Integration of periodic functions

[Jacques Gélinas <jacquesg00@hotmail.com>]

Responding to http://pari.math.u-bordeaux.fr/archives/pari-dev-1812/msg00003.html

For periodic functions integrated over a period, the trapezoidal or midpoint methods can

give much more accurate results than other numerical integration methods.

Compare, for example, the number of bits LOST in

exr(x,y) = [exponent( if(!x,y,1-y/x) ), 0] - exponent(0.)*[1,1];

{

localbitprec(64*4);

rm = intnumromb(X=0,2*Pi,sin(X)/(5+3*sin(X)));

de = intnum(X=0,2*Pi,sin(X)/(5+3*sin(X)));

exr(-Pi/6,rm) == [9,256] && exr(-Pi/6,de) == [158,256]

}

This carries over to infinite intervals, as pointed out by Alan Turing

http://www.turingarchive.org/viewer/?id=471&title=4

where “ the method of Romberg does rather spoil than improve that

excellent convergence” of the trapezoidal method.

(Bauer+Rutishauser+Stiefel 1963 New aspects in numerical quadrature)

Ref: Weideman 2002 Numerical Integration of Periodic Functions - A Few Examples (AMM 109)

Jacques Gélinas