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111 lines
3.5 KiB
Python
111 lines
3.5 KiB
Python
import numpy as np
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from scipy.linalg import svd
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__all__ = ['polar']
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def polar(a, side="right"):
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"""
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Compute the polar decomposition.
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Returns the factors of the polar decomposition [1]_ `u` and `p` such
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that ``a = up`` (if `side` is "right") or ``a = pu`` (if `side` is
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"left"), where `p` is positive semidefinite. Depending on the shape
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of `a`, either the rows or columns of `u` are orthonormal. When `a`
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is a square array, `u` is a square unitary array. When `a` is not
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square, the "canonical polar decomposition" [2]_ is computed.
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Parameters
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----------
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a : (m, n) array_like
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The array to be factored.
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side : {'left', 'right'}, optional
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Determines whether a right or left polar decomposition is computed.
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If `side` is "right", then ``a = up``. If `side` is "left", then
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``a = pu``. The default is "right".
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Returns
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-------
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u : (m, n) ndarray
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If `a` is square, then `u` is unitary. If m > n, then the columns
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of `a` are orthonormal, and if m < n, then the rows of `u` are
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orthonormal.
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p : ndarray
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`p` is Hermitian positive semidefinite. If `a` is nonsingular, `p`
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is positive definite. The shape of `p` is (n, n) or (m, m), depending
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on whether `side` is "right" or "left", respectively.
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References
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----------
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.. [1] R. A. Horn and C. R. Johnson, "Matrix Analysis", Cambridge
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University Press, 1985.
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.. [2] N. J. Higham, "Functions of Matrices: Theory and Computation",
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SIAM, 2008.
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Examples
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--------
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>>> from scipy.linalg import polar
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>>> a = np.array([[1, -1], [2, 4]])
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>>> u, p = polar(a)
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>>> u
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array([[ 0.85749293, -0.51449576],
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[ 0.51449576, 0.85749293]])
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>>> p
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array([[ 1.88648444, 1.2004901 ],
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[ 1.2004901 , 3.94446746]])
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A non-square example, with m < n:
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>>> b = np.array([[0.5, 1, 2], [1.5, 3, 4]])
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>>> u, p = polar(b)
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>>> u
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array([[-0.21196618, -0.42393237, 0.88054056],
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[ 0.39378971, 0.78757942, 0.4739708 ]])
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>>> p
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array([[ 0.48470147, 0.96940295, 1.15122648],
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[ 0.96940295, 1.9388059 , 2.30245295],
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[ 1.15122648, 2.30245295, 3.65696431]])
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>>> u.dot(p) # Verify the decomposition.
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array([[ 0.5, 1. , 2. ],
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[ 1.5, 3. , 4. ]])
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>>> u.dot(u.T) # The rows of u are orthonormal.
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array([[ 1.00000000e+00, -2.07353665e-17],
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[ -2.07353665e-17, 1.00000000e+00]])
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Another non-square example, with m > n:
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>>> c = b.T
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>>> u, p = polar(c)
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>>> u
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array([[-0.21196618, 0.39378971],
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[-0.42393237, 0.78757942],
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[ 0.88054056, 0.4739708 ]])
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>>> p
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array([[ 1.23116567, 1.93241587],
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[ 1.93241587, 4.84930602]])
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>>> u.dot(p) # Verify the decomposition.
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array([[ 0.5, 1.5],
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[ 1. , 3. ],
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[ 2. , 4. ]])
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>>> u.T.dot(u) # The columns of u are orthonormal.
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array([[ 1.00000000e+00, -1.26363763e-16],
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[ -1.26363763e-16, 1.00000000e+00]])
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"""
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if side not in ['right', 'left']:
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raise ValueError("`side` must be either 'right' or 'left'")
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a = np.asarray(a)
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if a.ndim != 2:
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raise ValueError("`a` must be a 2-D array.")
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w, s, vh = svd(a, full_matrices=False)
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u = w.dot(vh)
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if side == 'right':
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# a = up
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p = (vh.T.conj() * s).dot(vh)
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else:
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# a = pu
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p = (w * s).dot(w.T.conj())
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return u, p
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