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 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 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
