def central_diff_weights(Np,ndiv=1):
"""Return weights for an Np-point central derivative of order ndiv
assuming equally-spaced function points.
If weights are in the vector w, then
derivative is w[0] * f(x-ho*dx) + ... + w[-1] * f(x+h0*dx)
Can be inaccurate for large number of points.
"""
assert (Np >= ndiv+1), "Number of points must be at least the derivative order + 1."
assert (Np % 2 == 1), "Odd-number of points only."
ho = Np >> 1
x = arange(-ho,ho+1.0)
x = x[:,NewAxis]
X = x**0.0
for k in range(1,Np):
X = hstack([X,x**k])
w = product(arange(1,ndiv+1))*linalg.inv(X)[ndiv]
return w
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])