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 derivative(func,x0,dx=1.0,n=1,args=(),order=3):
"""Given a function, use a central difference formula with spacing dx to
compute the nth derivative at x0.
order is the number of points to use and must be odd.
Warning: Decreasing the step size too small can result in
round-off error.
"""
assert (order >= n+1), "Number of points must be at least the derivative order + 1."
assert (order % 2 == 1), "Odd number of points only."
# pre-computed for n=1 and 2 and low-order for speed.
if n==1:
if order == 3:
weights = array([-1,0,1])/2.0
elif order == 5:
weights = array([1,-8,0,8,-1])/12.0
elif order == 7:
weights = array([-1,9,-45,0,45,-9,1])/60.0
elif order == 9:
weights = array([3,-32,168,-672,0,672,-168,32,-3])/840.0
else:
weights = central_diff_weights(order,1)
elif n==2:
if order == 3:
weights = array([1,-2.0,1])
elif order == 5:
weights = array([-1,16,-30,16,-1])/12.0
elif order == 7:
weights = array([2,-27,270,-490,270,-27,2])/180.0
elif order == 9:
weights = array([-9,128,-1008,8064,-14350,8064,-1008,128,-9])/5040.0
else:
weights = central_diff_weights(order,2)
else:
weights = central_diff_weights(order, n)
val = 0.0
ho = order >> 1
for k in range(order):
val += weights[k]*func(x0+(k-ho)*dx,*args)
return val / product((dx,)*n)