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 $f$ on a compact interval $[a,b]$. If $n$ is omitted, make a
 default choice $n \approx B / 4$, where $B$ is
 \kbd{realbitprecision}, suitable for analytic functions on $[-1,1]$.
 The error is bounded by
 $$
    \dfrac{(b-a)^{2n+1} (n!)^{4}}{(2n+1)!(2n)!} \dfrac{f^{(2n)}}{(2n)!} (\xi) ,
    \qquad a < \xi < b.
 $$
 If $r$ denotes the distance of the nearest pole to the interval $[a,b]$,
 then this is of the order of $((b-a) / (4r))^{2n}$. In particular, the
 integral must be subdivided if the interval length $b - a$ becomes close to
 $4r$. The default choice $n \approx B / 4$ makes this quantity of order
 $2^{-B}$ when $b - a = r$, as is the case when integrating $1/(1+t)$ on
 $[0,1]$ for instance. If the interval length increases, $n$ should be
 increased as well.

 Specifically, the function returns a pair of vectors $[x,w]$, where $x$
 contains the nonnegative roots of the $n$-th Legendre polynomial $P_{n}$ and
 $w$ the corresponding Gaussian integration weights
 $Q_{n}(x_{j})/P'_{n}(x_{j}) = 2 / ((1-x_{j}^{2})P'_{n}(x_{j}))^{2}$  such that
 $$ \int_{-1}^{1} f(t)\, dt \approx \sum_{j} w_{j} f(x_{j})\;. $$

 \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 \\ b - a = 2r
 %3 = -9.490148553624725335 E-22 \\ ... loses half the accuracy

 ? 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}.
