def pinvh(a,
cond=None,
rcond=None,
lower=True,
return_rank=False,
check_finite=True):
"""Compute the (Moore-Penrose) pseudo-inverse of a hermetian matrix.
Calculate a generalized inverse of a symmetric matrix using its
eigenvalue decomposition and including all 'large' eigenvalues.
Parameters
----------
a : array, shape (N, N)
Real symmetric or complex hermetian matrix to be pseudo-inverted
cond, rcond : float or None
Cutoff for 'small' eigenvalues.
Singular values smaller than rcond * largest_eigenvalue are considered
zero.
If None or -1, suitable machine precision is used.
lower : boolean
Whether the pertinent array data is taken from the lower or upper
triangle of a. (Default: lower)
return_rank : bool, optional
if True, return the effective rank of the matrix
check_finite : boolean, optional
Whether to check the input matrixes contain only finite numbers.
Disabling may give a performance gain, but may result to problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
B : array, shape (N, N)
The pseudo-inverse of matrix `a`.
rank : int
The effective rank of the matrix. Returned if return_rank == True
Raises
------
LinAlgError
If eigenvalue does not converge
Examples
--------
>>> from numpy import *
>>> a = random.randn(9, 6)
>>> a = np.dot(a, a.T)
>>> B = pinvh(a)
>>> allclose(a, dot(a, dot(B, a)))
True
>>> allclose(B, dot(B, dot(a, B)))
True
"""
if check_finite:
a = np.asarray_chkfinite(a)
else:
a = np.asarray(a)
s, u = decomp.eigh(a, lower=lower)
if rcond is not None:
cond = rcond
if cond in [None, -1]:
t = u.dtype.char.lower()
factor = {'f': 1E3, 'd': 1E6}
cond = factor[t] * np.finfo(t).eps
# For hermitian matrices, singular values equal abs(eigenvalues)
above_cutoff = (abs(s) > cond * np.max(abs(s)))
psigma_diag = np.zeros_like(s)
psigma_diag[above_cutoff] = 1.0 / s[above_cutoff]
B = np.dot(u * psigma_diag, np.conjugate(u).T)
if return_rank:
return B, np.sum(above_cutoff)
else:
return B
def pinvh(a, cond=None, rcond=None, lower=True, return_rank=False,
check_finite=True):
"""Compute the (Moore-Penrose) pseudo-inverse of a Hermitian matrix.
Calculate a generalized inverse of a Hermitian or real symmetric matrix
using its eigenvalue decomposition and including all eigenvalues with
'large' absolute value.
Parameters
----------
a : array, shape (N, N)
Real symmetric or complex hermetian matrix to be pseudo-inverted
cond, rcond : float or None
Cutoff for 'small' eigenvalues.
Singular values smaller than rcond * largest_eigenvalue are considered
zero.
If None or -1, suitable machine precision is used.
lower : boolean
Whether the pertinent array data is taken from the lower or upper
triangle of a. (Default: lower)
return_rank : bool, optional
if True, return the effective rank of the matrix
check_finite : boolean, optional
Whether to check the input matrixes contain only finite numbers.
Disabling may give a performance gain, but may result to problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
B : array, shape (N, N)
The pseudo-inverse of matrix `a`.
rank : int
The effective rank of the matrix. Returned if return_rank == True
Raises
------
LinAlgError
If eigenvalue does not converge
Examples
--------
>>> from numpy import *
>>> a = random.randn(9, 6)
>>> a = np.dot(a, a.T)
>>> B = pinvh(a)
>>> allclose(a, dot(a, dot(B, a)))
True
>>> allclose(B, dot(B, dot(a, B)))
True
"""
if check_finite:
a = np.asarray_chkfinite(a)
else:
a = np.asarray(a)
s, u = decomp.eigh(a, lower=lower)
if rcond is not None:
cond = rcond
if cond in [None, -1]:
t = u.dtype.char.lower()
factor = {'f': 1E3, 'd': 1E6}
cond = factor[t] * np.finfo(t).eps
# For Hermitian matrices, singular values equal abs(eigenvalues)
above_cutoff = (abs(s) > cond * np.max(abs(s)))
psigma_diag = 1.0 / s[above_cutoff]
u = u[:, above_cutoff]
B = np.dot(u * psigma_diag, np.conjugate(u).T)
if return_rank:
return B, len(psigma_diag)
else:
return B
def pinvh(a, cond=None, rcond=None, lower=True, return_rank=False):
"""Compute the (Moore-Penrose) pseudo-inverse of a hermetian matrix.
Calculate a generalized inverse of a symmetric matrix using its
eigenvalue decomposition and including all 'large' eigenvalues.
Parameters
----------
a : array, shape (N, N)
Real symmetric or complex hermetian matrix to be pseudo-inverted
cond, rcond : float or None
Cutoff for 'small' eigenvalues.
Singular values smaller than rcond * largest_eigenvalue are considered
zero.
If None or -1, suitable machine precision is used.
lower : boolean
Whether the pertinent array data is taken from the lower or upper
triangle of a. (Default: lower)
return_rank : bool, optional
if True, return the effective rank of the matrix
Returns
-------
B : array, shape (N, N)
The pseudo-inverse of matrix `a`.
rank : int
The effective rank of the matrix. Returned if return_rank == True
Raises
------
LinAlgError
If eigenvalue does not converge
Examples
--------
>>> from numpy import *
>>> a = random.randn(9, 6)
>>> a = np.dot(a, a.T)
>>> B = pinvh(a)
>>> allclose(a, dot(a, dot(B, a)))
True
>>> allclose(B, dot(B, dot(a, B)))
True
"""
a = np.asarray_chkfinite(a)
s, u = decomp.eigh(a, lower=lower)
if rcond is not None:
cond = rcond
if cond in [None, -1]:
t = u.dtype.char.lower()
factor = {'f': 1E3, 'd': 1E6}
cond = factor[t] * np.finfo(t).eps
# For hermitian matrices, singular values equal abs(eigenvalues)
above_cutoff = (abs(s) > cond * np.max(abs(s)))
psigma_diag = np.zeros_like(s)
psigma_diag[above_cutoff] = 1.0 / s[above_cutoff]
B = np.dot(u * psigma_diag, np.conjugate(u).T)
if return_rank:
return B, np.sum(above_cutoff)
else:
return B
