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 rsf2csf(T, Z):
"""Convert real schur form to complex schur form.
Description:
If A is a real-valued matrix, then the real schur form is
quasi-upper triangular. 2x2 blocks extrude from the main-diagonal
corresponding to any complex-valued eigenvalues.
This function converts this real schur form to a complex schur form
which is upper triangular.
"""
Z,T = map(asarray_chkfinite, (Z,T))
if len(Z.shape) !=2 or Z.shape[0] != Z.shape[1]:
raise ValueError, "matrix must be square."
if len(T.shape) !=2 or T.shape[0] != T.shape[1]:
raise ValueError, "matrix must be square."
if T.shape[0] != Z.shape[0]:
raise ValueError, "matrices must be same dimension."
N = T.shape[0]
arr = scipy_base.array
t = _commonType(Z, T, arr([3.0],'F'))
Z, T = _castCopy(t, Z, T)
conj = scipy_base.conj
dot = scipy_base.dot
r_ = scipy_base.r_
transp = scipy_base.transpose
for m in range(N-1,0,-1):
if abs(T[m,m-1]) > eps*(abs(T[m-1,m-1]) + abs(T[m,m])):
k = slice(m-1,m+1)
mu = eigvals(T[k,k]) - T[m,m]
r = basic.norm([mu[0], T[m,m-1]])
c = mu[0] / r
s = T[m,m-1] / r
G = r_[arr([[conj(c),s]],typecode=t),arr([[-s,c]],typecode=t)]
Gc = conj(transp(G))
j = slice(m-1,N)
T[k,j] = dot(G,T[k,j])
i = slice(0,m+1)
T[i,k] = dot(T[i,k], Gc)
i = slice(0,N)
Z[i,k] = dot(Z[i,k], Gc)
T[m,m-1] = 0.0;
return T, Z
def pade(an, m):
"""Given Taylor series coefficients in an, return a Pade approximation to
the function as the ratio of two polynomials p / q where the order of q is m.
"""
an = asarray(an)
N = len(an) - 1
n = N-m
if (n < 0):
raise ValueError, \
"Order of q <m> must be smaller than len(an)-1."
Akj = eye(N+1,n+1)
Bkj = zeros((N+1,m),'d')
for row in range(1,m+1):
Bkj[row,:row] = -(an[:row])[::-1]
for row in range(m+1,N+1):
Bkj[row,:] = -(an[row-m:row])[::-1]
C = hstack((Akj,Bkj))
pq = dot(linalg.inv(C),an)
p = pq[:n+1]
q = r_[1.0,pq[n+1:]]
return poly1d(p[::-1]), poly1d(q[::-1])
def type1(self, x):
return sb.dot(x, self.obj)