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basic.py
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basic.py
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#
# Author: Pearu Peterson, March 2002
#
# w/ additions by Travis Oliphant, March 2002
__all__ = ['solve','inv','det','lstsq','norm','pinv','pinv2',
'tri','tril','triu','toeplitz','hankel','lu_solve',
'cho_solve','solve_banded','LinAlgError']
#from blas import get_blas_funcs
from lapack import get_lapack_funcs
from flinalg import get_flinalg_funcs
from scipy_base import asarray,zeros,sum,NewAxis,greater_equal,subtract,arange,\
conjugate,ravel,r_,mgrid,take,ones,dot,transpose,diag,sqrt,add
import Matrix
import scipy_base
from scipy_base import asarray_chkfinite
import calc_lwork
class LinAlgError(Exception):
pass
def lu_solve((lu, piv), b, trans=0, overwrite_b=0):
""" lu_solve((lu, piv), b, trans=0, overwrite_b=0) -> x
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
Inputs:
(c,lower) -- The factored matrix, a (the output of cho_factor)
b -- a set of right-hand sides
Outputs:
x -- the solution to the system a*x = b
"""
b1 = asarray_chkfinite(b)
overwrite_b = overwrite_b or (b1 is not b and not hasattr(b,'__array__'))
if c.shape[0] != b1.shape[0]:
raise ValuError, "incompatible dimensions."
potrs, = get_lapack_funcs(('potrs',),(c,b1))
x,info = potrs(c,b1,lower=lower,overwrite_b=overwrite_b)
if info==0:
return x
raise ValueError,\
'illegal value in %-th argument of internal gesv|posv'%(-info)
# Linear equations
def solve(a, b, sym_pos=0, lower=0, overwrite_a=0, overwrite_b=0,
debug = 0):
""" solve(a, b, sym_pos=0, lower=0, overwrite_a=0, overwrite_b=0) -> x
Solve a linear system of equations a * x = b for x.
Inputs:
a -- An N x N matrix.
b -- An N x nrhs matrix or N vector.
sym_pos -- Assume a is symmetric and positive definite.
lower -- Assume a is lower triangular, otherwise upper one.
Only used if sym_pos is true.
overwrite_y - Discard data in y, where y is a or b.
Outputs:
x -- The solution to the system a * x = b
"""
a1, b1 = map(asarray_chkfinite,(a,b))
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError, 'expected square matrix'
if a1.shape[0] != b1.shape[0]:
raise ValueError, 'incompatible dimensions'
overwrite_a = overwrite_a or (a1 is not a and not hasattr(a,'__array__'))
overwrite_b = overwrite_b or (b1 is not b and not hasattr(b,'__array__'))
if debug:
print 'solve:overwrite_a=',overwrite_a
print 'solve:overwrite_b=',overwrite_b
if sym_pos:
posv, = get_lapack_funcs(('posv',),(a1,b1),debug=debug)
c,x,info = posv(a1,b1,
lower = lower,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
else:
gesv, = get_lapack_funcs(('gesv',),(a1,b1))
lu,piv,x,info = gesv(a1,b1,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
if info==0:
return x
if info>0:
raise LinAlgError, "singular matrix"
raise ValueError,\
'illegal value in %-th argument of internal gesv|posv'%(-info)
def solve_banded((l,u), ab, b, overwrite_ab=0, overwrite_b=0,
debug = 0):
""" solve_banded((l,u), ab, b, overwrite_ab=0, overwrite_b=0) -> x
Solve a linear system of equations a * x = b for x where
a is a banded matrix stored in diagonal orded form
* * a1u
* a12 a23 ...
a11 a22 a33 ...
a21 a32 a43 ...
.
al1 .. *
Inputs:
(l,u) -- number of non-zero lower and upper diagonals, respectively.
a -- An N x (l+u+1) matrix.
b -- An N x nrhs matrix or N vector.
overwrite_y - Discard data in y, where y is ab or b.
Outputs:
x -- The solution to the system a * x = b
"""
a1, b1 = map(asarray_chkfinite,(ab,b))
overwrite_b = overwrite_b or (b1 is not b and not hasattr(b,'__array__'))
gbsv, = get_lapack_funcs(('gbsv',),(a1,b1))
a2 = zeros((2*l+u+1,a1.shape[1]),gbsv.typecode)
a2[l:,:] = a1
lu,piv,x,info = gbsv(l,u,a2,b1,
overwrite_ab=1,
overwrite_b=overwrite_b)
if info==0:
return x
if info>0:
raise LinAlgError, "singular matrix"
raise ValueError,\
'illegal value in %-th argument of internal gbsv'%(-info)
# matrix inversion
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
## matrix and Vector norm
import decomp
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."
### Determinant
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
### Linear Least Squares
def lstsq(a, b, cond=None, overwrite_a=0, overwrite_b=0):
""" lstsq(a, b, cond=None, overwrite_a=0, overwrite_b=0) -> x,resids,rank,s
Return least-squares solution of a * x = b.
Inputs:
a -- An M x N matrix.
b -- An M x nrhs matrix or M vector.
cond -- Used to determine effective rank of a.
Outputs:
x -- The solution (N x nrhs matrix) to the minimization problem:
2-norm(| b - a * x |) -> min
resids -- The residual sum-of-squares for the solution matrix x
(only if M>N and rank==N).
rank -- The effective rank of a.
s -- Singular values of a in decreasing order. The condition number
of a is abs(s[0]/s[-1]).
"""
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))
if n>m:
# need to extend b matrix as it will be filled with
# a larger solution matrix
b2 = zeros((n,nrhs),gelss.typecode)
if len(b1.shape)==2: b2[:m,:] = b1
else: b2[:m,0] = b1
b1 = b2
overwrite_a = overwrite_a or (a1 is not a and not hasattr(a,'__array__'))
overwrite_b = overwrite_b or (b1 is not b and not hasattr(b,'__array__'))
if gelss.module_name[:7] == 'flapack':
lwork = calc_lwork.gelss(gelss.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' % (gelss.module_name)
if info>0: raise LinAlgError, "SVD did not converge in Linear Least Squares"
if info<0: raise ValueError,\
'illegal value in %-th argument of internal gelss'%(-info)
resids = asarray([],x.typecode())
if n<m:
x1 = x[:n]
if rank==n: resids = sum(x[n:]**2)
x = x1
return x,resids,rank,s
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]
eps = scipy_base.limits.double_epsilon
feps = scipy_base.limits.float_epsilon
_array_precision = {'f': 0, 'd': 1, 'F': 0, 'D': 1}
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)))
#-----------------------------------------------------------------------------
# matrix construction functions
#-----------------------------------------------------------------------------
def tri(N, M=None, k=0, typecode=None):
""" returns a N-by-M matrix where all the diagonals starting from
lower left corner up to the k-th are all ones.
"""
if M is None: M = N
if type(M) == type('d'):
typecode = M
M = N
m = greater_equal(subtract.outer(arange(N), arange(M)),-k)
if typecode is None:
return m
else:
return m.astype(typecode)
def tril(m, k=0):
""" returns the elements on and below the k-th diagonal of m. k=0 is the
main diagonal, k > 0 is above and k < 0 is below the main diagonal.
"""
svsp = m.spacesaver()
m = asarray(m,savespace=1)
out = tri(m.shape[0], m.shape[1], k=k, typecode=m.typecode())*m
out.savespace(svsp)
return out
def triu(m, k=0):
""" returns the elements on and above the k-th diagonal of m. k=0 is the
main diagonal, k > 0 is above and k < 0 is below the main diagonal.
"""
svsp = m.spacesaver()
m = asarray(m,savespace=1)
out = (1-tri(m.shape[0], m.shape[1], k-1, m.typecode()))*m
out.savespace(svsp)
return out
def toeplitz(c,r=None):
""" Construct a toeplitz matrix (i.e. a matrix with constant diagonals).
Description:
toeplitz(c,r) is a non-symmetric Toeplitz matrix with c as its first
column and r as its first row.
toeplitz(c) is a symmetric (Hermitian) Toeplitz matrix (r=c).
See also: hankel
"""
isscalar = scipy_base.isscalar
if isscalar(c) or isscalar(r):
return c
if r is None:
r = c
r[0] = conjugoate(r[0])
c = conjugate(c)
r,c = map(asarray_chkfinite,(r,c))
r,c = map(ravel,(r,c))
rN,cN = map(len,(r,c))
if r[0] != c[0]:
print "Warning: column and row values don't agree; column value used."
vals = r_[r[rN-1:0:-1], c]
cols = mgrid[0:cN]
rows = mgrid[rN:0:-1]
indx = cols[:,NewAxis]*ones((1,rN)) + \
rows[NewAxis,:]*ones((cN,1)) - 1
return take(vals, indx)
def hankel(c,r=None):
""" Construct a hankel matrix (i.e. matrix with constant anti-diagonals).
Description:
hankel(c,r) is a Hankel matrix whose first column is c and whose
last row is r.
hankel(c) is a square Hankel matrix whose first column is C.
Elements below the first anti-diagonal are zero.
See also: toeplitz
"""
isscalar = scipy_base.isscalar
if isscalar(c) or isscalar(r):
return c
if r is None:
r = zeros(len(c))
elif r[0] != c[-1]:
print "Warning: column and row values don't agree; column value used."
r,c = map(asarray_chkfinite,(r,c))
r,c = map(ravel,(r,c))
rN,cN = map(len,(r,c))
vals = r_[c, r[1:rN]]
cols = mgrid[1:cN+1]
rows = mgrid[0:rN]
indx = cols[:,NewAxis]*ones((1,rN)) + \
rows[NewAxis,:]*ones((cN,1)) - 1
return take(vals, indx)
def all_mat(args):
return map(Matrix.Matrix,args)