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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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>>> import numpy as np |
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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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p = (vh.T.conj() * s).dot(vh) |
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else: |
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p = (w * s).dot(w.T.conj()) |
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return u, p |
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