Re: Integration of periodic functions

[Brad Klee <bradklee@gmail.com>]

> For periodic functions integrated over a period, the trapezoidal or midpoint methods can
> give much more accurate results than other numerical integration methods.

This may be true with sufficiently simple functions, but in general if
curvature is not
equally distributed, then an accurate algorithm would need to account
for statistical
variation by deciding on an unequal partition of the domain.

If it's possible to calculate the Fourier transform and the
coefficients decrease
regularly, then we can write a 100% accurate summation and control precision
with a truncation parameter. In this particular case, try:

1.0*sum(n=1,100,binomial(2*n,n)*(-12/10)*(-3/10)^(2*n-1))

Another technique for arbitrary precision is to quadrisect the
function on intervals
[ n*Pi/2 , (n+1)*Pi/2 ], n=0,1,2,3; construct polynomial
representations on each part,
and sum the integrals of all monomials. On an earlier thread, I gave
an example for
function sinc(x)^2 :

> http://pari.math.u-bordeaux.fr/archives/pari-users-1810/msg00037.html
> > http://pari.math.u-bordeaux.fr/archives/pari-users-1810/msg00041.html

The Polya walk coefficients are a better example for geometric-numeric
integration,
because the integrand reaches a pole on the real domain. See for example:

http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
https://oeis.org/A086232
https://oeis.org/A039699

Cheers,

Brad