112 lines
3.5 KiB
Python
112 lines
3.5 KiB
Python
|
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
|