Function: intnumgaussinit
Section: sums
C-Name: intnumgaussinit
Prototype: D0,L,p
Help: intnumgaussinit({n}): initialize tables for n-point Gauss-Legendre
 integration on a compact interval.
Doc: initialize tables for $n$-point Gauss-Legendre integration of
 a smooth function $f$ lon a compact
 interval $[a,b]$ at current \kbd{realprecision}. If $n$ is omitted, make a
 default choice $n \approx \kbd{realprecision}$, suitable for analytic
 functions on $[-1,1]$. The error is bounded by
 $$
    \dfrac{(b-a)^{2n+1} (n!)^4}{(2n+1)[(2n)!]^3} f^{(2n)} (\xi) ,
    \qquad a < \xi < b
 $$
 so, if the interval length increases, $n$ should be increased as well.
 \bprog
 ? T = intnumgaussinit();
 ? intnumgauss(t=-1,1,exp(t), T) - exp(1)+exp(-1)
 %1 = -5.877471754111437540 E-39
 ? intnumgauss(t=-10,10,exp(t), T) - exp(10)+exp(-10)
 %2 = -8.358367809712546836 E-35
 ? intnumgauss(t=-1,1,1/(1+t^2), T) - Pi/2
 %3 = -9.490148553624725335 E-22

 ? T = intnumgaussinit(50);
 ? intnumgauss(t=-1,1,1/(1+t^2), T) - Pi/2
 %5 = -1.1754943508222875080 E-38
 ? intnumgauss(t=-5,5,1/(1+t^2), T) - 2*atan(5)
 %6 = -1.2[...]E-8
 @eprog
 On the other hand, we recommend to split the integral and change variables
 rather than increasing $n$ too much, see \tet{intnumgauss}.