I have a function for with i need to do an infinite summation on (over all the integers) numerically. The summation doesn't always need to converge as I can change internal parameters. The function looks like,
m(g, x, q0) = sum(abs(g(x - n*q0))^2 for n in Integers)
m(g, q0) = minimize(m(g, x, q0) for x in [0, q0])
using a Pythonic pseudo-code
Using Scipy integration methods, I was just flooring the n and integrating like for a fixed x,
m(g, z, q0) = integrate.quad(lambda n:
abs(g(x - int(n)*q0))**2,
-inf, +inf)[0]
This works pretty well, but then I have to do optimization on the x as a function of x, and then do another summation on that which yields a integral of a optimization of an integral. Pretty much it takes a really long time.
Do you know of a better way to do the summation that is faster? Hand coding it seemed to go slower.
Currently, I am working with
g(x) = (2/sqrt(3))*pi**(-0.25)*(1 - x**2)*exp(-x**2/2)
but the solution should be general
The paper this comes from is "The Wavelet Transform, Time-Frequency Localization and Signal Analysis" by Daubechies (IEEE 1990)
Thank you
Thanks to all the useful comment, I wrote my own summator that seems to run pretty fast. It anyone has any recommendations to make it better, I will gladly take them.
I will test this on the problem I am working on and once it demonstrates success, I will claim it functional.
def integers(blk_size=100):
x = arange(0, blk_size)
while True:
yield x
yield -x -1
x += blk_size
#
# For convergent summation
# on not necessarily finite sequences
# processes in blocks which can be any size
# shape that the function can handle
#
def converge_sum(f, x_strm, eps=1e-5, axis=0):
total = sum(f(x_strm.next()), axis=axis)
for x_blk in x_strm:
diff = sum(f(x_blk), axis=axis)
if abs(linalg.norm(diff)) <= eps:
# Converged
return total + diff
else:
total += diff