ 1.23116567,  1.93241587],
           [ 1.93241587,  4.84930602]])
    >>> u.dot(p)   # Verify the decomposition.
    array([[ 0.5,  1.5],
           [ 1. ,  3. ],
           [ 2. ,  4. ]])
    >>> u.T.dot(u)  # The columns of u are orthonormal.
    array([[  1.00000000e+00,  -1.26363763e-16],
           [ -1.26363763e-16,   1.00000000e+00]])

    """
    if side not in ['right', 'left']:
        raise ValueError("`side` must be either 'right' or 'left'")
    a = np.asarray(a)
    if a.ndim != 2:
        raise ValueError("`a` must be a 2-D array.")

    w, s, vh = svd(a, full_matrices=False)
    u = w.dot(vh)
    if side == 'right':
        # a = up
        p = (vh.T.conj() * s).dot(vh)
    else:
        # a = pu
        p = (w * s).dot(w.T.conj())
    return u, p