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] = conjugate(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 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 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."
