def pinv2(a, cond=None, rcond=None):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using its
singular-value decomposition and including all 'large' singular
values.
Parameters
----------
a : array, shape (M, N)
Matrix to be pseudo-inverted.
cond, rcond : float or None
Cutoff for 'small' singular values.
Singular values smaller than ``rcond*largest_singular_value``
are considered zero.
If None or -1, suitable machine precision is used.
Returns
-------
B : array, shape (N, M)
The pseudo-inverse of matrix `a`.
Raises
------
LinAlgError
If SVD computation does not converge.
Examples
--------
>>> a = np.random.randn(9, 6)
>>> B = linalg.pinv2(a)
>>> np.allclose(a, dot(a, dot(B, a)))
True
>>> np.allclose(B, dot(B, dot(a, B)))
True
"""
a = np.asarray_chkfinite(a)
u, s, vh = decomp_svd.svd(a)
t = u.dtype.char
if rcond is not None:
cond = rcond
if cond in [None,-1]:
eps = np.finfo(np.float).eps
feps = np.finfo(np.single).eps
_array_precision = {'f': 0, 'd': 1, 'F': 0, 'D': 1}
cond = {0: feps*1e3, 1: eps*1e6}[_array_precision[t]]
m, n = a.shape
cutoff = cond*np.maximum.reduce(s)
psigma = np.zeros((m, n), t)
for i in range(len(s)):
if s[i] > cutoff:
psigma[i,i] = 1.0/np.conjugate(s[i])
#XXX: use lapack/blas routines for dot
return np.transpose(np.conjugate(np.dot(np.dot(u,psigma),vh)))
def pinv2(a, cond=None, rcond=None):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using its
singular-value decomposition and including all 'large' singular
values.
Parameters
----------
a : array, shape (M, N)
Matrix to be pseudo-inverted.
cond, rcond : float or None
Cutoff for 'small' singular values.
Singular values smaller than ``rcond*largest_singular_value``
are considered zero.
If None or -1, suitable machine precision is used.
Returns
-------
B : array, shape (N, M)
The pseudo-inverse of matrix `a`.
Raises
------
LinAlgError
If SVD computation does not converge.
Examples
--------
>>> a = np.random.randn(9, 6)
>>> B = linalg.pinv2(a)
>>> np.allclose(a, dot(a, dot(B, a)))
True
>>> np.allclose(B, dot(B, dot(a, B)))
True
"""
a = np.asarray_chkfinite(a)
u, s, vh = decomp_svd.svd(a)
t = u.dtype.char
if rcond is not None:
cond = rcond
if cond in [None, -1]:
eps = np.finfo(np.float).eps
feps = np.finfo(np.single).eps
_array_precision = {'f': 0, 'd': 1, 'F': 0, 'D': 1}
cond = {0: feps * 1e3, 1: eps * 1e6}[_array_precision[t]]
m, n = a.shape
cutoff = cond * np.maximum.reduce(s)
psigma = np.zeros((m, n), t)
for i in range(len(s)):
if s[i] > cutoff:
psigma[i, i] = 1.0 / np.conjugate(s[i])
#XXX: use lapack/blas routines for dot
return np.transpose(np.conjugate(np.dot(np.dot(u, psigma), vh)))
def pinv2(a, cond=None, rcond=None, return_rank=False, check_finite=True):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using its
singular-value decomposition and including all 'large' singular
values.
Parameters
----------
a : array, shape (M, N)
Matrix to be pseudo-inverted.
cond, rcond : float or None
Cutoff for 'small' singular values.
Singular values smaller than ``rcond*largest_singular_value``
are considered zero.
If None or -1, suitable machine precision is used.
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, M)
The pseudo-inverse of matrix `a`.
rank : int
The effective rank of the matrix. Returned if return_rank == True
Raises
------
LinAlgError
If SVD computation does not converge.
Examples
--------
>>> a = np.random.randn(9, 6)
>>> B = linalg.pinv2(a)
>>> np.allclose(a, dot(a, dot(B, a)))
True
>>> np.allclose(B, dot(B, dot(a, B)))
True
"""
if check_finite:
a = np.asarray_chkfinite(a)
else:
a = np.asarray(a)
u, s, vh = decomp_svd.svd(a, full_matrices=False)
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
rank = np.sum(s > cond * np.max(s))
psigma_diag = 1.0 / s[: rank]
B = np.transpose(np.conjugate(np.dot(u[:, : rank] *
psigma_diag, vh[: rank])))
if return_rank:
return B, rank
else:
return B
def pinv2(a, cond=None, rcond=None, return_rank=False, check_finite=True):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using its
singular-value decomposition and including all 'large' singular
values.
Parameters
----------
a : array, shape (M, N)
Matrix to be pseudo-inverted.
cond, rcond : float or None
Cutoff for 'small' singular values.
Singular values smaller than ``rcond*largest_singular_value``
are considered zero.
If None or -1, suitable machine precision is used.
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, M)
The pseudo-inverse of matrix `a`.
rank : int
The effective rank of the matrix. Returned if return_rank == True
Raises
------
LinAlgError
If SVD computation does not converge.
Examples
--------
>>> a = np.random.randn(9, 6)
>>> B = linalg.pinv2(a)
>>> np.allclose(a, dot(a, dot(B, a)))
True
>>> np.allclose(B, dot(B, dot(a, B)))
True
"""
if check_finite:
a = np.asarray_chkfinite(a)
else:
a = np.asarray(a)
u, s, vh = decomp_svd.svd(a, full_matrices=False)
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
above_cutoff = (s > cond * np.max(s))
psigma_diag = np.zeros_like(s)
psigma_diag[above_cutoff] = 1.0 / s[above_cutoff]
B = np.transpose(np.conjugate(np.dot(u * psigma_diag, vh)))
if return_rank:
return B, np.sum(above_cutoff)
else:
return B
def signm(a, disp=True):
"""Matrix sign function.
Extension of the scalar sign(x) to matrices.
Parameters
----------
A : array, shape(M,M)
Matrix at which to evaluate the sign function
disp : boolean
Print warning if error in the result is estimated large
instead of returning estimated error. (Default: True)
Returns
-------
sgnA : array, shape(M,M)
Value of the sign function at A
(if disp == False)
errest : float
1-norm of the estimated error, ||err||_1 / ||A||_1
Examples
--------
>>> from scipy.linalg import signm, eigvals
>>> a = [[1,2,3], [1,2,1], [1,1,1]]
>>> eigvals(a)
array([ 4.12488542+0.j, -0.76155718+0.j, 0.63667176+0.j])
>>> eigvals(signm(a))
array([-1.+0.j, 1.+0.j, 1.+0.j])
"""
def rounded_sign(x):
rx = real(x)
if rx.dtype.char=='f':
c = 1e3*feps*amax(x)
else:
c = 1e3*eps*amax(x)
return sign( (absolute(rx) > c) * rx )
result,errest = funm(a, rounded_sign, disp=0)
errtol = {0:1e3*feps, 1:1e3*eps}[_array_precision[result.dtype.char]]
if errest < errtol:
return result
# Handle signm of defective matrices:
# See "E.D.Denman and J.Leyva-Ramos, Appl.Math.Comp.,
# 8:237-250,1981" for how to improve the following (currently a
# rather naive) iteration process:
a = asarray(a)
#a = result # sometimes iteration converges faster but where??
# Shifting to avoid zero eigenvalues. How to ensure that shifting does
# not change the spectrum too much?
vals = svd(a,compute_uv=0)
max_sv = np.amax(vals)
#min_nonzero_sv = vals[(vals>max_sv*errtol).tolist().count(1)-1]
#c = 0.5/min_nonzero_sv
c = 0.5/max_sv
S0 = a + c*np.identity(a.shape[0])
prev_errest = errest
for i in range(100):
iS0 = inv(S0)
S0 = 0.5*(S0 + iS0)
Pp=0.5*(dot(S0,S0)+S0)
errest = norm(dot(Pp,Pp)-Pp,1)
if errest < errtol or prev_errest==errest:
break
prev_errest = errest
if disp:
if not isfinite(errest) or errest >= errtol:
print "Result may be inaccurate, approximate err =", errest
return S0
else:
return S0, errest
def pinv2(a, cond=None, rcond=None, return_rank=False):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using its
singular-value decomposition and including all 'large' singular
values.
Parameters
----------
a : array, shape (M, N)
Matrix to be pseudo-inverted.
cond, rcond : float or None
Cutoff for 'small' singular values.
Singular values smaller than ``rcond*largest_singular_value``
are considered zero.
If None or -1, suitable machine precision is used.
return_rank : bool, optional
if True, return the effective rank of the matrix
Returns
-------
B : array, shape (N, M)
The pseudo-inverse of matrix `a`.
rank : int
The effective rank of the matrix. Returned if return_rank == True
Raises
------
LinAlgError
If SVD computation does not converge.
Examples
--------
>>> a = np.random.randn(9, 6)
>>> B = linalg.pinv2(a)
>>> np.allclose(a, dot(a, dot(B, a)))
True
>>> np.allclose(B, dot(B, dot(a, B)))
True
"""
a = np.asarray_chkfinite(a)
u, s, vh = decomp_svd.svd(a, full_matrices=False)
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
above_cutoff = (s > cond * np.max(s))
psigma_diag = np.zeros_like(s)
psigma_diag[above_cutoff] = 1.0 / s[above_cutoff]
B = np.transpose(np.conjugate(np.dot(u * psigma_diag, vh)))
if return_rank:
return B, np.sum(above_cutoff)
else:
return B
def signm(a, disp=True):
"""Matrix sign function.
Extension of the scalar sign(x) to matrices.
Parameters
----------
A : array, shape(M,M)
Matrix at which to evaluate the sign function
disp : boolean
Print warning if error in the result is estimated large
instead of returning estimated error. (Default: True)
Returns
-------
sgnA : array, shape(M,M)
Value of the sign function at A
(if disp == False)
errest : float
1-norm of the estimated error, ||err||_1 / ||A||_1
Examples
--------
>>> from scipy.linalg import signm, eigvals
>>> a = [[1,2,3], [1,2,1], [1,1,1]]
>>> eigvals(a)
array([ 4.12488542+0.j, -0.76155718+0.j, 0.63667176+0.j])
>>> eigvals(signm(a))
array([-1.+0.j, 1.+0.j, 1.+0.j])
"""
def rounded_sign(x):
rx = real(x)
if rx.dtype.char=='f':
c = 1e3*feps*amax(x)
else:
c = 1e3*eps*amax(x)
return sign( (absolute(rx) > c) * rx )
result,errest = funm(a, rounded_sign, disp=0)
errtol = {0:1e3*feps, 1:1e3*eps}[_array_precision[result.dtype.char]]
if errest < errtol:
return result
# Handle signm of defective matrices:
# See "E.D.Denman and J.Leyva-Ramos, Appl.Math.Comp.,
# 8:237-250,1981" for how to improve the following (currently a
# rather naive) iteration process:
a = asarray(a)
#a = result # sometimes iteration converges faster but where??
# Shifting to avoid zero eigenvalues. How to ensure that shifting does
# not change the spectrum too much?
vals = svd(a,compute_uv=0)
max_sv = np.amax(vals)
#min_nonzero_sv = vals[(vals>max_sv*errtol).tolist().count(1)-1]
#c = 0.5/min_nonzero_sv
c = 0.5/max_sv
S0 = a + c*np.identity(a.shape[0])
prev_errest = errest
for i in range(100):
iS0 = inv(S0)
S0 = 0.5*(S0 + iS0)
Pp=0.5*(dot(S0,S0)+S0)
errest = norm(dot(Pp,Pp)-Pp,1)
if errest < errtol or prev_errest==errest:
break
prev_errest = errest
if disp:
if not isfinite(errest) or errest >= errtol:
print "Result may be inaccurate, approximate err =", errest
return S0
else:
return S0, errest