Task: Numerical integration_Gauss-Legendre Quadrature
C: #include <stdio.h>
#include <math.h>

#define N 5
double Pi;
double lroots[N];
double weight[N];
double lcoef[N + 1][N + 1] = {{0}};

void lege_coef()
{
	int n, i;
	lcoef[0][0] = lcoef[1][1] = 1;
	for (n = 2; n <= N; n++) {
		lcoef[n][0] = -(n - 1) * lcoef[n - 2][0] / n;
		for (i = 1; i <= n; i++)
			lcoef[n][i] = ((2 * n - 1) * lcoef[n - 1][i - 1]
					 - (n - 1) * lcoef[n - 2][i] ) / n;
	}
}

double lege_eval(int n, double x)
{
	int i;
	double s = lcoef[n][n];
	for (i = n; i; i--)
		s = s * x + lcoef[n][i - 1];
	return s;
}

double lege_diff(int n, double x)
{
	return n * (x * lege_eval(n, x) - lege_eval(n - 1, x)) / (x * x - 1);
}

void lege_roots()
{
	int i;
	double x, x1;
	for (i = 1; i <= N; i++) {
		x = cos(Pi * (i - .25) / (N + .5));
		do {
			x1 = x;
			x -= lege_eval(N, x) / lege_diff(N, x);
		} while ( fdim( x, x1) > 2e-16 );
		
		lroots[i - 1] = x;

		x1 = lege_diff(N, x);
		weight[i - 1] = 2 / ((1 - x * x) * x1 * x1);
	}
}

double lege_inte(double (*f)(double), double a, double b)
{
	double c1 = (b - a) / 2, c2 = (b + a) / 2, sum = 0;
	int i;
	for (i = 0; i < N; i++)
		sum += weight[i] * f(c1 * lroots[i] + c2);
	return c1 * sum;
}

int main()
{
	int i;
	Pi = atan2(1, 1) * 4;

	lege_coef();
	lege_roots();

	printf("Roots: ");
	for (i = 0; i < N; i++)
		printf(" %g", lroots[i]);

	printf("\nWeight:");
	for (i = 0; i < N; i++)
		printf(" %g", weight[i]);

	printf("\nintegrating Exp(x) over [-3, 3]:\n\t%10.8f,\n"
		"compred to actual\n\t%10.8f\n",
		lege_inte(exp, -3, 3), exp(3) - exp(-3));
	return 0;
}

Go: package main

import (
    "fmt"
    "math"
)


type cFunc func(float64) float64

func main() {
    fmt.Println("integral:", glq(math.Exp, -3, 3, 5))
}





func glq(f cFunc, a, b float64, n int) float64 {
    x, w := glqNodes(n, f)
    show := func(label string, vs []float64) {
        fmt.Printf("%8s: ", label)
        for _, v := range vs {
            fmt.Printf("%8.5f ", v)
        }
        fmt.Println()
    }
    show("nodes", x)
    show("weights", w)
    var sum float64
    bma2 := (b - a) * .5
    bpa2 := (b + a) * .5
    for i, xi := range x {
        sum += w[i] * f(bma2*xi+bpa2)
    }
    return bma2 * sum
}





func glqNodes(n int, f cFunc) (node []float64, weight []float64) {
    p := legendrePoly(n)
    pn := p[n]
    n64 := float64(n)
    dn := func(x float64) float64 {
        return (x*pn(x) - p[n-1](x)) * n64 / (x*x - 1)
    }
    node = make([]float64, n)
    for i := range node {
        x0 := math.Cos(math.Pi * (float64(i+1) - .25) / (n64 + .5))
        node[i] = newtonRaphson(pn, dn, x0)
    }
    weight = make([]float64, n)
    for i, x := range node {
        dnx := dn(x)
        weight[i] = 2 / ((1 - x*x) * dnx * dnx)
    }
    return
}




func legendrePoly(n int) []cFunc {
    r := make([]cFunc, n+1)
    r[0] = func(float64) float64 { return 1 }
    r[1] = func(x float64) float64 { return x }
    for i := 2; i <= n; i++ {
        i2m1 := float64(i*2 - 1)
        im1 := float64(i - 1)
        rm1 := r[i-1]
        rm2 := r[i-2]
        invi := 1 / float64(i)
        r[i] = func(x float64) float64 {
            return (i2m1*x*rm1(x) - im1*rm2(x)) * invi
        }
    }
    return r
}





func newtonRaphson(f, df cFunc, x0 float64) float64 {
    for i := 0; i < 30; i++ {
        x1 := x0 - f(x0)/df(x0)
        if math.Abs(x1-x0) <= math.Abs(x0*1e-15) {
            return x1
        }
        x0 = x1
    }
    panic("no convergence")
}