def _geneig(a1, b, left, right, overwrite_a, overwrite_b):
b1 = asarray(b)
overwrite_b = overwrite_b or _datacopied(b1, b)
if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
raise ValueError('expected square matrix')
ggev, = get_lapack_funcs(('ggev', ), (a1, b1))
cvl, cvr = left, right
ggev_info = get_func_info(ggev)
if ggev_info.module_name[:7] == 'clapack':
raise NotImplementedError('calling ggev from %s' %
get_func_info(ggev).module_name)
res = ggev(a1, b1, lwork=-1)
lwork = res[-2][0]
if ggev_info.prefix in 'cz':
alpha, beta, vl, vr, work, info = ggev(a1, b1, cvl, cvr, lwork,
overwrite_a, overwrite_b)
w = alpha / beta
else:
alphar, alphai, beta, vl, vr, work, info = ggev(
a1, b1, cvl, cvr, lwork, overwrite_a, overwrite_b)
w = (alphar + _I * alphai) / beta
if info < 0:
raise ValueError('illegal value in %d-th argument of internal ggev' %
-info)
if info > 0:
raise LinAlgError(
"generalized eig algorithm did not converge (info=%d)" % info)
only_real = numpy.logical_and.reduce(numpy.equal(w.imag, 0.0))
if not (ggev_info.prefix in 'cz' or only_real):
t = w.dtype.char
if left:
vl = _make_complex_eigvecs(w, vl, t)
if right:
vr = _make_complex_eigvecs(w, vr, t)
if not (left or right):
return w
if left:
if right:
return w, vl, vr
return w, vl
return w, vr
def _geneig(a1, b, left, right, overwrite_a, overwrite_b):
b1 = asarray(b)
overwrite_b = overwrite_b or _datacopied(b1, b)
if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
raise ValueError('expected square matrix')
ggev, = get_lapack_funcs(('ggev',), (a1, b1))
cvl, cvr = left, right
ggev_info = get_func_info(ggev)
if ggev_info.module_name[:7] == 'clapack':
raise NotImplementedError('calling ggev from %s' % get_func_info(ggev).module_name)
res = ggev(a1, b1, lwork=-1)
lwork = res[-2][0]
if ggev_info.prefix in 'cz':
alpha, beta, vl, vr, work, info = ggev(a1, b1, cvl, cvr, lwork,
overwrite_a, overwrite_b)
w = alpha / beta
else:
alphar, alphai, beta, vl, vr, work, info = ggev(a1, b1, cvl, cvr, lwork,
overwrite_a,overwrite_b)
w = (alphar + _I * alphai) / beta
if info < 0:
raise ValueError('illegal value in %d-th argument of internal ggev'
% -info)
if info > 0:
raise LinAlgError("generalized eig algorithm did not converge (info=%d)"
% info)
only_real = numpy.logical_and.reduce(numpy.equal(w.imag, 0.0))
if not (ggev_info.prefix in 'cz' or only_real):
t = w.dtype.char
if left:
vl = _make_complex_eigvecs(w, vl, t)
if right:
vr = _make_complex_eigvecs(w, vr, t)
if not (left or right):
return w
if left:
if right:
return w, vl, vr
return w, vl
return w, vr
def lstsq(a, b, cond=None, overwrite_a=False, overwrite_b=False):
"""
Compute least-squares solution to equation Ax = b.
Compute a vector x such that the 2-norm ``|b - A x|`` is minimized.
Parameters
----------
a : array, shape (M, N)
Left hand side matrix (2-D array).
b : array, shape (M,) or (M, K)
Right hand side matrix or vector (1-D or 2-D array).
cond : float, optional
Cutoff for 'small' singular values; used to determine effective
rank of a. Singular values smaller than
``rcond * largest_singular_value`` are considered zero.
overwrite_a : bool, optional
Discard data in `a` (may enhance performance). Default is False.
overwrite_b : bool, optional
Discard data in `b` (may enhance performance). Default is False.
Returns
-------
x : array, shape (N,) or (N, K) depending on shape of b
Least-squares solution.
residues : ndarray, shape () or (1,) or (K,)
Sums of residues, squared 2-norm for each column in ``b - a x``.
If rank of matrix a is < N or > M this is an empty array.
If b was 1-D, this is an (1,) shape array, otherwise the shape is (K,).
rank : int
Effective rank of matrix `a`.
s : array, shape (min(M,N),)
Singular values of `a`. The condition number of a is
``abs(s[0]/s[-1])``.
Raises
------
LinAlgError :
If computation does not converge.
See Also
--------
optimize.nnls : linear least squares with non-negativity constraint
"""
a1, b1 = map(asarray_chkfinite, (a, b))
if len(a1.shape) != 2:
raise ValueError('expected matrix')
m, n = a1.shape
if len(b1.shape) == 2:
nrhs = b1.shape[1]
else:
nrhs = 1
if m != b1.shape[0]:
raise ValueError('incompatible dimensions')
gelss, = get_lapack_funcs(('gelss', ), (a1, b1))
gelss_info = get_func_info(gelss)
if n > m:
# need to extend b matrix as it will be filled with
# a larger solution matrix
b2 = zeros((n, nrhs), dtype=gelss_info.dtype)
if len(b1.shape) == 2:
b2[:m, :] = b1
else:
b2[:m, 0] = b1
b1 = b2
overwrite_a = overwrite_a or _datacopied(a1, a)
overwrite_b = overwrite_b or _datacopied(b1, b)
if gelss_info.module_name[:7] == 'flapack':
lwork = calc_lwork.gelss(gelss_info.prefix, m, n, nrhs)[1]
v, x, s, rank, info = gelss(a1,
b1,
cond=cond,
lwork=lwork,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
else:
raise NotImplementedError('calling gelss from %s' %
get_func_info(gelss).module_name)
if info > 0:
raise LinAlgError("SVD did not converge in Linear Least Squares")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gelss' %
-info)
resids = asarray([], dtype=x.dtype)
if n < m:
x1 = x[:n]
if rank == n:
resids = sum(abs(x[n:])**2, axis=0)
x = x1
return x, resids, rank, s
def inv(a, overwrite_a=False):
"""
Compute the inverse of a matrix.
Parameters
----------
a : array_like
Square matrix to be inverted.
overwrite_a : bool, optional
Discard data in `a` (may improve performance). Default is False.
Returns
-------
ainv : ndarray
Inverse of the matrix `a`.
Raises
------
LinAlgError :
If `a` is singular.
ValueError :
If `a` is not square, or not 2-dimensional.
Examples
--------
>>> a = np.array([[1., 2.], [3., 4.]])
>>> sp.linalg.inv(a)
array([[-2. , 1. ],
[ 1.5, -0.5]])
>>> np.dot(a, sp.linalg.inv(a))
array([[ 1., 0.],
[ 0., 1.]])
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or _datacopied(a1, a)
#XXX: I found no advantage or disadvantage of using finv.
## finv, = get_flinalg_funcs(('inv',),(a1,))
## if finv is not None:
## a_inv,info = finv(a1,overwrite_a=overwrite_a)
## if info==0:
## return a_inv
## if info>0: raise LinAlgError, "singular matrix"
## if info<0: raise ValueError,\
## 'illegal value in %d-th argument of internal inv.getrf|getri'%(-info)
getrf, getri = get_lapack_funcs(('getrf', 'getri'), (a1, ))
getrf_info = get_func_info(getrf)
getri_info = get_func_info(getri)
#XXX: C ATLAS versions of getrf/i have rowmajor=1, this could be
# exploited for further optimization. But it will be probably
# a mess. So, a good testing site is required before trying
# to do that.
if (getrf_info.module_name[:7] == 'clapack' != getri_info.module_name[:7]):
# ATLAS 3.2.1 has getrf but not getri.
lu, piv, info = getrf(transpose(a1),
rowmajor=0,
overwrite_a=overwrite_a)
lu = transpose(lu)
else:
lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
if info == 0:
if getri_info.module_name[:7] == 'flapack':
lwork = calc_lwork.getri(getri_info.prefix, a1.shape[0])
lwork = lwork[1]
# XXX: the following line fixes curious SEGFAULT when
# benchmarking 500x500 matrix inverse. This seems to
# be a bug in LAPACK ?getri routine because if lwork is
# minimal (when using lwork[0] instead of lwork[1]) then
# all tests pass. Further investigation is required if
# more such SEGFAULTs occur.
lwork = int(1.01 * lwork)
inv_a, info = getri(lu, piv, lwork=lwork, overwrite_lu=1)
else: # clapack
inv_a, info = getri(lu, piv, overwrite_lu=1)
if info > 0:
raise LinAlgError("singular matrix")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal '
'getrf|getri' % -info)
return inv_a
"""
a1, b1 = map(asarray_chkfinite, (ab, b))
# Validate shapes.
if a1.shape[-1] != b1.shape[0]:
raise ValueError("shapes of ab and b are not compatible.")
if l + u + 1 != a1.shape[0]:
raise ValueError(
"invalid values for the number of lower and upper diagonals:"
" l+u+1 (%d) does not equal ab.shape[0] (%d)" %
(l + u + 1, ab.shape[0]))
overwrite_b = overwrite_b or _datacopied(b1, b)
gbsv, = get_lapack_funcs(('gbsv', ), (a1, b1))
a2 = zeros((2 * l + u + 1, a1.shape[1]), dtype=get_func_info(gbsv).dtype)
a2[l:, :] = a1
lu, piv, x, info = gbsv(l,
u,
a2,
b1,
overwrite_ab=True,
overwrite_b=overwrite_b)
if info == 0:
return x
if info > 0:
raise LinAlgError("singular matrix")
raise ValueError('illegal value in %d-th argument of internal gbsv' %
-info)
def svd(a, full_matrices=True, compute_uv=True, overwrite_a=False):
"""Singular Value Decomposition.
Factorizes the matrix a into two unitary matrices U and Vh and
an 1d-array s of singular values (real, non-negative) such that
a == U S Vh if S is an suitably shaped matrix of zeros whose
main diagonal is s.
Parameters
----------
a : array, shape (M, N)
Matrix to decompose
full_matrices : boolean
If true, U, Vh are shaped (M,M), (N,N)
If false, the shapes are (M,K), (K,N) where K = min(M,N)
compute_uv : boolean
Whether to compute also U, Vh in addition to s (Default: true)
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
Returns
-------
U: array, shape (M,M) or (M,K) depending on full_matrices
s: array, shape (K,)
The singular values, sorted so that s[i] >= s[i+1]. K = min(M, N)
Vh: array, shape (N,N) or (K,N) depending on full_matrices
For compute_uv = False, only s is returned.
Raises LinAlgError if SVD computation does not converge
Examples
--------
>>> from scipy import random, linalg, allclose, dot
>>> a = random.randn(9, 6) + 1j*random.randn(9, 6)
>>> U, s, Vh = linalg.svd(a)
>>> U.shape, Vh.shape, s.shape
((9, 9), (6, 6), (6,))
>>> U, s, Vh = linalg.svd(a, full_matrices=False)
>>> U.shape, Vh.shape, s.shape
((9, 6), (6, 6), (6,))
>>> S = linalg.diagsvd(s, 6, 6)
>>> allclose(a, dot(U, dot(S, Vh)))
True
>>> s2 = linalg.svd(a, compute_uv=False)
>>> allclose(s, s2)
True
See also
--------
svdvals : return singular values of a matrix
diagsvd : return the Sigma matrix, given the vector s
"""
# A hack until full_matrices == 0 support is fixed here.
if full_matrices == 0:
import numpy.linalg
return numpy.linalg.svd(a, full_matrices=0, compute_uv=compute_uv)
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
m, n = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
gesdd, = get_lapack_funcs(('gesdd', ), (a1, ))
gesdd_info = get_func_info(gesdd)
if gesdd_info.module_name[:7] == 'flapack':
lwork = calc_lwork.gesdd(gesdd_info.prefix, m, n, compute_uv)[1]
u, s, v, info = gesdd(a1,
compute_uv=compute_uv,
lwork=lwork,
overwrite_a=overwrite_a)
else: # 'clapack'
raise NotImplementedError('calling gesdd from %s' %
gesdd_info.module_name)
if info > 0:
raise LinAlgError("SVD did not converge")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gesdd' %
-info)
if compute_uv:
return u, s, v
else:
return s
def svd(a, full_matrices=True, compute_uv=True, overwrite_a=False):
"""Singular Value Decomposition.
Factorizes the matrix a into two unitary matrices U and Vh and
an 1d-array s of singular values (real, non-negative) such that
a == U S Vh if S is an suitably shaped matrix of zeros whose
main diagonal is s.
Parameters
----------
a : array, shape (M, N)
Matrix to decompose
full_matrices : boolean
If true, U, Vh are shaped (M,M), (N,N)
If false, the shapes are (M,K), (K,N) where K = min(M,N)
compute_uv : boolean
Whether to compute also U, Vh in addition to s (Default: true)
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
Returns
-------
U: array, shape (M,M) or (M,K) depending on full_matrices
s: array, shape (K,)
The singular values, sorted so that s[i] >= s[i+1]. K = min(M, N)
Vh: array, shape (N,N) or (K,N) depending on full_matrices
For compute_uv = False, only s is returned.
Raises LinAlgError if SVD computation does not converge
Examples
--------
>>> from scipy import random, linalg, allclose, dot
>>> a = random.randn(9, 6) + 1j*random.randn(9, 6)
>>> U, s, Vh = linalg.svd(a)
>>> U.shape, Vh.shape, s.shape
((9, 9), (6, 6), (6,))
>>> U, s, Vh = linalg.svd(a, full_matrices=False)
>>> U.shape, Vh.shape, s.shape
((9, 6), (6, 6), (6,))
>>> S = linalg.diagsvd(s, 6, 6)
>>> allclose(a, dot(U, dot(S, Vh)))
True
>>> s2 = linalg.svd(a, compute_uv=False)
>>> allclose(s, s2)
True
See also
--------
svdvals : return singular values of a matrix
diagsvd : return the Sigma matrix, given the vector s
"""
# A hack until full_matrices == 0 support is fixed here.
if full_matrices == 0:
import numpy.linalg
return numpy.linalg.svd(a, full_matrices=0, compute_uv=compute_uv)
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
m,n = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
gesdd, = get_lapack_funcs(('gesdd',), (a1,))
gesdd_info = get_func_info(gesdd)
if gesdd_info.module_name[:7] == 'flapack':
lwork = calc_lwork.gesdd(gesdd_info.prefix, m, n, compute_uv)[1]
u,s,v,info = gesdd(a1,compute_uv = compute_uv, lwork = lwork,
overwrite_a = overwrite_a)
else: # 'clapack'
raise NotImplementedError('calling gesdd from %s' % gesdd_info.module_name)
if info > 0:
raise LinAlgError("SVD did not converge")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gesdd'
% -info)
if compute_uv:
return u, s, v
else:
return s
def lstsq(a, b, cond=None, overwrite_a=False, overwrite_b=False):
"""
Compute least-squares solution to equation Ax = b.
Compute a vector x such that the 2-norm ``|b - A x|`` is minimized.
Parameters
----------
a : array, shape (M, N)
Left hand side matrix (2-D array).
b : array, shape (M,) or (M, K)
Right hand side matrix or vector (1-D or 2-D array).
cond : float, optional
Cutoff for 'small' singular values; used to determine effective
rank of a. Singular values smaller than
``rcond * largest_singular_value`` are considered zero.
overwrite_a : bool, optional
Discard data in `a` (may enhance performance). Default is False.
overwrite_b : bool, optional
Discard data in `b` (may enhance performance). Default is False.
Returns
-------
x : array, shape (N,) or (N, K) depending on shape of b
Least-squares solution.
residues : ndarray, shape () or (1,) or (K,)
Sums of residues, squared 2-norm for each column in ``b - a x``.
If rank of matrix a is < N or > M this is an empty array.
If b was 1-D, this is an (1,) shape array, otherwise the shape is (K,).
rank : int
Effective rank of matrix `a`.
s : array, shape (min(M,N),)
Singular values of `a`. The condition number of a is
``abs(s[0]/s[-1])``.
Raises
------
LinAlgError :
If computation does not converge.
See Also
--------
optimize.nnls : linear least squares with non-negativity constraint
"""
a1, b1 = map(asarray_chkfinite, (a, b))
if len(a1.shape) != 2:
raise ValueError('expected matrix')
m, n = a1.shape
if len(b1.shape) == 2:
nrhs = b1.shape[1]
else:
nrhs = 1
if m != b1.shape[0]:
raise ValueError('incompatible dimensions')
gelss, = get_lapack_funcs(('gelss',), (a1, b1))
gelss_info = get_func_info(gelss)
if n > m:
# need to extend b matrix as it will be filled with
# a larger solution matrix
b2 = zeros((n, nrhs), dtype=gelss_info.dtype)
if len(b1.shape) == 2:
b2[:m,:] = b1
else:
b2[:m,0] = b1
b1 = b2
overwrite_a = overwrite_a or _datacopied(a1, a)
overwrite_b = overwrite_b or _datacopied(b1, b)
if gelss_info.module_name[:7] == 'flapack':
lwork = calc_lwork.gelss(gelss_info.prefix, m, n, nrhs)[1]
v, x, s, rank, info = gelss(a1, b1, cond=cond, lwork=lwork,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
else:
raise NotImplementedError('calling gelss from %s' % get_func_info(gelss).module_name)
if info > 0:
raise LinAlgError("SVD did not converge in Linear Least Squares")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gelss'
% -info)
resids = asarray([], dtype=x.dtype)
if n < m:
x1 = x[:n]
if rank == n:
resids = sum(abs(x[n:])**2, axis=0)
x = x1
return x, resids, rank, s
def inv(a, overwrite_a=False):
"""
Compute the inverse of a matrix.
Parameters
----------
a : array_like
Square matrix to be inverted.
overwrite_a : bool, optional
Discard data in `a` (may improve performance). Default is False.
Returns
-------
ainv : ndarray
Inverse of the matrix `a`.
Raises
------
LinAlgError :
If `a` is singular.
ValueError :
If `a` is not square, or not 2-dimensional.
Examples
--------
>>> a = np.array([[1., 2.], [3., 4.]])
>>> sp.linalg.inv(a)
array([[-2. , 1. ],
[ 1.5, -0.5]])
>>> np.dot(a, sp.linalg.inv(a))
array([[ 1., 0.],
[ 0., 1.]])
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or _datacopied(a1, a)
#XXX: I found no advantage or disadvantage of using finv.
## finv, = get_flinalg_funcs(('inv',),(a1,))
## if finv is not None:
## a_inv,info = finv(a1,overwrite_a=overwrite_a)
## if info==0:
## return a_inv
## if info>0: raise LinAlgError, "singular matrix"
## if info<0: raise ValueError,\
## 'illegal value in %d-th argument of internal inv.getrf|getri'%(-info)
getrf, getri = get_lapack_funcs(('getrf','getri'), (a1,))
getrf_info = get_func_info(getrf)
getri_info = get_func_info(getri)
#XXX: C ATLAS versions of getrf/i have rowmajor=1, this could be
# exploited for further optimization. But it will be probably
# a mess. So, a good testing site is required before trying
# to do that.
if (getrf_info.module_name[:7] == 'clapack' !=
getri_info.module_name[:7]):
# ATLAS 3.2.1 has getrf but not getri.
lu, piv, info = getrf(transpose(a1), rowmajor=0,
overwrite_a=overwrite_a)
lu = transpose(lu)
else:
lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
if info == 0:
if getri_info.module_name[:7] == 'flapack':
lwork = calc_lwork.getri(getri_info.prefix, a1.shape[0])
lwork = lwork[1]
# XXX: the following line fixes curious SEGFAULT when
# benchmarking 500x500 matrix inverse. This seems to
# be a bug in LAPACK ?getri routine because if lwork is
# minimal (when using lwork[0] instead of lwork[1]) then
# all tests pass. Further investigation is required if
# more such SEGFAULTs occur.
lwork = int(1.01 * lwork)
inv_a, info = getri(lu, piv, lwork=lwork, overwrite_lu=1)
else: # clapack
inv_a, info = getri(lu, piv, overwrite_lu=1)
if info > 0:
raise LinAlgError("singular matrix")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal '
'getrf|getri' % -info)
return inv_a
The solution to the system a x = b
"""
a1, b1 = map(asarray_chkfinite, (ab, b))
# Validate shapes.
if a1.shape[-1] != b1.shape[0]:
raise ValueError("shapes of ab and b are not compatible.")
if l + u + 1 != a1.shape[0]:
raise ValueError("invalid values for the number of lower and upper diagonals:"
" l+u+1 (%d) does not equal ab.shape[0] (%d)" % (l+u+1, ab.shape[0]))
overwrite_b = overwrite_b or _datacopied(b1, b)
gbsv, = get_lapack_funcs(('gbsv',), (a1, b1))
a2 = zeros((2*l+u+1, a1.shape[1]), dtype=get_func_info(gbsv).dtype)
a2[l:,:] = a1
lu, piv, x, info = gbsv(l, u, a2, b1, overwrite_ab=True,
overwrite_b=overwrite_b)
if info == 0:
return x
if info > 0:
raise LinAlgError("singular matrix")
raise ValueError('illegal value in %d-th argument of internal gbsv' % -info)
def solveh_banded(ab, b, overwrite_ab=False, overwrite_b=False, lower=False):
"""Solve equation a x = b. a is Hermitian positive-definite banded matrix.
The matrix a is stored in ab either in lower diagonal or upper
diagonal ordered form:
def eig(a,
b=None,
left=False,
right=True,
overwrite_a=False,
overwrite_b=False):
"""Solve an ordinary or generalized eigenvalue problem of a square matrix.
Find eigenvalues w and right or left eigenvectors of a general matrix::
a vr[:,i] = w[i] b vr[:,i]
a.H vl[:,i] = w[i].conj() b.H vl[:,i]
where .H is the Hermitean conjugation.
Parameters
----------
a : array, shape (M, M)
A complex or real matrix whose eigenvalues and eigenvectors
will be computed.
b : array, shape (M, M)
Right-hand side matrix in a generalized eigenvalue problem.
If omitted, identity matrix is assumed.
left : boolean
Whether to calculate and return left eigenvectors
right : boolean
Whether to calculate and return right eigenvectors
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
overwrite_b : boolean
Whether to overwrite data in b (may improve performance)
Returns
-------
w : double or complex array, shape (M,)
The eigenvalues, each repeated according to its multiplicity.
(if left == True)
vl : double or complex array, shape (M, M)
The normalized left eigenvector corresponding to the eigenvalue w[i]
is the column v[:,i].
(if right == True)
vr : double or complex array, shape (M, M)
The normalized right eigenvector corresponding to the eigenvalue w[i]
is the column vr[:,i].
Raises LinAlgError if eigenvalue computation does not converge
See Also
--------
eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
if b is not None:
b = asarray_chkfinite(b)
if b.shape != a1.shape:
raise ValueError('a and b must have the same shape')
return _geneig(a1, b, left, right, overwrite_a, overwrite_b)
geev, = get_lapack_funcs(('geev', ), (a1, ))
geev_info = get_func_info(geev)
compute_vl, compute_vr = left, right
if geev_info.module_name[:7] == 'flapack':
lwork = calc_lwork.geev(geev_info.prefix, a1.shape[0], compute_vl,
compute_vr)[1]
if geev_info.prefix in 'cz':
w, vl, vr, info = geev(a1,
lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
else:
wr, wi, vl, vr, info = geev(a1,
lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
t = {'f': 'F', 'd': 'D'}[wr.dtype.char]
w = wr + _I * wi
else: # 'clapack'
if geev_info.prefix in 'cz':
w, vl, vr, info = geev(a1,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
else:
wr, wi, vl, vr, info = geev(a1,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
t = {'f': 'F', 'd': 'D'}[wr.dtype.char]
w = wr + _I * wi
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geev' %
-info)
if info > 0:
raise LinAlgError("eig algorithm did not converge (only eigenvalues "
"with order >= %d have converged)" % info)
only_real = numpy.logical_and.reduce(numpy.equal(w.imag, 0.0))
if not (geev_info.prefix in 'cz' or only_real):
t = w.dtype.char
if left:
vl = _make_complex_eigvecs(w, vl, t)
if right:
vr = _make_complex_eigvecs(w, vr, t)
if not (left or right):
return w
if left:
if right:
return w, vl, vr
return w, vl
return w, vr
def hessenberg(a, calc_q=False, overwrite_a=False):
"""Compute Hessenberg form of a matrix.
The Hessenberg decomposition is
A = Q H Q^H
where Q is unitary/orthogonal and H has only zero elements below the first
subdiagonal.
Parameters
----------
a : array, shape (M,M)
Matrix to bring into Hessenberg form
calc_q : boolean
Whether to compute the transformation matrix
overwrite_a : boolean
Whether to ovewrite data in a (may improve performance)
Returns
-------
H : array, shape (M,M)
Hessenberg form of A
(If calc_q == True)
Q : array, shape (M,M)
Unitary/orthogonal similarity transformation matrix s.t. A = Q H Q^H
"""
a1 = asarray(a)
if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
gehrd, gebal = get_lapack_funcs(('gehrd', 'gebal'), (a1, ))
ba, lo, hi, pivscale, info = gebal(a1, permute=1, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gebal '
'(hessenberg)' % -info)
n = len(a1)
lwork = calc_lwork.gehrd(get_func_info(gehrd).prefix, n, lo, hi)[0]
hq, tau, info = gehrd(ba, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gehrd '
'(hessenberg)' % -info)
if not calc_q:
for i in range(lo, hi):
hq[i + 2:hi + 1, i] = 0.0
return hq
# XXX: Use ORGHR routines to compute q.
ger, gemm = get_blas_funcs(('ger', 'gemm'), (hq, ))
typecode = hq.dtype.char
q = None
for i in range(lo, hi):
if tau[i] == 0.0:
continue
v = zeros(n, dtype=typecode)
v[i + 1] = 1.0
v[i + 2:hi + 1] = hq[i + 2:hi + 1, i]
hq[i + 2:hi + 1, i] = 0.0
h = ger(-tau[i], v, v, a=diag(ones(n, dtype=typecode)), overwrite_a=1)
if q is None:
q = h
else:
q = gemm(1.0, q, h)
if q is None:
q = diag(ones(n, dtype=typecode))
return hq, q
def eig(a, b=None, left=False, right=True, overwrite_a=False, overwrite_b=False):
"""Solve an ordinary or generalized eigenvalue problem of a square matrix.
Find eigenvalues w and right or left eigenvectors of a general matrix::
a vr[:,i] = w[i] b vr[:,i]
a.H vl[:,i] = w[i].conj() b.H vl[:,i]
where .H is the Hermitean conjugation.
Parameters
----------
a : array, shape (M, M)
A complex or real matrix whose eigenvalues and eigenvectors
will be computed.
b : array, shape (M, M)
Right-hand side matrix in a generalized eigenvalue problem.
If omitted, identity matrix is assumed.
left : boolean
Whether to calculate and return left eigenvectors
right : boolean
Whether to calculate and return right eigenvectors
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
overwrite_b : boolean
Whether to overwrite data in b (may improve performance)
Returns
-------
w : double or complex array, shape (M,)
The eigenvalues, each repeated according to its multiplicity.
(if left == True)
vl : double or complex array, shape (M, M)
The normalized left eigenvector corresponding to the eigenvalue w[i]
is the column v[:,i].
(if right == True)
vr : double or complex array, shape (M, M)
The normalized right eigenvector corresponding to the eigenvalue w[i]
is the column vr[:,i].
Raises LinAlgError if eigenvalue computation does not converge
See Also
--------
eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
if b is not None:
b = asarray_chkfinite(b)
if b.shape != a1.shape:
raise ValueError('a and b must have the same shape')
return _geneig(a1, b, left, right, overwrite_a, overwrite_b)
geev, = get_lapack_funcs(('geev',), (a1,))
geev_info = get_func_info(geev)
compute_vl, compute_vr = left, right
if geev_info.module_name[:7] == 'flapack':
lwork = calc_lwork.geev(geev_info.prefix, a1.shape[0],
compute_vl, compute_vr)[1]
if geev_info.prefix in 'cz':
w, vl, vr, info = geev(a1, lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
else:
wr, wi, vl, vr, info = geev(a1, lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
t = {'f':'F','d':'D'}[wr.dtype.char]
w = wr + _I * wi
else: # 'clapack'
if geev_info.prefix in 'cz':
w, vl, vr, info = geev(a1,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
else:
wr, wi, vl, vr, info = geev(a1,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
t = {'f':'F','d':'D'}[wr.dtype.char]
w = wr + _I * wi
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geev'
% -info)
if info > 0:
raise LinAlgError("eig algorithm did not converge (only eigenvalues "
"with order >= %d have converged)" % info)
only_real = numpy.logical_and.reduce(numpy.equal(w.imag, 0.0))
if not (geev_info.prefix in 'cz' or only_real):
t = w.dtype.char
if left:
vl = _make_complex_eigvecs(w, vl, t)
if right:
vr = _make_complex_eigvecs(w, vr, t)
if not (left or right):
return w
if left:
if right:
return w, vl, vr
return w, vl
return w, vr
def hessenberg(a, calc_q=False, overwrite_a=False):
"""Compute Hessenberg form of a matrix.
The Hessenberg decomposition is
A = Q H Q^H
where Q is unitary/orthogonal and H has only zero elements below the first
subdiagonal.
Parameters
----------
a : array, shape (M,M)
Matrix to bring into Hessenberg form
calc_q : boolean
Whether to compute the transformation matrix
overwrite_a : boolean
Whether to ovewrite data in a (may improve performance)
Returns
-------
H : array, shape (M,M)
Hessenberg form of A
(If calc_q == True)
Q : array, shape (M,M)
Unitary/orthogonal similarity transformation matrix s.t. A = Q H Q^H
"""
a1 = asarray(a)
if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
gehrd,gebal = get_lapack_funcs(('gehrd','gebal'), (a1,))
ba, lo, hi, pivscale, info = gebal(a1, permute=1, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gebal '
'(hessenberg)' % -info)
n = len(a1)
lwork = calc_lwork.gehrd(get_func_info(gehrd).prefix, n, lo, hi)[0]
hq, tau, info = gehrd(ba, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gehrd '
'(hessenberg)' % -info)
if not calc_q:
for i in range(lo, hi):
hq[i+2:hi+1, i] = 0.0
return hq
# XXX: Use ORGHR routines to compute q.
ger,gemm = get_blas_funcs(('ger','gemm'), (hq,))
typecode = hq.dtype.char
q = None
for i in range(lo, hi):
if tau[i]==0.0:
continue
v = zeros(n, dtype=typecode)
v[i+1] = 1.0
v[i+2:hi+1] = hq[i+2:hi+1, i]
hq[i+2:hi+1, i] = 0.0
h = ger(-tau[i], v, v,a=diag(ones(n, dtype=typecode)), overwrite_a=1)
if q is None:
q = h
else:
q = gemm(1.0, q, h)
if q is None:
q = diag(ones(n, dtype=typecode))
return hq, q