intnumgauss

Function: intnumgauss
Section: sums
C-Name: intnumgauss0
Prototype: V=GGEDGp
Help: intnumgauss(X=a,b,expr,{tab}): numerical integration of expr from
 a to b, a compact interval, with respect to X using Gauss-Legendre
 quadrature. tab is either omitted (and will be recomputed) or
 precomputed with intnumgaussinit.
Wrapper: (,,G)
Description:
  (gen,gen,gen,?gen):gen:prec intnumgauss(${3 cookie}, ${3 wrapper}, $1, $2, $4, $prec)
Doc: numerical integration of \var{expr} on the compact interval $[a,b]$ with
 respect to $X$ using Gauss-Legendre quadrature; \kbd{tab} is either omitted
 or precomputed with \kbd{intnumgaussinit}. As a convenience, it can be an
 integer $n$ in which case we call
 \kbd{intnumgaussinit}$(n)$ and use $n$-point quadrature.
 \bprog
 ? test(n, b = 1) = T=intnumgaussinit(n);\
     intnumgauss(x=-b,b, 1/(1+x^2),T) - 2*atan(b);
 ? test(0) \\ default
 %1 = -9.490148553624725335 E-22
 ? test(40)
 %2 = -6.186629001816965717 E-31
 ? test(50)
 %3 = -1.1754943508222875080 E-38
 ? test(50, 2) \\ double interval length
 %4 = -4.891779568527713636 E-21
 ? test(90, 2) \\ n must almost be doubled as well!
 %5 = -9.403954806578300064 E-38
 @eprog\noindent On the other hand, we recommend to split the integral
 and change variables rather than increasing $n$ too much:
 \bprog
 ? f(x) = 1/(1+x^2);
 ? b = 100;
 ? intnumgauss(x=0,1, f(x)) + intnumgauss(x=1,1/b, f(1/x)*(-1/x^2)) - atan(b)
 %3 = -1.0579449157400587572 E-37
 @eprog