def schur(a,output='real',lwork=None,overwrite_a=0):
"""Compute Schur decomposition of matrix a.
Description:
Return T, Z such that a = Z * T * (Z**H) where Z is a
unitary matrix and T is either upper-triangular or quasi-upper
triangular for output='real'
"""
if not output in ['real','complex','r','c']:
raise ValueError, "argument must be 'real', or 'complex'"
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
raise ValueError, 'expected square matrix'
N = a1.shape[0]
typ = a1.typecode()
if output in ['complex','c'] and typ not in ['F','D']:
if typ in _double_precision:
a1 = a1.astype('D')
typ = 'D'
else:
a1 = a1.astype('F')
typ = 'F'
overwrite_a = overwrite_a or (a1 is not a and not hasattr(a,'__array__'))
gees, = get_lapack_funcs(('gees',),(a1,))
if lwork is None or lwork == -1:
# get optimal work array
result = gees(lambda x: None,a,lwork=-1)
lwork = result[-2][0]
result = gees(lambda x: None,a,lwork=result[-2][0],overwrite_a=overwrite_a)
info = result[-1]
if info<0: raise ValueError,\
'illegal value in %-th argument of internal gees'%(-info)
elif info>0: raise LinAlgError, "Schur form not found. Possibly ill-conditioned."
return result[0], result[-3]
def pinv(a, cond=None):
""" pinv(a, cond=None) -> a_pinv
Compute generalized inverse of A using least-squares solver.
"""
a = asarray_chkfinite(a)
t = a.typecode()
b = scipy_base.identity(a.shape[0],t)
return lstsq(a, b, cond=cond)[0]
def cho_solve(clow, b):
"""Solve a previously factored symmetric system of equations.
First input is a tuple (LorU, lower) which is the output to cho_factor.
Second input is the right-hand side.
"""
c, lower = clow
c = asarray_chkfinite(c)
_assert_squareness(c)
b = asarray_chkfinite(b)
potrs, = get_lapack_funcs(('potrs',),(c,))
b, info = potrs(c,b,lower)
if info < 0:
msg = "Argument %d to lapack's ?potrs() has an illegal value." % info
raise TypeError, msg
if info > 0:
msg = "Unknown error occured int ?potrs(): error code = %d" % info
raise TypeError, msg
return b
def norm(x, ord=2):
""" norm(x, ord=2) -> n
Matrix and vector norm.
Inputs:
x -- a rank-1 (vector) or rank-2 (matrix) array
ord -- the order of norm.
Comments:
For vectors ord can be any real number including Inf or -Inf.
ord = Inf, computes the maximum of the magnitudes
ord = -Inf, computes minimum of the magnitudes
ord is finite, computes sum(abs(x)**ord)**(1.0/ord)
For matrices ord can only be + or - 1, 2, Inf.
ord = 2 computes the largest singular value
ord = -2 computes the smallest singular value
ord = 1 computes the largest column sum of absolute values
ord = -1 computes the smallest column sum of absolute values
ord = Inf computes the largest row sum of absolute values
ord = -Inf computes the smallest row sum of absolute values
ord = 'fro' computes the frobenius norm sqrt(sum(diag(X.H * X)))
"""
x = asarray_chkfinite(x)
nd = len(x.shape)
Inf = scipy_base.Inf
if nd == 1:
if ord == Inf:
return scipy_base.amax(abs(x))
elif ord == -Inf:
return scipy_base.amin(abs(x))
else:
return scipy_base.sum(abs(x)**ord)**(1.0/ord)
elif nd == 2:
if ord == 2:
return scipy_base.amax(decomp.svd(x,compute_uv=0))
elif ord == -2:
return scipy_base.amin(decomp.svd(x,compute_uv=0))
elif ord == 1:
return scipy_base.amax(scipy_base.sum(abs(x)))
elif ord == Inf:
return scipy_base.amax(scipy_base.sum(abs(x),axis=1))
elif ord == -1:
return scipy_base.amin(scipy_base.sum(abs(x)))
elif ord == -Inf:
return scipy_base.amin(scipy_base.sum(abs(x),axis=1))
elif ord in ['fro','f']:
val = real((conjugate(x)*x).flat)
return sqrt(add.reduce(val))
else:
raise ValueError, "Invalid norm order for matrices."
else:
raise ValueError, "Improper number of dimensions to norm."
def lu_solve(a_lu_pivots,b):
"""Solve a previously factored system. First input is a tuple (lu, pivots)
which is the output to lu_factor. Second input is the right hand side.
"""
a_lu, pivots = a_lu_pivots
a_lu = asarray_chkfinite(a_lu)
pivots = asarray_chkfinite(pivots)
b = asarray_chkfinite(b)
_assert_squareness(a_lu)
getrs, = get_lapack_funcs(('getrs',),(a,))
b, info = getrs(a_lu,pivots,b)
if info < 0:
msg = "Argument %d to lapack's ?getrs() has an illegal value." % info
raise TypeError, msg
if info > 0:
msg = "Unknown error occured int ?getrs(): error code = %d" % info
raise TypeError, msg
return b
def det(a, overwrite_a=0):
""" det(a, overwrite_a=0) -> d
Return determinant of a square matrix.
"""
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 (a1 is not a and not hasattr(a,'__array__'))
fdet, = get_flinalg_funcs(('det',),(a1,))
a_det,info = fdet(a1,overwrite_a=overwrite_a)
if info<0: raise ValueError,\
'illegal value in %-th argument of internal det.getrf'%(-info)
return a_det
def cho_factor(a, lower=0, overwrite_a=0):
""" Compute Cholesky decomposition of matrix and return an object
to be used for solving a linear system using cho_solve.
"""
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 (a1 is not a and not hasattr(a,'__array__'))
potrf, = get_lapack_funcs(('potrf',),(a1,))
c,info = potrf(a1,lower=lower,overwrite_a=overwrite_a,clean=0)
if info>0: raise LinAlgError, "matrix not positive definite"
if info<0: raise ValueError,\
'illegal value in %-th argument of internal potrf'%(-info)
return c, lower
def inv(a, overwrite_a=0):
""" inv(a, overwrite_a=0) -> a_inv
Return inverse of square matrix a.
"""
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 (a1 is not a and not hasattr(a,'__array__'))
#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 %-th argument of internal inv.getrf|getri'%(-info)
getrf,getri = get_lapack_funcs(('getrf','getri'),(a1,))
#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.module_name[:7]=='clapack'!=getri.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.module_name[:7] == 'flapack':
lwork = calc_lwork.getri(getri.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 %-th argument of internal getrf|getri'%(-info)
return inv_a
def qr(a,overwrite_a=0,lwork=None):
"""QR decomposition of an M x N matrix a.
Description:
Find a unitary matrix, q, and an upper-trapezoidal 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 (a1 is not a and not hasattr(a,'__array__'))
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.typecode()
R = basic.triu(qr)
Q = scipy_base.identity(M,typecode=t)
ident = scipy_base.identity(M,typecode=t)
zeros = scipy_base.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 pinv2(a, cond=None):
""" pinv2(a, cond=None) -> a_pinv
Compute the generalized inverse of A using svd.
"""
a = asarray_chkfinite(a)
u, s, vh = decomp.svd(a)
t = u.typecode()
if cond in [None,-1]:
cond = {0: feps*1e3, 1: eps*1e6}[_array_precision[t]]
m,n = a.shape
cutoff = cond*scipy_base.maximum.reduce(s)
psigma = zeros((m,n),t)
for i in range(len(s)):
if s[i] > cutoff:
psigma[i,i] = 1.0/conjugate(s[i])
#XXX: use lapack/blas routines for dot
return transpose(conjugate(dot(dot(u,psigma),vh)))
def svd(a,compute_uv=1,overwrite_a=0):
"""Compute singular value decomposition (SVD) of matrix a.
Description:
Singular value decomposition of a matrix a is
a = u * sigma * v^H,
where v^H denotes conjugate(transpose(v)), u,v are unitary
matrices, sigma is zero matrix with a main diagonal containing
real non-negative singular values of the matrix a.
Inputs:
a -- An M x N matrix.
compute_uv -- If zero, then only the vector of singular values
is returned.
Outputs:
u -- An M x M unitary matrix [compute_uv=1].
s -- An min(M,N) vector of singular values in descending order,
sigma = diagsvd(s).
vh -- An N x N unitary matrix [compute_uv=1], vh = v^H.
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError, 'expected matrix'
m,n = a1.shape
overwrite_a = overwrite_a or (a1 is not a and not hasattr(a,'__array__'))
gesdd, = get_lapack_funcs(('gesdd',),(a1,))
if gesdd.module_name[:7] == 'flapack':
lwork = calc_lwork.gesdd(gesdd.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.module_name)
if info>0: raise LinAlgError, "SVD did not converge"
if info<0: raise ValueError,\
'illegal value in %-th argument of internal gesdd'%(-info)
if compute_uv:
return u,s,v
else:
return s
def cholesky(a,lower=0,overwrite_a=0):
"""Compute Cholesky decomposition of matrix.
Description:
For a hermitian positive-definite matrix a return the
upper-triangular (or lower-triangular if lower==1) matrix,
u such that u^H * u = a (or l * l^H = a).
"""
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 (a1 is not a and not hasattr(a,'__array__'))
potrf, = get_lapack_funcs(('potrf',),(a1,))
c,info = potrf(a1,lower=lower,overwrite_a=overwrite_a,clean=1)
if info>0: raise LinAlgError, "matrix not positive definite"
if info<0: raise ValueError,\
'illegal value in %-th argument of internal potrf'%(-info)
return c
def lu(a,permute_l=0,overwrite_a=0):
"""Return LU decompostion of a matrix.
Inputs:
a -- An M x N matrix.
permute_l -- Perform matrix multiplication p * l [disabled].
Outputs:
p,l,u -- LU decomposition matrices of a [permute_l=0]
pl,u -- LU decomposition matrices of a [permute_l=1]
Definitions:
a = p * l * u [permute_l=0]
a = pl * u [permute_l=1]
p - An M x M permutation matrix
l - An M x K lower triangular or trapezoidal matrix
with unit-diagonal
u - An K x N upper triangular or trapezoidal matrix
K = min(M,N)
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError, 'expected matrix'
m,n = a1.shape
overwrite_a = overwrite_a or (a1 is not a and not hasattr(a,'__array__'))
flu, = get_flinalg_funcs(('lu',),(a1,))
p,l,u,info = flu(a1,permute_l=permute_l,overwrite_a = overwrite_a)
if info<0: raise ValueError,\
'illegal value in %-th argument of internal lu.getrf'%(-info)
if permute_l:
return l,u
return p,l,u
Solve a system of equations given a previously factored matrix
Inputs:
(lu,piv) -- The factored matrix, a (the output of lu_factor)
b -- a set of right-hand sides
trans -- type of system to solve:
0 : a * x = b (no transpose)
1 : a^T * x = b (transpose)
2 a^H * x = b (conjugate transpose)
Outputs:
x -- the solution to the system
"""
b1 = asarray_chkfinite(b)
overwrite_b = overwrite_b or (b1 is not b and not hasattr(b,'__array__'))
if lu.shape[0] != b1.shape[0]:
raise ValuError, "incompatible dimensions."
getrs, = get_lapack_funcs(('getrs',),(lu,b1))
x,info = getrs(lu,piv,b1,trans=trans,overwrite_b=overwrite_b)
if info==0:
return x
raise ValueError,\
'illegal value in %-th argument of internal gesv|posv'%(-info)
def cho_solve((c, lower), b, overwrite_b=0):
""" cho_solve((c, lower), b, overwrite_b=0) -> x
Solve a system of equations given a previously cholesky factored matrix
def eig(a,b=None,left=0,right=1,overwrite_a=0,overwrite_b=0):
""" Solve ordinary and generalized eigenvalue problem
of a square matrix.
Inputs:
a -- An N x N matrix.
b -- An N x N matrix [default is identity(N)].
left -- Return left eigenvectors [disabled].
right -- Return right eigenvectors [enabled].
Outputs:
w -- eigenvalues [left==right==0].
w,vr -- w and right eigenvectors [left==0,right=1].
w,vl -- w and left eigenvectors [left==1,right==0].
w,vl,vr -- [left==right==1].
Definitions:
a * vr[:,i] = w[i] * b * vr[:,i]
a^H * vl[:,i] = conjugate(w[i]) * b^H * vl[:,i]
where a^H denotes transpose(conjugate(a)).
"""
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 (a1 is not a and not hasattr(a,'__array__'))
if b is not None:
b = asarray_chkfinite(b)
return _geneig(a1,b,left,right,overwrite_a,overwrite_b)
geev, = get_lapack_funcs(('geev',),(a1,))
compute_vl,compute_vr=left,right
if geev.module_name[:7] == 'flapack':
lwork = calc_lwork.geev(geev.prefix,a1.shape[0],
compute_vl,compute_vr)[1]
if geev.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.typecode()]
w = wr+_I*wi
else: # 'clapack'
if geev.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.typecode()]
w = wr+_I*wi
if info<0: raise ValueError,\
'illegal value in %-th argument of internal geev'%(-info)
if info>0: raise LinAlgError,"eig algorithm did not converge"
only_real = scipy_base.logical_and.reduce(scipy_base.equal(w.imag,0.0))
if not (geev.prefix in 'cz' or only_real):
t = w.typecode()
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