def rq(a, overwrite_a=False, lwork=None):
"""Compute RQ decomposition of a square real matrix.
Calculate the decomposition :lm:`A = R Q` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, M)
Square real matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
econ : boolean
Returns
-------
R : double array, shape (M, N) or (K, N) for econ==True
Size K = min(M, N)
Q : double or complex array, shape (M, M) or (M, K) for econ==True
Raises LinAlgError if decomposition fails
"""
# TODO: implement support for non-square and complex arrays
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
M, N = a1.shape
if M != N:
raise ValueError('expected square matrix')
if issubclass(a1.dtype.type, complexfloating):
raise ValueError('expected real (non-complex) matrix')
overwrite_a = overwrite_a or (_datanotshared(a1, a))
gerqf, = get_lapack_funcs(('gerqf', ), (a1, ))
if lwork is None or lwork == -1:
# get optimal work array
rq, tau, work, info = gerqf(a1, lwork=-1, overwrite_a=1)
lwork = work[0]
rq, tau, work, info = gerqf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geqrf' %
-info)
gemm, = get_blas_funcs(('gemm', ), (rq, ))
t = rq.dtype.char
R = special_matrices.triu(rq)
Q = numpy.identity(M, dtype=t)
ident = numpy.identity(M, dtype=t)
zeros = numpy.zeros
k = min(M, N)
for i in range(k):
v = zeros((M, ), t)
v[N - k + i] = 1
v[0:N - k + i] = rq[M - k + i, 0:N - k + i]
H = gemm(-tau[i], v, v, 1 + 0j, ident, trans_b=2)
Q = gemm(1, Q, H)
return R, Q
def rq(a, overwrite_a=False, lwork=None):
"""Compute RQ decomposition of a square real matrix.
Calculate the decomposition :lm:`A = R Q` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, M)
Square real matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
econ : boolean
Returns
-------
R : double array, shape (M, N) or (K, N) for econ==True
Size K = min(M, N)
Q : double or complex array, shape (M, M) or (M, K) for econ==True
Raises LinAlgError if decomposition fails
"""
# TODO: implement support for non-square and complex arrays
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
M,N = a1.shape
if M != N:
raise ValueError('expected square matrix')
if issubclass(a1.dtype.type, complexfloating):
raise ValueError('expected real (non-complex) matrix')
overwrite_a = overwrite_a or (_datanotshared(a1, a))
gerqf, = get_lapack_funcs(('gerqf',), (a1,))
if lwork is None or lwork == -1:
# get optimal work array
rq, tau, work, info = gerqf(a1, lwork=-1, overwrite_a=1)
lwork = work[0]
rq, tau, work, info = gerqf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geqrf'
% -info)
gemm, = get_blas_funcs(('gemm',), (rq,))
t = rq.dtype.char
R = special_matrices.triu(rq)
Q = numpy.identity(M, dtype=t)
ident = numpy.identity(M, dtype=t)
zeros = numpy.zeros
k = min(M, N)
for i in range(k):
v = zeros((M,), t)
v[N-k+i] = 1
v[0:N-k+i] = rq[M-k+i, 0:N-k+i]
H = gemm(-tau[i], v, v, 1+0j, ident, trans_b=2)
Q = gemm(1, Q, H)
return R, Q
def qr_old(a, overwrite_a=False, lwork=None, check_finite=True):
"""Compute QR decomposition of a matrix.
Calculate the decomposition :lm:`A = Q R` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, N)
Matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
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
-------
Q : float or complex array, shape (M, M)
R : float or complex array, shape (M, N)
Size K = min(M, N)
Raises LinAlgError if decomposition fails
"""
if check_finite:
a1 = numpy.asarray_chkfinite(a)
else:
a1 = numpy.asarray(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
M,N = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
geqrf, = get_lapack_funcs(('geqrf',), (a1,))
if lwork is None or lwork == -1:
# get optimal work array
qr, tau, work, info = geqrf(a1, lwork=-1, overwrite_a=1)
lwork = work[0]
qr, tau, work, info = geqrf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geqrf'
% -info)
gemm, = get_blas_funcs(('gemm',), (qr,))
t = qr.dtype.char
R = numpy.triu(qr)
Q = numpy.identity(M, dtype=t)
ident = numpy.identity(M, dtype=t)
zeros = numpy.zeros
for i in range(min(M, N)):
v = zeros((M,), t)
v[i] = 1
v[i+1:M] = qr[i+1:M, i]
H = gemm(-tau[i], v, v, 1+0j, ident, trans_b=2)
Q = gemm(1, Q, H)
return Q, R
def qr_old(a, overwrite_a=False, lwork=None, check_finite=True):
"""Compute QR decomposition of a matrix.
Calculate the decomposition :lm:`A = Q R` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, N)
Matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
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
-------
Q : float or complex array, shape (M, M)
R : float or complex array, shape (M, N)
Size K = min(M, N)
Raises LinAlgError if decomposition fails
"""
if check_finite:
a1 = numpy.asarray_chkfinite(a)
else:
a1 = numpy.asarray(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
M, N = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
geqrf, = get_lapack_funcs(('geqrf', ), (a1, ))
if lwork is None or lwork == -1:
# get optimal work array
qr, tau, work, info = geqrf(a1, lwork=-1, overwrite_a=1)
lwork = work[0]
qr, tau, work, info = geqrf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geqrf' %
-info)
gemm, = get_blas_funcs(('gemm', ), (qr, ))
t = qr.dtype.char
R = numpy.triu(qr)
Q = numpy.identity(M, dtype=t)
ident = numpy.identity(M, dtype=t)
zeros = numpy.zeros
for i in range(min(M, N)):
v = zeros((M, ), t)
v[i] = 1
v[i + 1:M] = qr[i + 1:M, i]
H = gemm(-tau[i], v, v, 1 + 0j, ident, trans_b=2)
Q = gemm(1, Q, H)
return Q, R
def hessenberg(a,calc_q=0,overwrite_a=0):
""" Compute Hessenberg form of a matrix.
Inputs:
a -- the matrix
calc_q -- if non-zero then calculate unitary similarity
transformation matrix q.
overwrite_a=0 -- if non-zero then discard the contents of a,
i.e. a is used as a work array if possible.
Outputs:
h -- Hessenberg form of a [calc_q=0]
h, q -- matrices such that a = q * h * q^T [calc_q=1]
"""
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 (_datanotshared(a1,a))
gehrd,gebal = get_lapack_funcs(('gehrd','gebal'),(a1,))
ba,lo,hi,pivscale,info = gebal(a,permute=1,overwrite_a = overwrite_a)
if info<0: raise ValueError,\
'illegal value in %-th argument of internal gebal (hessenberg)'%(-info)
n = len(a1)
lwork = calc_lwork.gehrd(gehrd.prefix,n,lo,hi)
hq,tau,info = gehrd(ba,lo=lo,hi=hi,lwork=lwork,overwrite_a=1)
if info<0: raise ValueError,\
'illegal value in %-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 rq(a,overwrite_a=0,lwork=None):
"""RQ decomposition of an M x N matrix a.
Description:
Find an upper-triangular matrix r and a unitary (orthogonal)
matrix q such that r * q = a
Inputs:
a -- the matrix
overwrite_a=0 -- if non-zero then discard the contents of a,
i.e. a is used as a work array if possible.
lwork=None -- >= shape(a)[1]. If None (or -1) compute optimal
work array size.
Outputs:
r, q -- matrices such that r * q = a
"""
# TODO: implement support for non-square and complex arrays
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError, 'expected matrix'
M,N = a1.shape
if M != N:
raise ValueError, 'expected square matrix'
if issubclass(a1.dtype.type,complexfloating):
raise ValueError, 'expected real (non-complex) matrix'
overwrite_a = overwrite_a or (_datanotshared(a1,a))
gerqf, = get_lapack_funcs(('gerqf',),(a1,))
if lwork is None or lwork == -1:
# get optimal work array
rq,tau,work,info = gerqf(a1,lwork=-1,overwrite_a=1)
lwork = work[0]
rq,tau,work,info = gerqf(a1,lwork=lwork,overwrite_a=overwrite_a)
if info<0: raise ValueError, \
'illegal value in %-th argument of internal geqrf'%(-info)
gemm, = get_blas_funcs(('gemm',),(rq,))
t = rq.dtype.char
R = basic.triu(rq)
Q = numpy.identity(M,dtype=t)
ident = numpy.identity(M,dtype=t)
zeros = numpy.zeros
k = min(M,N)
for i in range(k):
v = zeros((M,),t)
v[N-k+i] = 1
v[0:N-k+i] = rq[M-k+i,0:N-k+i]
H = gemm(-tau[i],v,v,1+0j,ident,trans_b=2)
Q = gemm(1,Q,H)
return R, Q
def qr_old(a, overwrite_a=False, lwork=None):
"""Compute QR decomposition of a matrix.
Calculate the decomposition :lm:`A = Q R` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, N)
Matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
Returns
-------
Q : double or complex array, shape (M, M)
R : double or complex array, shape (M, N)
Size K = min(M, N)
Raises LinAlgError if decomposition fails
"""
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))
geqrf, = get_lapack_funcs(('geqrf',), (a1,))
if lwork is None or lwork == -1:
# get optimal work array
qr, tau, work, info = geqrf(a1, lwork=-1, overwrite_a=1)
lwork = work[0]
qr, tau, work, info = geqrf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geqrf'
% -info)
gemm, = get_blas_funcs(('gemm',), (qr,))
t = qr.dtype.char
R = special_matrices.triu(qr)
Q = numpy.identity(M, dtype=t)
ident = numpy.identity(M, dtype=t)
zeros = numpy.zeros
for i in range(min(M, N)):
v = zeros((M,), t)
v[i] = 1
v[i+1:M] = qr[i+1:M, i]
H = gemm(-tau[i], v, v, 1, ident, trans_b=2)
Q = gemm(1, Q, H)
return Q, R
def qr_old(a, overwrite_a=False, lwork=None):
"""Compute QR decomposition of a matrix.
Calculate the decomposition :lm:`A = Q R` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, N)
Matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
Returns
-------
Q : double or complex array, shape (M, M)
R : double or complex array, shape (M, N)
Size K = min(M, N)
Raises LinAlgError if decomposition fails
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
M,N = a1.shape
overwrite_a = overwrite_a or (_datanotshared(a1, a))
geqrf, = get_lapack_funcs(('geqrf',), (a1,))
if lwork is None or lwork == -1:
# get optimal work array
qr, tau, work, info = geqrf(a1, lwork=-1, overwrite_a=1)
lwork = work[0]
qr, tau, work, info = geqrf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geqrf'
% -info)
gemm, = get_blas_funcs(('gemm',), (qr,))
t = qr.dtype.char
R = special_matrices.triu(qr)
Q = numpy.identity(M, dtype=t)
ident = numpy.identity(M, dtype=t)
zeros = numpy.zeros
for i in range(min(M, N)):
v = zeros((M,), t)
v[i] = 1
v[i+1:M] = qr[i+1:M, i]
H = gemm(-tau[i], v, v, 1+0j, ident, trans_b=2)
Q = gemm(1, Q, H)
return Q, R
def qr_old(a,overwrite_a=0,lwork=None):
"""QR decomposition of an M x N matrix a.
Description:
Find a unitary (orthogonal) matrix, q, and an upper-triangular
matrix r such that q * r = a
Inputs:
a -- the matrix
overwrite_a=0 -- if non-zero then discard the contents of a,
i.e. a is used as a work array if possible.
lwork=None -- >= shape(a)[1]. If None (or -1) compute optimal
work array size.
Outputs:
q, r -- matrices such that q * r = a
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError, 'expected matrix'
M,N = a1.shape
overwrite_a = overwrite_a or (_datanotshared(a1,a))
geqrf, = get_lapack_funcs(('geqrf',),(a1,))
if lwork is None or lwork == -1:
# get optimal work array
qr,tau,work,info = geqrf(a1,lwork=-1,overwrite_a=1)
lwork = work[0]
qr,tau,work,info = geqrf(a1,lwork=lwork,overwrite_a=overwrite_a)
if info<0: raise ValueError,\
'illegal value in %-th argument of internal geqrf'%(-info)
gemm, = get_blas_funcs(('gemm',),(qr,))
t = qr.dtype.char
R = basic.triu(qr)
Q = numpy.identity(M,dtype=t)
ident = numpy.identity(M,dtype=t)
zeros = numpy.zeros
for i in range(min(M,N)):
v = zeros((M,),t)
v[i] = 1
v[i+1:M] = qr[i+1:M,i]
H = gemm(-tau[i],v,v,1+0j,ident,trans_b=2)
Q = gemm(1,Q,H)
return Q, R
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 sub-diagonal.
Parameters
----------
a : ndarray
Matrix to bring into Hessenberg form, of shape ``(M,M)``.
calc_q : bool, optional
Whether to compute the transformation matrix. Default is False.
overwrite_a : bool, optional
Whether to overwrite `a`; may improve performance.
Default is False.
Returns
-------
H : ndarray
Hessenberg form of `a`, of shape (M,M).
Q : ndarray
Unitary/orthogonal similarity transformation matrix ``A = Q H Q^H``.
Only returned if ``calc_q=True``. Of shape (M,M).
"""
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(gehrd.prefix, n, lo, hi)
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.
typecode = hq.dtype
ger,gemm = get_blas_funcs(('ger','gemm'), dtype=typecode)
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 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(gehrd.prefix, n, lo, hi)
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.
typecode = hq.dtype
ger, gemm = get_blas_funcs(("ger", "gemm"), dtype=typecode)
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 check_blas(self):
a = array([[1,1,1]])
b = array([[1],[1],[1]])
gemm, = get_blas_funcs(('gemm',),(a,b))
assert_array_almost_equal(gemm(1,a,b),[[3]],15)
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(gehrd.prefix, n, lo, hi)
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