def polar(a, side="<STR_LIT:right>"):
if side not in ['<STR_LIT:right>', '<STR_LIT:left>']:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>a = np.asarray(a)<EOL>if a.ndim != <NUM_LIT:2>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>w, s, vh = svd(a, full_matrices=False)<EOL>u = w.dot(vh)<EOL>if side == '<STR_LIT:right>':<EOL><INDENT>p = (vh.T.conj() * s).dot(vh)<EOL><DEDENT>else:<EOL><INDENT>p = (w * s).dot(w.T.conj())<EOL><DEDENT>return u, p<EOL>
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 : string, 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
--------
>>> 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]])