You are encouraged to solve this task according to the task description, using any language you may know.
 In a general Gaussian quadrature rule, an definite integral of ${\displaystyle f(x)}$ is first approximated over the interval ${\displaystyle [-1,1]}$ by a polynomial approximable function ${\displaystyle g(x)}$ and a known weighting function ${\displaystyle W(x)}$. ${\displaystyle \int _{-1}^{1}f(x)\,dx=\int _{-1}^{1}W(x)g(x)\,dx}$ Those are then approximated by a sum of function values at specified points ${\displaystyle x_{i}}$ multiplied by some weights ${\displaystyle w_{i}}$: ${\displaystyle \int _{-1}^{1}W(x)g(x)\,dx\approx \sum _{i=1}^{n}w_{i}g(x_{i})}$ In the case of Gauss-Legendre quadrature, the weighting function ${\displaystyle W(x)=1}$, so we can approximate an integral of ${\displaystyle f(x)}$ with: ${\displaystyle \int _{-1}^{1}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})}$

For this, we first need to calculate the nodes and the weights, but after we have them, we can reuse them for numerious integral evaluations, which greatly speeds up the calculation compared to more simple numerical integration methods.

 The ${\displaystyle n}$ evaluation points ${\displaystyle x_{i}}$ for a n-point rule, also called "nodes", are roots of n-th order Legendre Polynomials ${\displaystyle P_{n}(x)}$. Legendre polynomials are defined by the following recursive rule: ${\displaystyle P_{0}(x)=1}$ ${\displaystyle P_{1}(x)=x}$ ${\displaystyle nP_{n}(x)=(2n-1)xP_{n-1}(x)-(n-1)P_{n-2}(x)}$ There is also a recursive equation for their derivative: ${\displaystyle P_{n}'(x)={\frac {n}{x^{2}-1}}\left(xP_{n}(x)-P_{n-1}(x)\right)}$ The roots of those polynomials are in general not analytically solvable, so they have to be approximated numerically, for example by Newton-Raphson iteration: ${\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}}$ The first guess ${\displaystyle x_{0}}$ for the ${\displaystyle i}$-th root of a ${\displaystyle n}$-order polynomial ${\displaystyle P_{n}}$ can be given by ${\displaystyle x_{0}=\cos \left(\pi \,{\frac {i-{\frac {1}{4}}}{n+{\frac {1}{2}}}}\right)}$ After we get the nodes ${\displaystyle x_{i}}$, we compute the appropriate weights by: ${\displaystyle w_{i}={\frac {2}{\left(1-x_{i}^{2}\right)[P'_{n}(x_{i})]^{2}}}}$ After we have the nodes and the weights for a n-point quadrature rule, we can approximate an integral over any interval ${\displaystyle [a,b]}$ by ${\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{2}}\sum _{i=1}^{n}w_{i}f\left({\frac {b-a}{2}}x_{i}+{\frac {a+b}{2}}\right)}$

Similar to the task Numerical Integration, the task here is to calculate the definite integral of a function ${\displaystyle f(x)}$, but by applying an n-point Gauss-Legendre quadrature rule, as described here, for example. The input values should be an function f to integrate, the bounds of the integration interval a and b, and the number of gaussian evaluation points n. An reference implementation in Common Lisp is provided for comparison.

To demonstrate the calculation, compute the weights and nodes for an 5-point quadrature rule and then use them to compute:

         ${\displaystyle \int _{-3}^{3}\exp(x)\,dx\approx \sum _{i=1}^{5}w_{i}\;\exp(x_{i})\approx 20.036}$


## 11l

Translation of: Nim
F legendreIn(x, n)
F prev1(idx, pn1)
R (2 * idx - 1) * @x * pn1
F prev2(idx, pn2)
R (idx - 1) * pn2

I n == 0
R 1.0
E I n == 1
R x
E
V result = 0.0
V p1 = x
V p2 = 1.0
L(i) 2 .. n
result = (prev1(i, p1) - prev2(i, p2)) / i
p2 = p1
p1 = result
R result

F deriveLegendreIn(x, n)
F calcresult(curr, prev)
R Float(@n) / (@x ^ 2 - 1) * (@x * curr - prev)
R calcresult(legendreIn(x, n), legendreIn(x, n - 1))

F guess(n, i)
R cos(math:pi * (i - 0.25) / (n + 0.5))

F nodes(n)
V result = [(0.0, 0.0)] * n
F calc(x)
R legendreIn(x, @n) / deriveLegendreIn(x, @n)

L(i) 0 .< n
V x = guess(n, i + 1)
V x0 = x
x -= calc(x)
L abs(x - x0) > 1e-12
x0 = x
x -= calc(x)

result[i] = (x, 2 / ((1.0 - x ^ 2) * (deriveLegendreIn(x, n)) ^ 2))

R result

F integ(f, ns, p1, p2)
F dist()
R (@p2 - @p1) / 2
F avg()
R (@p1 + @p2) / 2
V result = dist()
V sum = 0.0
V thenodes = [0.0] * ns
V weights  = [0.0] * ns
L(nw) nodes(ns)
sum += nw[1] * f(dist() * nw[0] + avg())
thenodes[L.index] = nw[0]
weights[L.index] = nw[1]

print(‘   nodes:’, end' ‘’)
L(n) thenodes
print(‘ #.5’.format(n), end' ‘’)
print()
print(‘ weights:’, end' ‘’)
L(w) weights
print(‘ #.5’.format(w), end' ‘’)
print()
R result * sum

print(‘integral: ’integ(x -> exp(x), 5, -3, 3))
Output:
   nodes: 0.90618 0.53847 0.00000 -0.53847 -0.90618
weights: 0.23693 0.47863 0.56889 0.47863 0.23693
integral: 20.035577718


## ATS

Translation of: Common Lisp

This is a very close translation of the Common Lisp.

(A lot of the "ATS-ism" is completely optional. For instance, you can use arrszref instead of arrayref, if you want bounds checking at runtime instead of compile-time. But then debugging and regression-prevention become harder, and in that particular case the code will almost surely be slower.

And, if I may grumble a bit: Some of us do not think "turning off bounds checking for production" is acceptable. It is at best something to tolerate grudgingly.)

#include "share/atspre_staload.hats"

%{^
#include <float.h>
#include <math.h>
%}

extern fn {tk : tkind} g0float_pi : () -<> g0float tk
extern fn {tk : tkind} g0float_cos : g0float tk -<> g0float tk
extern fn {tk : tkind} g0float_exp : g0float tk -<> g0float tk
implement g0float_pi<dblknd> () = $extval (double, "M_PI") implement g0float_cos<dblknd> x =$extfcall (double, "cos", x)
implement g0float_exp<dblknd> x = $extfcall (double, "exp", x) macdef PI = g0float_pi () overload cos with g0float_cos overload exp with g0float_exp macdef NAN = g0f2f ($extval (float, "NAN"))
macdef Zero = g0i2f 0
macdef One = g0i2f 1
macdef Two = g0i2f 2

(* Computes the initial guess for the root i of a n-order Legendre
polynomial. *)
fn {tk : tkind}
guess {n, i : int | 1 <= i; i <= n}
(n : int n, i : int i) :<> g0float tk =
cos (PI * ((g0i2f i - g0f2f 0.25) / (g0i2f n + g0f2f 0.5)))

(* Computes and evaluates the degree-n Legendre polynomial at the
point x. *)
fn {tk : tkind}
legpoly {n : pos}
(n : int n, x : g0float tk) :<> g0float tk =
let
fun
loop {i  : int | 2 <= i; i <= n + 1} .<n + 1 - i>.
(i  : int i, pa : g0float tk, pb : g0float tk)
:<> g0float tk =
if i = succ n then
pb
else
let
val iflt = (g0i2f i) : g0float tk
val pn = (((iflt + iflt - One) / iflt) * x * pb)
- (((iflt - One) / iflt) * pa)
in
loop (succ i, pb, pn)
end
in
if n = 0 then
One
else if n = 1 then
x
else
loop (2, One, x)
end

(* Computes and evaluates the derivative of an n-order Legendre
polynomial at point x. *)
fn {tk : tkind}
legdiff {n : int | 2 <= n}
(n : int n, x : g0float tk) :<> g0float tk =
(g0i2f n / ((x * x) - One))
* ((x * legpoly<tk> (n, x)) - legpoly<tk> (pred n, x))

(* Computes the n nodes for an n-point quadrature rule (the n roots of
a degree-n polynomial). *)
fn {tk : tkind}
nodes {n : int | 2 <= n}
(n : int n) :<!refwrt> arrayref (g0float tk, n) =
let
val x = arrayref_make_elt<g0float tk> (i2sz n, Zero)
fn
v_update (v : g0float tk) :<> g0float tk =
v - (legpoly<tk> (n, v) / legdiff<tk> (n, v))
var i : Int
in
for* {i : nat | i <= n} .<n - i>.
(i : int i) =>
(i := 0; i <> n; i := succ i)
let
val v = guess<tk> (n, succ i)
val v = v_update v
val v = v_update v
val v = v_update v
val v = v_update v
val v = v_update v
in
x[i] := v
end;
x
end

(* Computes the weight for an degree-n polynomial at the node x. *)
fn {tk : tkind}
legwts {n : int | 2 <= n}
(n : int n, x : g0float tk) :<> g0float tk =
(* Here I am having slightly excessive fun with notation: *)
Two / ((One - (x * x)) * (y * y where {val y = legdiff<tk> (n, x)}))
(* Normally I would not write code in such fashion. :) Nevertheless,
it is interesting that this works. *)

(* Takes an array of nodes x and computes an array of corresponding
weights w. Note that x is an arrayref, not an arrszref, and so
(unlike in the Common Lisp) we have to tell the function the size
of the new array w. That information is not otherwise stored AT
RUNTIME. The ATS compiler, however, will force us AT COMPILE TIME
to pass the correct size. *)
fn {tk : tkind}
weights {n : int | 2 <= n}
(n : int n, x : arrayref (g0float tk, n))
:<!refwrt> arrayref (g0float tk, n) =
let
val w = arrayref_make_elt<g0float tk> (i2sz n, Zero)
var i : Int
in
for* {i : nat | i <= n} .<n - i>.
(i : int i) =>
(i := 0; i <> n; i := succ i)
w[i] := legwts (n, x[i]);
w
end

(* Estimates the definite integral of a function on [a,b], using an
fn {tk : tkind}
quad {n : int | 2 <= n}
(f : g0float tk -<> g0float tk,
n : int n,
a : g0float tk,
b : g0float tk) :<> g0float tk =
let
val x = $effmask_ref ($effmask_wrt (nodes<tk> n))
val w = $effmask_ref ($effmask_wrt (weights<tk> (n, x)))

val ahalf = g0f2f 0.5 * a and bhalf = g0f2f 0.5 * b
val C1 = bhalf - ahalf and C2 = ahalf + bhalf

fun
loop {i : nat | i <= n} .<n - i>.
(i : int i, sum : g0float tk) :<> g0float tk =
if i = n then
sum
else
let
val y = $effmask_ref (w[i] * f ((C1 * x[i]) + C2)) in loop (succ i, sum + y) end in C1 * loop (0, Zero) end implement main () = let val outf = stdout_ref in fprintln! (outf, "nodes<dblknd> 5"); fprint_arrayref_sep<double> (outf, nodes<dblknd> (5), i2sz 5, " "); fprintln! (outf); fprintln! (outf); fprintln! (outf, "weights (nodes<dblknd> 5)"); fprint_arrayref_sep<double> (outf, weights (5, nodes<dblknd> (5)), i2sz 5, " "); fprintln! (outf); fprintln! (outf); fprintln! (outf, "quad (lam x => exp x, 5, ~3.0, 3.0) = ", quad (lam x => exp x, 5, ~3.0, 3.0)); fprintln! (outf); fprintln! (outf, "More examples, borrowed from the Common Lisp:"); fprintln! (outf, "quad (lam x => x ** 3, 5, 0.0, 1.0) = ", quad (lam x => x ** 3, 5, 0.0, 1.0)); fprintln! (outf, "quad (lam x => 1.0 / x, 5, 1.0, 100.0) = ", quad (lam x => 1.0 / x, 5, 1.0, 100.0)); fprintln! (outf, "quad (lam x => x, 5, 0.0, 5000.0) = ", quad (lam x => x, 5, 0.0, 5000.0)); fprintln! (outf, "quad (lam x => x, 5, 0.0, 6000.0) = ", quad (lam x => x, 5, 0.0, 6000.0)); 0 end Output: $ patscc -std=gnu2x -g -O2 -DATS_MEMALLOC_GCBDW gauss_legendre_task.dats -lgc -lm && ./a.out
nodes<dblknd> 5
0.906180 0.538469 0.000000 -0.538469 -0.906180

weights (nodes<dblknd> 5)
0.236927 0.478629 0.568889 0.478629 0.236927

quad (lam x => exp x, 5, ~3.0, 3.0) = 20.035578

More examples, borrowed from the Common Lisp:
quad (lam x => x ** 3, 5, 0.0, 1.0) = 0.250000
quad (lam x => 1.0 / x, 5, 1.0, 100.0) = 4.059148
quad (lam x => x, 5, 0.0, 5000.0) = 12500000.000000
quad (lam x => x, 5, 0.0, 6000.0) = 18000000.000000

## Axiom

Translation of: Maxima

Axiom provides Legendre polynomials and related solvers.

NNI ==> NonNegativeInteger
RECORD ==> Record(x : List Fraction Integer, w : List Fraction Integer)

gaussCoefficients(n : NNI, eps : Fraction Integer) : RECORD ==
p := legendreP(n,z)
q := n/2*D(p, z)*legendreP(subtractIfCan(n,1)::NNI, z)
x := map(rhs,solve(p,eps))
w := [subst(1/q, z=xi) for xi in x]
[x,w]

gaussIntegrate(e : Expression Float, segbind : SegmentBinding(Float), n : NNI) : Float ==
eps := 1/10^100
u := gaussCoefficients(n,eps)
interval := segment segbind
var := variable segbind
a := lo interval
b := hi interval
c := (a+b)/2
h := (b-a)/2
h*reduce(+,[wi*subst(e,var=c+xi*h) for xi in u.x for wi in u.w])

Example:

digits(50)
gaussIntegrate(4/(1+x^2), x=0..1, 20)

(1)  3.1415926535_8979323846_2643379815_9534002592_872901276
Type: Float
% - %pi

(2)  - 0.3463549483_9378821092_475 E -26

## 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 );
/*  fdim( ) was introduced in C99, if it isn't available
*  on your system, try fabs( ) */
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;
}

Output:
Roots:  0.90618 0.538469 0 -0.538469 -0.90618
Weight: 0.236927 0.478629 0.568889 0.478629 0.236927
integrating Exp(x) over [-3, 3]:
20.03557772,
compred to actual
20.03574985

## C++

Derived from various sources already here.

Does not quite perform the task quite as specified since the node count, N, is set at compile time (to avoid heap allocation) so cannot be passed as a parameter.

#include <iostream>
#include <iomanip>
#include <cmath>

namespace Rosetta {

*
*/
template <int N>
public:
enum {eDEGREE = N};

/*! Compute the integral of a functor
*
*   @param a    lower limit of integration
*   @param b    upper limit of integration
*   @param f    the function to integrate
*   @param err  callback in case of problems
*/
template <typename Function>
double integrate(double a, double b, Function f) {
double p = (b - a) / 2;
double q = (b + a) / 2;
const LegendrePolynomial& legpoly = s_LegendrePolynomial;

double sum = 0;
for (int i = 1; i <= eDEGREE; ++i) {
sum += legpoly.weight(i) * f(p * legpoly.root(i) + q);
}

return p * sum;
}

/*! Print out roots and weights for information
*/
void print_roots_and_weights(std::ostream& out) const {
const LegendrePolynomial& legpoly = s_LegendrePolynomial;
out << "Roots:  ";
for (int i = 0; i <= eDEGREE; ++i) {
out << ' ' << legpoly.root(i);
}
out << std::endl;
out << "Weights:";
for (int i = 0; i <= eDEGREE; ++i) {
out << ' ' << legpoly.weight(i);
}
out << std::endl;
}
private:
/*! Implementation of the Legendre polynomials that form
*   the basis of this quadrature
*/
class LegendrePolynomial {
public:
LegendrePolynomial () {
// Solve roots and weights
for (int i = 0; i <= eDEGREE; ++i) {
double dr = 1;

// Find zero
Evaluation eval(cos(M_PI * (i - 0.25) / (eDEGREE + 0.5)));
do {
dr = eval.v() / eval.d();
eval.evaluate(eval.x() - dr);
} while (fabs (dr) > 2e-16);

this->_r[i] = eval.x();
this->_w[i] = 2 / ((1 - eval.x() * eval.x()) * eval.d() * eval.d());
}
}

double root(int i) const { return this->_r[i]; }
double weight(int i) const { return this->_w[i]; }
private:
double _r[eDEGREE + 1];
double _w[eDEGREE + 1];

/*! Evaluate the value *and* derivative of the
*   Legendre polynomial
*/
class Evaluation {
public:
explicit Evaluation (double x) : _x(x), _v(1), _d(0) {
this->evaluate(x);
}

void evaluate(double x) {
this->_x = x;

double vsub1 = x;
double vsub2 = 1;
double f     = 1 / (x * x - 1);

for (int i = 2; i <= eDEGREE; ++i) {
this->_v = ((2 * i - 1) * x * vsub1 - (i - 1) * vsub2) / i;
this->_d = i * f * (x * this->_v - vsub1);

vsub2 = vsub1;
vsub1 = this->_v;
}
}

double v() const { return this->_v; }
double d() const { return this->_d; }
double x() const { return this->_x; }

private:
double _x;
double _v;
double _d;
};
};

/*! Pre-compute the weights and abscissae of the Legendre polynomials
*/
static LegendrePolynomial s_LegendrePolynomial;
};

template <int N>
}

// This to avoid issues with exp being a templated function
double RosettaExp(double x) {
return exp(x);
}

int main() {

std::cout << std::setprecision(10);

gl5.print_roots_and_weights(std::cout);
std::cout << "Integrating Exp(X) over [-3, 3]: " << gl5.integrate(-3., 3., RosettaExp) << std::endl;
std::cout << "Actual value:                    " << RosettaExp(3) - RosettaExp(-3) << std::endl;
}

Output:
Roots:   0.9061798459 0.9061798459 0.5384693101 0 -0.5384693101 -0.9061798459
Weights: 0.2369268851 0.2369268851 0.4786286705 0.5688888889 0.4786286705 0.2369268851
Integrating Exp(X) over [-3, 3]: 20.03557772
Actual value:                    20.03574985


## C#

Derived from the C++ and Java versions here.

using System;
//Works in .NET 6+
//Tested using https://dotnetfiddle.net because im lazy

public class Program {

public static double[][] legeCoef(int N) {
//Initialising Jagged Array
double[][] lcoef = new double[N+1][];
for (int i=0; i < lcoef.Length; ++i)
lcoef[i] = new double[N+1];

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

static double legeEval(double[][] lcoef, int N, double x) {
double s = lcoef[N][N];
for (int i = N; i > 0; --i)
s = s * x + lcoef[N][i-1];
return s;
}

static double legeDiff(double[][] lcoef, int N, double x) {
return N * (x * legeEval(lcoef, N, x) - legeEval(lcoef, N-1, x)) / (x*x - 1);
}

static void legeRoots(double[][] lcoef, int N, out double[] lroots,  out double[] weight) {
lroots = new double[N];
weight = new double[N];

double x, x1;
for (int i = 1; i <= N; i++) {
x = Math.Cos(Math.PI * (i - 0.25) / (N + 0.5));
do {
x1 = x;
x -= legeEval(lcoef, N, x) / legeDiff(lcoef, N, x);
}
while (x != x1);
lroots[i-1] = x;

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

static double legeInte(Func<Double, Double> f, int N, double[] weights, double[] lroots, double a, double b) {
double c1 = (b - a) / 2, c2 = (b + a) / 2, sum = 0;
for (int i = 0; i < N; i++)
sum += weights[i] * f.Invoke(c1 * lroots[i] + c2);
return c1 * sum;
}

//..................Main...............................
public static string Combine(double[] arrayD) {
return string.Join(", ", arrayD);
}

public static void Main() {
int N = 5;

var lcoeff = legeCoef(N);

double[] roots;
double[] weights;
legeRoots(lcoeff, N, out roots, out weights);

var integrateResult = legeInte(x=>Math.Exp(x), N, weights, roots, -3, 3);

Console.WriteLine("Roots:   " + Combine(roots));
Console.WriteLine("Weights: " + Combine(weights)+ "\n" );
Console.WriteLine("integral: " + integrateResult );
Console.WriteLine("actual:   " + (Math.Exp(3)-Math.Exp(-3)) );
}

}

Output:
Roots:   0.906179845938664, 0.538469310105683, 0, -0.538469310105683, -0.906179845938664
Weights: 0.236926885056189, 0.478628670499367, 0.568888888888889, 0.478628670499367, 0.236926885056189

integral: 20.0355777183856
actual:   20.0357498548198


## Common Lisp

;; Computes the initial guess for the root i of a n-order Legendre polynomial.
(defun guess (n i)
(cos (* pi
(/ (- i 0.25d0)
(+ n 0.5d0)))))

;; Computes and evaluates the n-order Legendre polynomial at the point x.
(defun legpoly (n x)
(let ((pa 1.0d0)
(pb x)
(pn))
(cond ((= n 0) pa)
((= n 1) pb)
(t (loop for i from 2 to n do
(setf pn (- (* (/ (- (* 2 i) 1) i) x pb)
(* (/ (- i 1) i) pa)))
(setf pa pb)
(setf pb pn)
finally (return pn))))))

;; Computes and evaluates the derivative of an n-order Legendre polynomial at point x.
(defun legdiff (n x)
(* (/ n (- (* x x) 1))
(- (* x (legpoly n x))
(legpoly (- n 1) x))))

;; Computes the n nodes for an n-point quadrature rule. (i.e. n roots of a n-order polynomial)
(defun nodes (n)
(let ((x (make-array n :initial-element 0.0d0)))
(loop for i from 0 to (- n 1) do
(let ((val (guess n (+ i 1))) ;Nullstellen-Schätzwert.
(itermax 5))
(dotimes (j itermax)
(setf val (- val
(/ (legpoly n val)
(legdiff n val)))))
(setf (aref x i) val)))
x))

;; Computes the weight for an n-order polynomial at the point (node) x.
(defun legwts (n x)
(/ 2
(* (- 1 (* x x))
(expt (legdiff n x) 2))))

;; Takes a array of nodes x and computes an array of corresponding weights w.
(defun weights (x)
(let* ((n (car (array-dimensions x)))
(w (make-array n :initial-element 0.0d0)))
(loop for i from 0 to (- n 1) do
(setf (aref w i) (legwts n (aref x i))))
w))

;; Integrates a function f with a n-point Gauss-Legendre quadrature rule over the interval [a,b].
(defun int (f n a b)
(let* ((x (nodes n))
(w (weights x)))
(* (/ (- b a) 2.0d0)
(loop for i from 0 to (- n 1)
sum (* (aref w i)
(funcall f (+ (* (/ (- b a) 2.0d0)
(aref x i))
(/ (+ a b) 2.0d0))))))))

Example:
(nodes 5)
#(0.906179845938664d0 0.5384693101056831d0 2.996272867003007d-95 -0.5384693101056831d0 -0.906179845938664d0)

(weights (nodes 5))
#(0.23692688505618917d0 0.47862867049936647d0 0.5688888888888889d0 0.47862867049936647d0 0.23692688505618917d0)

(int #'exp 5 -3 3)
20.035577718385568d0


Comparison of the 5-point rule with simpler, but more costly methods from the task Numerical Integration:

(int #'(lambda (x) (expt x 3)) 5 0 1)
0.24999999999999997d0

(int #'(lambda (x) (/ 1 x)) 5 1 100)
4.059147508941519d0

(int #'(lambda (x) x) 5 0 5000)
1.25d7

(int #'(lambda (x) x) 5 0 6000)
1.8000000000000004d7


## D

Translation of: C
import std.stdio, std.math;

size_t NBITS=50) {
immutable static double[N] lroots, weight;
alias FP[N + 1][N + 1] CoefMat;

pure nothrow @safe @nogc static this() {
static FP legendreEval(in ref FP[N + 1][N + 1] lcoef,
in int n, in FP x) pure nothrow {
FP s = lcoef[n][n];
foreach_reverse (immutable i; 1 .. n+1)
s = s * x + lcoef[n][i - 1];
return s;
}

static FP legendreDiff(in ref CoefMat lcoef,
in int n, in FP x)
pure nothrow @safe @nogc {
return n * (x * legendreEval(lcoef, n, x) -
legendreEval(lcoef, n - 1, x)) /
(x ^^ 2 - 1);
}

CoefMat lcoef = 0.0;
legendreCoefInit(/*ref*/ lcoef);

// Legendre roots:
foreach (immutable i; 1 .. N + 1) {
FP x = cos(PI * (i - 0.25) / (N + 0.5));
FP x1;
do {
x1 = x;
x -= legendreEval(lcoef, N, x) /
legendreDiff(lcoef, N, x);
} while (feqrel(x, x1) < NBITS);
lroots[i - 1] = x;
x1 = legendreDiff(lcoef, N, x);
weight[i - 1] = 2 / ((1 - x ^^ 2) * (x1 ^^ 2));
}
}

static private void legendreCoefInit(ref CoefMat lcoef)
pure nothrow @safe @nogc {
lcoef[0][0] = lcoef[1][1] = 1;
foreach (immutable int n; 2 .. N + 1) { // n must be signed.
lcoef[n][0] = -(n - 1) * lcoef[n - 2][0] / n;
foreach (immutable i; 1 .. n + 1)
lcoef[n][i] = ((2 * n - 1) * lcoef[n - 1][i - 1] -
(n - 1) * lcoef[n - 2][i]) / n;
}
}

static public FP integrate(in FP function(/*in*/ FP x) pure nothrow @safe @nogc f,
in FP a, in FP b)
pure nothrow @safe @nogc {
immutable FP c1 = (b - a) / 2;
immutable FP c2 = (b + a) / 2;
FP sum = 0.0;
foreach (immutable i; 0 .. N)
sum += weight[i] * f(c1 * lroots[i] + c2);
return c1 * sum;
}
}

void main() {
writeln("Roots:  ", glq.lroots);
writeln("Weight: ", glq.weight);
writefln("Integrating exp(x) over [-3, 3]: %10.12f",
glq.integrate(&exp, -3, 3));
writefln("Compred to actual:               %10.12f",
3.0.exp - exp(-3.0));
}

Output:
Roots:  [0.90618, 0.538469, 0, -0.538469, -0.90618]
Weight: [0.236927, 0.478629, 0.568889, 0.478629, 0.236927]
Integrating exp(x) over [-3, 3]: 20.035577718386
Compred to actual:               20.035749854820

## Delphi

program Legendre;

{$APPTYPE CONSOLE} const Order = 5; Epsilon = 1E-12; var Roots : array[0..Order-1] of double; Weight : array[0..Order-1] of double; LegCoef : array [0..Order,0..Order] of double; function F(X:double) : double; begin Result := Exp(X); end; procedure PrepCoef; var I, N : integer; begin for I:=0 to Order do for N := 0 to Order do LegCoef[I,N] := 0; LegCoef[0,0] := 1; LegCoef[1,1] := 1; For N:=2 to Order do begin LegCoef[N,0] := -(N-1) * LegCoef[N-2,0] / N; For I := 1 to Order do LegCoef[N,I] := ((2*N-1) * LegCoef[N-1,I-1] - (N-1)*LegCoef[N-2,I]) / N; end; end; function LegEval(N:integer; X:double) : double; var I : integer; begin Result := LegCoef[n][n]; for I := N-1 downto 0 do Result := Result * X + LegCoef[N][I]; end; function LegDiff(N:integer; X:double) : double; begin Result := N * (X * LegEval(N,X) - LegEval(N-1,X)) / (X*X-1); end; procedure LegRoots; var I : integer; X, X1 : double; begin for I := 1 to Order do begin X := Cos(Pi * (I-0.25) / (Order+0.5)); repeat X1 := X; X := X - LegEval(Order,X) / LegDiff(Order, X); until Abs (X-X1) < Epsilon; Roots[I-1] := X; X1 := LegDiff(Order,X); Weight[I-1] := 2 / ((1-X*X) * X1*X1); end; end; function LegInt(A,B:double) : double; var I : integer; C1, C2 : double; begin C1 := (B-A)/2; C2 := (B+A)/2; Result := 0; For I := 0 to Order-1 do Result := Result + Weight[I] * F(C1*Roots[I] + C2); Result := C1 * Result; end; var I : integer; begin PrepCoef; LegRoots; Write('Roots: '); for I := 0 to Order-1 do Write (' ',Roots[I]:13:10); Writeln; Write('Weight: '); for I := 0 to Order-1 do Write (' ', Weight[I]:13:10); writeln; Writeln('Integrating Exp(x) over [-3, 3]: ',LegInt(-3,3):13:10); Writeln('Actual value: ',Exp(3)-Exp(-3):13:10); Readln; end.  Roots: 0.9061798459 0.5384693101 0.0000000000 -0.5384693101 -0.9061798459 Weight: 0.2369268851 0.4786286705 0.5688888889 0.4786286705 0.2369268851 Integrating Exp(X) over [-3, 3]: 20.0355777184 Actual value: 20.0357498548  ## Fortran ! Works with gfortran but needs the option ! -assume realloc_lhs ! when compiled with Intel Fortran. program gauss implicit none integer, parameter :: p = 16 ! quadruple precision integer :: n = 10, k real(kind=p), allocatable :: r(:,:) real(kind=p) :: z, a, b, exact do n = 1,20 a = -3; b = 3 r = gaussquad(n) z = (b-a)/2*dot_product(r(2,:),exp((a+b)/2+r(1,:)*(b-a)/2)) exact = exp(3.0_p)-exp(-3.0_p) print "(i0,1x,g0,1x,g10.2)",n, z, z-exact end do contains function gaussquad(n) result(r) integer :: n real(kind=p), parameter :: pi = 4*atan(1._p) real(kind=p) :: r(2, n), x, f, df, dx integer :: i, iter real(kind = p), allocatable :: p0(:), p1(:), tmp(:) p0 = [1._p] p1 = [1._p, 0._p] do k = 2, n tmp = ((2*k-1)*[p1,0._p]-(k-1)*[0._p, 0._p,p0])/k p0 = p1; p1 = tmp end do do i = 1, n x = cos(pi*(i-0.25_p)/(n+0.5_p)) do iter = 1, 10 f = p1(1); df = 0._p do k = 2, size(p1) df = f + x*df f = p1(k) + x * f end do dx = f / df x = x - dx if (abs(dx)<10*epsilon(dx)) exit end do r(1,i) = x r(2,i) = 2/((1-x**2)*df**2) end do end function end program  n numerical integral error -------------------------------------------------- 1 6.00000000000000000000000000000000 -14. 2 17.4874646410555689643606840462449 -2.5 3 19.8536919968055821921309108927158 -.18 4 20.0286883952907008527738054439858 -.71E-02 5 20.0355777183855621539285357252751 -.17E-03 6 20.0357469750923438830654575585499 -.29E-05 7 20.0357498197266007755718729372892 -.35E-07 8 20.0357498544945172882260918041684 -.33E-09 9 20.0357498548174338368864419454859 -.24E-11 10 20.0357498548197898711175766908548 -.14E-13 11 20.0357498548198037305529147159695 -.67E-16 12 20.0357498548198037976759531014464 -.27E-18 13 20.0357498548198037979482458119095 -.94E-21 14 20.0357498548198037979491844483597 -.28E-23 15 20.0357498548198037979491872317190 -.72E-26 16 20.0357498548198037979491872388913 -.40E-28 17 20.0357498548198037979491872389166 -.15E-28 18 20.0357498548198037979491872389259 -.58E-29 19 20.0357498548198037979491872388910 -.41E-28 20 20.0357498548198037979491872388495 -.82E-28  ## FreeBASIC Translation of: Wren #define PI 4 * Atn(1) Const As Double LIM = 5 Dim Shared As Double lroots(LIM - 1) Dim Shared As Double weight(LIM - 1) Dim Shared As Double lcoef(LIM, LIM) For i As Integer = 0 To LIM For j As Integer = 0 To LIM lcoef(i, j) = 0 Next j Next i Sub legeCoef() lcoef(0, 0) = 1 lcoef(1, 1) = 1 For n As Integer = 2 To LIM lcoef(n, 0) = -(n - 1) * lcoef(n - 2, 0) / n For i As Integer = 1 To n lcoef(n, i) = ((2 * n - 1) * lcoef(n - 1, i - 1) - (n - 1) * lcoef(n - 2, i)) / n Next i Next n End Sub Function legeEval(n As Integer, x As Double) As Double Dim As Double s = lcoef(n, n) For i As Integer = n To 1 Step -1 s = s * x + lcoef(n, i - 1) Next i Return s End Function Function legeDiff(n As Integer, x As Double) As Double Return n * (x * legeEval(n, x) - legeEval(n - 1, x)) / (x * x - 1) End Function Sub legeRoots() Dim As Double x = 0 Dim As Double x1 = 0 For i As Integer = 1 To LIM x = Cos(PI * (i - 0.25) / (LIM + 0.5)) Do x1 = x x = x - legeEval(LIM, x) / legeDiff(LIM, x) Loop Until x = x1 lroots(i - 1) = x x1 = legeDiff(LIM, x) weight(i - 1) = 2 / ((1 - x * x) * x1 * x1) Next i End Sub Function legeIntegrate(f As Function (As Double) As Double, a As Double, b As Double) As Double Dim As Double c1 = (b - a) / 2 Dim As Double c2 = (b + a) / 2 Dim As Double sum = 0 For i As Integer = 0 To LIM - 1 sum = sum + weight(i) * f(c1 * lroots(i) + c2) Next i Return c1 * sum End Function legeCoef() legeRoots() Print "Roots: "; For i As Integer = 0 To LIM - 1 Print Using " ##.######"; lroots(i); Next i Print Print "Weight:"; For i As Integer = 0 To LIM - 1 Print Using " ##.######"; weight(i); Next i Print Function f(x As Double) As Double Return Exp(x) End Function Dim As Double actual = Exp(3) - Exp(-3) Print Using !"Integrating exp(x) over [-3, 3]:\n\t########.######,\ncompared to actual\n\t########.######"; legeIntegrate(@f, -3, 3); actual Sleep  Output: Roots: 0.906180 0.538469 0.000000 -0.538469 -0.906180 Weight: 0.236927 0.478629 0.568889 0.478629 0.236927 Integrating exp(x) over [-3, 3]: 20.035578, compared to actual 20.035750 ## Go Implementation pretty much by the methods given in the task description. package main import ( "fmt" "math" ) // cFunc for continuous function. A type definition for convenience. type cFunc func(float64) float64 func main() { fmt.Println("integral:", glq(math.Exp, -3, 3, 5)) } // glq integrates f from a to b by Guass-Legendre quadrature using n nodes. // For the task, it also shows the intermediate values determining the nodes: // the n roots of the order n Legendre polynomal and the corresponding n // weights used for the integration. 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 } // glqNodes computes both nodes and weights for a Gauss-Legendre // Quadrature integration. Parameters are n, the number of nodes // to compute and f, a continuous function to integrate. Return // values have len n. 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 } // legendrePoly constructs functions that implement Lengendre polynomials. // This is done by function composition by recurrence relation (Bonnet's.) // For given n, n+1 functions are returned, computing P0 through Pn. 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 } // newtonRaphson is general purpose, although totally primitive, simply // panicking after a fixed number of iterations without convergence to // a fixed error. Parameter f must be a continuous function, // df its derivative, x0 an initial guess. 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") }  Output:  nodes: 0.90618 0.53847 0.00000 -0.53847 -0.90618 weights: 0.23693 0.47863 0.56889 0.47863 0.23693 integral: 20.035577718385564  ## Haskell Integration formula gaussLegendre n f a b = d*sum [ w x*f(m + d*x) | x <- roots ] where d = (b - a)/2 m = (b + a)/2 w x = 2/(1-x^2)/(legendreP' n x)^2 roots = map (findRoot (legendreP n) (legendreP' n) . x0) [1..n] x0 i = cos (pi*(i-1/4)/(n+1/2))  Calculation of Legendre polynomials legendreP n x = go n 1 x where go 0 p2 _ = p2 go 1 _ p1 = p1 go n p2 p1 = go (n-1) p1$ ((2*n-1)*x*p1 - (n-1)*p2)/n

legendreP' n x = n/(x^2-1)*(x*legendreP n x - legendreP (n-1) x)


Universal auxilary functions

findRoot f df = fixedPoint (\x -> x - f x / df x)

fixedPoint f x | abs (fx - x) < 1e-15 = x
| otherwise = fixedPoint f fx
where fx = f x


Integration on a given mesh using Gauss-Legendre quadrature:

integrate _ []     = 0
integrate f (m:ms) = sum $zipWith (gaussLegendre 5 f) (m:ms) ms  Output:  λ> integrate exp [-3,3] 20.035577718385547 λ> integrate exp [-3..3] 20.03574985481217 λ> gaussLegendre 10 exp (-3) 3 20.035749854819695  Analytical solution  λ> exp 3 - exp (-3) 20.035749854819805  ## J Solution: NB. returns coefficents for yth-order Legendre polynomial getLegendreCoeffs=: verb define M. if. y<:1 do. 1 {.~ - y+1 return. end. (%~ <:@(,~ +:) -/@:* (0;'') ,&> [: getLegendreCoeffs&.> -&1 2) y ) getPolyRoots=: 1 {:: p. NB. returns the roots of a polynomial getGaussLegendreWeights=: 2 % -.@*:@[ * (*:@p.~ p..) NB. form: roots getGaussLegendreWeights coeffs getGaussLegendrePoints=: (getPolyRoots ([ ,: getGaussLegendreWeights) ])@getLegendreCoeffs NB.*integrateGaussLegendre a Integrates a function u with a n-point Gauss-Legendre quadrature rule over the interval [a,b] NB. form: npoints function integrateGaussLegendre (a,b) integrateGaussLegendre=: adverb define : 'nodes wgts'=. getGaussLegendrePoints x -: (-~/ y) * wgts +/@:* u -: nodes p.~ (+/ , -~/) y )  Example use:  5 ^ integrateGaussLegendre _3 3 20.0356 -~/ ^ _3 3 NB. true value 20.0357  ## Java Translation of: C Works with: Java version 8 import static java.lang.Math.*; import java.util.function.Function; public class Test { final static int N = 5; static double[] lroots = new double[N]; static double[] weight = new double[N]; static double[][] lcoef = new double[N + 1][N + 1]; static void legeCoef() { lcoef[0][0] = lcoef[1][1] = 1; for (int n = 2; n <= N; n++) { lcoef[n][0] = -(n - 1) * lcoef[n - 2][0] / n; for (int i = 1; i <= n; i++) { lcoef[n][i] = ((2 * n - 1) * lcoef[n - 1][i - 1] - (n - 1) * lcoef[n - 2][i]) / n; } } } static double legeEval(int n, double x) { double s = lcoef[n][n]; for (int i = n; i > 0; i--) s = s * x + lcoef[n][i - 1]; return s; } static double legeDiff(int n, double x) { return n * (x * legeEval(n, x) - legeEval(n - 1, x)) / (x * x - 1); } static void legeRoots() { double x, x1; for (int i = 1; i <= N; i++) { x = cos(PI * (i - 0.25) / (N + 0.5)); do { x1 = x; x -= legeEval(N, x) / legeDiff(N, x); } while (x != x1); lroots[i - 1] = x; x1 = legeDiff(N, x); weight[i - 1] = 2 / ((1 - x * x) * x1 * x1); } } static double legeInte(Function<Double, Double> f, double a, double b) { double c1 = (b - a) / 2, c2 = (b + a) / 2, sum = 0; for (int i = 0; i < N; i++) sum += weight[i] * f.apply(c1 * lroots[i] + c2); return c1 * sum; } public static void main(String[] args) { legeCoef(); legeRoots(); System.out.print("Roots: "); for (int i = 0; i < N; i++) System.out.printf(" %f", lroots[i]); System.out.print("\nWeight:"); for (int i = 0; i < N; i++) System.out.printf(" %f", weight[i]); System.out.printf("%nintegrating Exp(x) over [-3, 3]:%n\t%10.8f,%n" + "compared to actual%n\t%10.8f%n", legeInte(x -> exp(x), -3, 3), exp(3) - exp(-3)); } }  Roots: 0,906180 0,538469 0,000000 -0,538469 -0,906180 Weight: 0,236927 0,478629 0,568889 0,478629 0,236927 integrating Exp(x) over [-3, 3]: 20,03557772, compared to actual 20,03574985 ## JavaScript const factorial = n => n <= 1 ? 1 : n * factorial(n - 1); const M = n => (n - (n % 2 !== 0)) / 2; const gaussLegendre = (fn, a, b, n) => { // coefficients of the Legendre polynomial const coef = [...Array(M(n) + 1)].map((v, m) => v = (-1) ** m * factorial(2 * n - 2 * m) / (2 ** n * factorial(m) * factorial(n - m) * factorial(n - 2 * m))); // the polynomial function const f = x => coef.map((v, i) => v * x ** (n - 2 * i)).reduce((sum, item) => sum + item, 0); const terms = coef.length - (n % 2 === 0); // coefficients of the derivative polybomial const dcoef = [...Array(terms)].map((v, i) => v = n - 2 * i).map((val, i) => val * coef[i]); // the derivative polynomial function const df = x => dcoef.map((v, i) => v * x ** (n - 1 - 2 * i)).reduce((sum, item) => sum + item, 0); const guess = [...Array(n)].map((v, i) => Math.cos(Math.PI * (i + 1 - 1 / 4) / (n + 1 / 2))); // Newton Raphson const roots = guess.map(xo => [...Array(100)].reduce(x => x - f(x) / df(x), xo)); const weights = roots.map(v => 2 / ((1 - v ** 2) * df(v) ** 2)); return (b - a) / 2 * weights.map((v, i) => v * fn((b - a) * roots[i] / 2 + (a + b) / 2)).reduce((sum, item) => sum + item, 0); } console.log(gaussLegendre(x => Math.exp(x), -3, 3, 5));  Output: 20.035577718385575  ## jq Adapted from Wren Works with: jq Also works with gojq, the Go implementation of jq, and with fq # output: an array def legendreCoef($N):
{lcoef: (reduce range(0;$N+1) as$i (null; .[$i] = [range(0;$N + 1)| 0]))}
| .lcoef[0][0] = 1
| .lcoef[1][1] = 1
| reduce range(2; $N+1) as$n (.;
.lcoef[$n][0] = -($n-1) * .lcoef[$n -2][0] /$n
| reduce range (1; $n+1) as$i (.;
.lcoef[$n][$i] = ((2*$n - 1) * .lcoef[$n-1][$i-1] - ($n - 1) * .lcoef[$n-2][$i]) / $n ) ) | .lcoef ; # input: lcoef # output: the value def legendreEval($n; $x): . as$lcoef
| reduce range($n; 0 ;-1) as$i ( $lcoef[$n][$n] ; . *$x + $lcoef[$n][$i-1] ) ; # input: lcoef def legendreDiff($n; $x):$n * ($x * legendreEval($n; $x) - legendreEval($n-1; $x)) / ($x*$x - 1) ; # input: lcoef # output: {lroots, weight} def legendreRoots($N):
def pi: 1|atan * 4;
. as $lcoef | { x: 0, x1: null} | reduce range(1; 1+$N) as $i (.; .x = ((pi * ($i - 0.25) / ($N + 0.5)) | cos ) | until (.x == .x1; .x1 = .x | .x as$x
| .x = .x - ($lcoef | (legendreEval($N; $x) / legendreDiff($N; $x) )) ) | .lroots[$i-1] = .x
| .x as $x | .x1 = ($lcoef|legendreDiff($N;$x))
| .weight[$i-1] = 2 / ((1 - .x*.x) * .x1 * .x1) ) ; # Input: {lroots, weight} def legendreIntegrate(f;$a; $b;$N):
.lroots as $lroots | .weight as$weight
| (($b -$a) / 2) as $c1 | (($b + $a) / 2) as$c2
| reduce range(0;$N) as$i (0; . + $weight[$i] * (($c1*$lroots[$i] +$c2)|f) )
| $c1 * .; def task($N):
def actual: 3|exp - ((-3)|exp);

legendreCoef($N) | legendreRoots($N)
| "Roots: ",
.lroots,
"\nWeight:",
.weight,

"\nIntegrating exp(x) over [-3, 3]: \(legendreIntegrate(exp; -3; 3; N))",
"compared to actual:              \(actual)" ;

task(5)

Invocation:

jq -ncr -f gauss-legendre-quadrature.jq

Output:
Roots:
[0.906179845938664,0.5384693101056831,0,-0.5384693101056831,-0.906179845938664]

Weight:
[0.23692688505618922,0.4786286704993667,0.5688888888888889,0.4786286704993667,0.23692688505618922]

Integrating exp(x) over [-3, 3]: 20.035577718385575
compared to actual:              20.035749854819805


## Julia

This function computes the points and weights of an N-point Gauss–Legendre quadrature rule on the interval (a,b). It uses the O(N2) algorithm described in Trefethen & Bau, Numerical Linear Algebra, which finds the points and weights by computing the eigenvalues and eigenvectors of a real-symmetric tridiagonal matrix:

using LinearAlgebra

function gauss(a, b, N)
λ, Q = eigen(SymTridiagonal(zeros(N), [n / sqrt(4n^2 - 1) for n = 1:N-1]))
@. (λ + 1) * (b - a) / 2 + a, [2Q[1, i]^2 for i = 1:N] * (b - a) / 2
end


(This code is a simplified version of the Base.gauss subroutine in the Julia standard library.)

Output:
julia> x, w = gauss(-3, 3, 5)
([-2.71854, -1.61541, 1.33227e-15, 1.61541, 2.71854], [0.710781, 1.43589, 1.70667, 1.43589, 0.710781])

julia> sum(exp.(x) .* w)
20.03557771838554


## Kotlin

Translation of: Java
import java.lang.Math.*

class Legendre(val N: Int) {
fun evaluate(n: Int, x: Double) = (n downTo 1).fold(c[n][n]) { s, i -> s * x + c[n][i - 1] }

fun diff(n: Int, x: Double) = n * (x * evaluate(n, x) - evaluate(n - 1, x)) / (x * x - 1)

fun integrate(f: (Double) -> Double, a: Double, b: Double): Double {
val c1 = (b - a) / 2
val c2 = (b + a) / 2
return c1 * (0 until N).fold(0.0) { s, i -> s + weights[i] * f(c1 * roots[i] + c2) }
}

private val roots = DoubleArray(N)
private val weights = DoubleArray(N)
private val c = Array(N + 1) { DoubleArray(N + 1) }    // coefficients

init {
// coefficients:
c[0][0] = 1.0
c[1][1] = 1.0
for (n in 2..N) {
c[n][0] = (1 - n) * c[n - 2][0] / n
for (i in 1..n)
c[n][i] = ((2 * n - 1) * c[n - 1][i - 1] - (n - 1) * c[n - 2][i]) / n
}

// roots:
var x: Double
var x1: Double
for (i in 1..N) {
x = cos(PI * (i - 0.25) / (N + 0.5))
do {
x1 = x
x -= evaluate(N, x) / diff(N, x)
} while (x != x1)

x1 = diff(N, x)
roots[i - 1] = x
weights[i - 1] = 2 / ((1 - x * x) * x1 * x1)
}

print("Roots:")
roots.forEach { print(" %f".format(it)) }
println()
print("Weights:")
weights.forEach { print(" %f".format(it)) }
println()
}
}

fun main(args: Array<String>) {
val legendre = Legendre(5)
println("integrating Exp(x) over [-3, 3]:")
println("\t%10.8f".format(legendre.integrate(Math::exp, -3.0, 3.0)))
println("compared to actual:")
println("\t%10.8f".format(exp(3.0) - exp(-3.0)))
}

Output:
Roots: 0.906180 0.538469 0.000000 -0.538469 -0.906180
Weights: 0.236927 0.478629 0.568889 0.478629 0.236927
integrating Exp(x) over [-3, 3]:
20.03557772
compared to actual:
20.03574985

## Lua

local order = 0

local legendreRoots = {}
local legendreWeights = {}

local function legendre(term, z)
if (term == 0) then
return 1
elseif (term == 1) then
return z
else
return ((2 * term - 1) * z * legendre(term - 1, z) - (term - 1) * legendre(term - 2, z)) / term
end
end

local function legendreDerivative(term, z)
if (term == 0) then
return 0
elseif (term == 1) then
return 1
else
return ( term * ((z * legendre(term, z)) - legendre(term - 1, z))) / (z * z - 1)
end
end

local function getLegendreRoots()
local y, y1

for index = 1, order do
y = math.cos(math.pi * (index - 0.25) / (order + 0.5))

repeat
y1 = y
y = y - (legendre(order, y) / legendreDerivative(order, y))
until y == y1

table.insert(legendreRoots, y)
end
end

local function getLegendreWeights()
for index = 1, order do
local weight = 2 / ((1 - (legendreRoots[index]) ^ 2) * (legendreDerivative(order, legendreRoots[index])) ^ 2)
table.insert(legendreWeights, weight)
end
end

order = n

do
getLegendreRoots()
getLegendreWeights()
end

local c1 = (upperLimit - lowerLimit) / 2
local c2 = (upperLimit + lowerLimit) / 2
local sum = 0

for i = 1, order do
sum = sum + legendreWeights[i] * f(c1 * legendreRoots[i] + c2)
end

return c1 * sum
end

do
print(gaussLegendreQuadrature(function(x) return math.exp(x) end, -3, 3, 5))
end

Output:
20.035577718386

## Mathematica/Wolfram Language

code assumes function to be integrated has attribute Listable which is true of most built in Mathematica functions

gaussLegendreQuadrature[func_, {a_, b_}, degree_: 5] :=
Block[{nodes, x, weights},
nodes = Cases[NSolve[LegendreP[degree, x] == 0, x], _?NumericQ, Infinity];
weights = 2 (1 - nodes^2)/(degree LegendreP[degree - 1, nodes])^2;
(b - a)/2 weights.func[(b - a)/2 nodes + (b + a)/2]]

Output:
20.0356

## MATLAB

Translated from the Python solution.

%Print the result.
disp(GLGD_int(@(x) exp(x), -3, 3, 5));

%Does almost the same as 'integral' in MATLAB
function y=GLGD_int(fun,xmin,xmax,n)
%fun: the intergrand as a function handle
%xmin: lower boundary of integration
%xmax: upper boundary of integration
%n: order of polynomials used (number of integration ponts)
[x_IP,weight]=GLGD_para(n);
%assign global coordinates to the integraton points
x_eval=x_IP*(xmax-xmin)/2+(xmax+xmin)/2;
y=0;
for aa=1:n
y=y+feval(fun,x_eval(aa))*weight(aa)*(xmax-xmin)/2;
end
end

function [x_IP,weight]=GLGD_para(n)
%n: the order of the polynomial
x_IP=legendreRoot(n,10^(-16));
weight=2./(1-x_IP.^2)./diff_legendrePoly(x_IP,n).^2;
end

%roots of the Legendre Polynomial using Newton-Raphson
function x_IP=legendreRoot(n,tol)
%n: order of the polynomial
%tol: tolerence of the error
if n<2
disp('No root can be found');
else
root=zeros(1,floor(n/2));
for aa=1:floor(n/2) %iterate to find half of the roots
x=cos(pi*(aa-0.25)/(n+0.5));
err=10*tol;
iter=0;
while (err>tol)&&(iter<1000)
dx=-legendrePoly(x,n)/diff_legendrePoly(x,n);
x=x+dx;
iter=iter+1;
err=abs(legendrePoly(x,n));
end
root(aa)=x;
end
if mod(n,2)==0
x_IP=[-1*root,root];
else
x_IP=[-1*root,0,root];
end
x_IP=sort(x_IP);
end
end

%derivative of the Legendre Polynomial
function y=diff_legendrePoly(x_IP,n)
%n: order of the polynomial
%x_IP: coordinates of the integration points
if n==0
y=0;
else
y=n./(x_IP.^2-1).*(x_IP.*legendrePoly(x_IP,n)-legendrePoly(x_IP,n-1));
end
end

%Produces Legendre Polynomials
function y=legendrePoly(x,n)
%n: order of polynomial
%x: input x
if n==0
y=1;
elseif n==1
y=x;
else
y=((2*n-1).*x.*legendrePoly(x,n-1)-(n-1)*legendrePoly(x,n-2))/n;
end
end

Output:
20.0356

## Maxima

gauss_coeff(n) := block([p, q, v, w],
p: expand(legendre_p(n, x)),
q: expand(n/2*diff(p, x)*legendre_p(n - 1, x)),
v: map(rhs, bfallroots(p)),
w: map(lambda([z], 1/subst([x = z], q)), v),
[map(bfloat, v), map(bfloat, w)])$gauss_int(f, a, b, n) := block([u, x, w, c, h], u: gauss_coeff(n), x: u[1], w: u[2], c: bfloat((a + b)/2), h: bfloat((b - a)/2), h*sum(w[i]*bfloat(f(c + x[i]*h)), i, 1, n))$

fpprec: 40$gauss_int(lambda([x], 4/(1 + x^2)), 0, 1, 20); /* 3.141592653589793238462643379852215927697b0 */ % - bfloat(%pi); /* -3.427286956499858315999116083264403489053b-27 */ gauss_int(exp, -3, 3, 5); /* 2.003557771838556215392853572527509393154b1 */ % - bfloat(integrate(exp(x), x, -3, 3)); /* -1.721364342416440206515136565621888185351b-4 */  ## Nim Translation of: Common Lisp import math, strformat proc legendreIn(x: float, n: int): float = template prev1(idx: int; pn1: float): float = (2*idx - 1).float * x * pn1 template prev2(idx: int; pn2: float): float = (idx-1).float * pn2 if n == 0: return 1.0 elif n == 1: return x else: var p1 = float x p2 = 1.0 for i in 2 .. n: result = (i.prev1(p1) - i.prev2(p2)) / i.float p2 = p1 p1 = result proc deriveLegendreIn(x: float, n: int): float = template calcresult(curr, prev: float): untyped = n.float / (x^2 - 1) * (x * curr - prev) result = calcresult(x.legendreIn n, x.legendreIn(n-1)) func guess(n, i: int): float = cos(PI * (i.float - 0.25) / (n.float + 0.5)) proc nodes(n: int): seq[(float, float)] = result = newseq[(float, float)](n) template calc(x: float): untyped = x.legendreIn(n) / x.deriveLegendreIn(n) for i in 0 .. result.high: var x = guess(n, i+1) block newton: var x0 = x x -= calc x while abs(x-x0) > 1e-12: x0 = x x -= calc x result[i][0] = x result[i][1] = 2 / ((1.0 - x^2) * (x.deriveLegendreIn n)^2) proc integ(f: proc(x: float): float; ns, p1, p2: int): float = template dist: untyped = (p2 - p1).float / 2.0 template avg: untyped = (p1 + p2).float / 2.0 result = dist() var sum = 0'f thenodes = newseq[float](ns) weights = newseq[float](ns) for i, nw in ns.nodes: sum += nw[1] * f(dist() * nw[0] + avg()) thenodes[i] = nw[0] weights[i] = nw[1] let apos = ":" stdout.write fmt"""{"nodes":>8}{apos}""" for n in thenodes: stdout.write &" {n:>6.5f}" stdout.write "\n" stdout.write &"""{"weights":>8}{apos}""" for w in weights: stdout.write &" {w:>6.5f}" stdout.write "\n" result *= sum proc main = echo "integral: ", integ(exp, 5, -3, 3) main()  Output:  nodes: 0.90618 0.53847 0.00000 -0.53847 -0.90618 weights: 0.23693 0.47863 0.56889 0.47863 0.23693 integral: 20.03557634353638  ## OCaml let rec leg n x = match n with (* Evaluate Legendre polynomial *) | 0 -> 1.0 | 1 -> x | k -> let u = 1.0 -. 1.0 /. float k in (1.0+.u)*.x*.(leg (k-1) x) -. u*.(leg (k-2) x);; let leg' n x = match n with (* derivative *) | 0 -> 0.0 | 1 -> 1.0 | _ -> ((leg (n-1) x) -. x*.(leg n x)) *. (float n)/.(1.0-.x*.x);; let approx_root k n = (* Reversed Francesco Tricomi: 1 <= k <= n *) let pi = acos (-1.0) and s = float(2*n) and t = 1.0 +. float(1-4*k)/.float(4*n+2) in (1.0 -. (float (n-1))/.(s*.s*.s))*.cos(pi*.t);; let rec refine r n = (* Newton-Raphson *) let r1 = r -. (leg n r)/.(leg' n r) in if abs_float (r-.r1) < 2e-16 then r1 else refine r1 n;; let root k n = refine (approx_root k n) n;; let node k n = (* Abscissa and weight *) let r = root k n in let deriv = leg' n r in let w = 2.0/.((1.0-.r*.r)*.(deriv*.deriv)) in (r,w);; let nodes n = let rec aux k = if k > n then [] else node k n :: aux (k+1) in aux 1;; let quadrature n f a b = let f1 x = f ((x*.(b-.a) +. a +. b)*.0.5) in let eval s (x,w) = s +. w*.(f1 x) in 0.5*.(b-.a)*.(List.fold_left eval 0.0 (nodes n));;  which can be used in: let calc n = Printf.printf "Gauss-Legendre %2d-point quadrature for exp over [-3..3] = %.16f\n" n (quadrature n exp (-3.0) 3.0);; calc 5;; calc 10;; calc 15;; calc 20;;  Output: Gauss-Legendre 5-point quadrature for exp over [-3..3] = 20.0355777183855608 Gauss-Legendre 10-point quadrature for exp over [-3..3] = 20.0357498548197839 Gauss-Legendre 15-point quadrature for exp over [-3..3] = 20.0357498548198052 Gauss-Legendre 20-point quadrature for exp over [-3..3] = 20.0357498548198052  This shows convergence to the correct double-precision value of the integral Printf.printf "%.16f\n" ((exp 3.0) -.(exp (-3.0)));; 20.0357498548198052  although going beyond 20 points starts reducing the accuracy, due to accumulated rounding errors. ## ooRexx /*--------------------------------------------------------------------- * 31.10.2013 Walter Pachl Translation from REXX (from PL/I) * using ooRexx' rxmath package * which limits the precision to 16 digits *--------------------------------------------------------------------*/ prec=60 Numeric Digits prec epsilon=1/10**prec pi=3.141592653589793238462643383279502884197169399375105820974944592307 exact = RxCalcExp(3,prec)-RxCalcExp(-3,prec) Do n = 1 To 20 a = -3; b = 3 r.=0 call gaussquad sum=0 Do j=1 To n sum=sum + r.2.j * RxCalcExp((a+b)/2+r.1.j*(b-a)/2,prec) End z = (b-a)/2 * sum Say right(n,2) format(z,2,40) format(z-exact,2,4,,0) End Say ' ' exact '(exact)' Exit gaussquad: p0.0=1; p0.1=1 p1.0=2; p1.1=1; p1.2=0 Do k = 2 To n tmp.0=p1.0+1 Do L = 1 To p1.0 tmp.l = p1.l End tmp.l=0 tmp2.0=p0.0+2 tmp2.1=0 tmp2.2=0 Do L = 1 To p0.0 l2=l+2 tmp2.l2=p0.l End Do j=1 To tmp.0 tmp.j = ((2*k-1)*tmp.j - (k-1)*tmp2.j)/k End p0.0=p1.0 Do j=1 To p0.0 p0.j = p1.j End p1.0=tmp.0 Do j=1 To p1.0 p1.j=tmp.j End End Do i = 1 To n x = RxCalcCos(pi*(i-0.25)/(n+0.5),prec,'R') Do iter = 1 To 10 f = p1.1; df = 0 Do k = 2 To p1.0 df = f + x*df f = p1.k + x * f End dx = f / df x = x - dx If abs(dx) < epsilon Then Leave End r.1.i = x r.2.i = 2/((1-x**2)*df**2) End Return ::requires 'rxmath' LIBRARY Output:  1 6.0000000000000000000000000000000000000000 -1.4036E+1 2 17.4874646410555686000000000000000000000000 -2.5483 3 19.8536919968055914500000000000000000000000 -1.8206E-1 4 20.0286883952907032246391703165575495371776 -7.0615E-3 5 20.0355777183855623345965085871972344078167 -1.7214E-4 6 20.0357469750923433031000982816859525440756 -2.8797E-6 7 20.0357498197266007450081506439422093510041 -3.5093E-8 8 20.0357498544945192648654062025059252571210 -3.2529E-10 9 20.0357498548174362426073138353882519240177 -2.3698E-12 10 20.0357498548197905075149387536361754813374 -1.5552E-14 11 20.0357498548198049052166074059523608613749 -1.1548E-15 12 20.0357498548198068119347633275378821700762 7.5193E-16 13 20.0357498548198063256375618073806663013152 2.6564E-16 14 20.0357498548198035202546245888922276792447 -2.5397E-15 15 20.0357498548198027919824444452012138941729 -3.2680E-15 16 20.0357498548198037471314715729442546019171 -2.3129E-15 17 20.0357498548198067452377635761033686644343 6.8524E-16 18 20.0357498548198042026084719530842757694873 -1.8574E-15 19 20.0357498548198042304714191024916472961732 -1.8295E-15 20 20.0357498548198034525095801113268011014944 -2.6075E-15 20.03574985481980606 (exact) ## PARI/GP Works with: PARI/GP version 2.4.2 and above This task is easy in GP thanks to built-in support for Legendre polynomials and efficient (Schonhage-Gourdon) polynomial root finding. GLq(f,a,b,n)={ my(P=pollegendre(n),Pp=P',x=polroots(P)); (b-a)*sum(i=1,n,f((b-a)*x[i]/2+(a+b)/2)/(1-x[i]^2)/subst(Pp,'x,x[i])^2) }; # \\ Turn on timer GLq(x->exp(x), -3, 3, 5) \\ As of version 2.4.4, this can be written GLq(exp, -3, 3, 5) Output: time = 0 ms. %1 = 20.035577718385562153928535725275093932 + 0.E-37*I Works with: PARI/GP version 2.9.0 and above Gauss-Legendre quadrature is built-in from 2.9 forward. intnumgauss(x=-3, 3, exp(x), intnumgaussinit(5)) intnumgauss(x=-3, 3, exp(x)) \\ determine number of points automatically; all digits shown should be accurate Output: %1 = 20.035746975092343883065457558549925374 %2 = 20.035749854819803797949187238931656120 ## Pascal Translation of: Delphi Works with: Free Pascal version 3.0.4 Works with: Multics Pascal version 8.0.4a program Legendre(output); const Order = 5; Order1 = Order - 1; Epsilon = 1E-12; Pi = 3.1415926; var Roots : array[0..Order1] of real; Weight : array[0..Order1] of real; LegCoef : array [0..Order,0..Order] of real; I : integer; function F(X:real) : real; begin F := Exp(X); end; procedure PrepCoef; var I, N : integer; begin for I:=0 to Order do for N := 0 to Order do LegCoef[I,N] := 0; LegCoef[0,0] := 1; LegCoef[1,1] := 1; For N:=2 to Order do begin LegCoef[N,0] := -(N-1) * LegCoef[N-2,0] / N; For I := 1 to Order do LegCoef[N,I] := ((2*N-1) * LegCoef[N-1,I-1] - (N-1)*LegCoef[N-2,I]) / N; end; end; function LegEval(N:integer; X:real) : real; var I : integer; Result : real; begin Result := LegCoef[n][n]; for I := N-1 downto 0 do Result := Result * X + LegCoef[N][I]; LegEval := Result; end; function LegDiff(N:integer; X:real) : real; begin LegDiff := N * (X * LegEval(N,X) - LegEval(N-1,X)) / (X*X-1); end; procedure LegRoots; var I : integer; X, X1 : real; begin for I := 1 to Order do begin X := Cos(Pi * (I-0.25) / (Order+0.5)); repeat X1 := X; X := X - LegEval(Order,X) / LegDiff(Order, X); until Abs (X-X1) < Epsilon; Roots[I-1] := X; X1 := LegDiff(Order,X); Weight[I-1] := 2 / ((1-X*X) * X1*X1); end; end; function LegInt(A,B:real) : real; var I : integer; C1, C2, Result : real; begin C1 := (B-A)/2; C2 := (B+A)/2; Result := 0; For I := 0 to Order-1 do Result := Result + Weight[I] * F(C1*Roots[I] + C2); Result := C1 * Result; LegInt := Result; end; begin PrepCoef; LegRoots; Write('Roots: '); for I := 0 to Order-1 do Write (' ',Roots[I]:13:10); Writeln; Write('Weight: '); for I := 0 to Order-1 do Write (' ', Weight[I]:13:10); writeln; Writeln('Integrating Exp(x) over [-3, 3]: ',LegInt(-3,3):13:10); Writeln('Actual value: ',Exp(3)-Exp(-3):13:10); end.  Output: Roots: 0.9061798459 0.5384693101 0.0000000000 -0.5384693101 -0.9061798459 Weight: 0.2369268851 0.4786286705 0.5688888889 0.4786286705 0.2369268851 Integrating Exp(x) over [-3, 3]: 20.0355777184 Actual value: 20.0357498548  ## Perl Translation of: Raku use List::Util qw(sum); use constant pi => 3.14159265; sub legendre_pair { my($n, $x) = @_; if ($n == 1) { return $x, 1 } my ($m1, $m2) = legendre_pair($n - 1, $x); my$u = 1 - 1 / $n; (1 +$u) * $x *$m1 - $u *$m2, $m1; } sub legendre { my($n, $x) = @_; (legendre_pair($n, $x))[0] } sub legendre_prime { my($n, $x) = @_; if ($n == 0) { return 0 }
if ($n == 1) { return 1 } my ($m0, $m1) = legendre_pair($n, $x); ($m1 - $x *$m0) * $n / (1 -$x**2);
}

sub approximate_legendre_root {
my($n,$k) = @_;
my $t = (4*$k - 1) / (4*$n + 2); (1 - ($n - 1) / (8 * $n**3)) * cos(pi *$t);
}

sub newton_raphson {
my($n,$r) = @_;
while (abs(my $dr = - legendre($n,$r) / legendre_prime($n,$r)) >= 2e-16) {$r += $dr; }$r;
}

sub legendre_root {
my($n,$k) = @_;
newton_raphson($n, approximate_legendre_root($n, $k)); } sub weight { my($n, $r) = @_; 2 / ((1 -$r**2) * legendre_prime($n,$r)**2)
}

sub nodes {
my($n) = @_; my %node;$node{'0'} = weight($n, 0) if 0 !=$n%2;
for (1 .. int $n/2) { my$r = legendre_root($n,$_);
my $w = weight($n, $r);$node{$r} =$w; $node{-$r} = $w; } return %node } sub quadrature { our($n, $a,$b) = @_;
sub scale { ($_[0] * ($b - $a) +$a + $b) / 2 } %nodes = nodes($n);
($b -$a) / 2 * sum map { $nodes{$_} * exp(scale($_)) } keys %nodes; } printf("Gauss-Legendre %2d-point quadrature ∫₋₃⁺³ exp(x) dx ≈ %.13f\n",$_, quadrature($_, -3, +3) ) for 5 .. 10, 20;  Output: Gauss-Legendre 5-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0355777183856 Gauss-Legendre 6-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357469750923 Gauss-Legendre 7-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357498197266 Gauss-Legendre 8-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357498544945 Gauss-Legendre 9-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357498548174 Gauss-Legendre 10-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357498548198 Gauss-Legendre 20-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357498548198 ## Phix Translation of: Lua with javascript_semantics integer order = 0 sequence legendreRoots = {}, legendreWeights = {} function legendre(integer term, atom z) if term=0 then return 1 elsif term=1 then return z else return ((2*term-1)*z*legendre(term-1,z)-(term-1)*legendre(term-2,z))/term end if end function function legendreDerivative(integer term, atom z) if term=0 or term=1 then return term end if return (term*(z*legendre(term,z)-legendre(term-1,z)))/(z*z-1) end function procedure getLegendreRoots() legendreRoots = {} for index=1 to order do atom y = cos(PI*(index-0.25)/(order+0.5)) while 1 do atom y1 = y y -= legendre(order,y)/legendreDerivative(order,y) if abs(y-y1)<2e-16 then exit end if end while legendreRoots &= y end for end procedure procedure getLegendreWeights() legendreWeights = {} for index=1 to order do atom lri = legendreRoots[index], diff = legendreDerivative(order,lri), weight = 2 / ((1-power(lri,2))*power(diff,2)) legendreWeights &= weight end for end procedure function gaussLegendreQuadrature(integer f, lowerLimit, upperLimit, n) order = n getLegendreRoots() getLegendreWeights() atom c1 = (upperLimit - lowerLimit) / 2 atom c2 = (upperLimit + lowerLimit) / 2 atom s = 0 for i = 1 to order do s += legendreWeights[i] * f(c1 * legendreRoots[i] + c2) end for return c1 * s end function string fmt = iff(machine_bits()=32?"%.13f":"%.14f"), res for i=5 to 11 by 6 do res = sprintf(fmt,{gaussLegendreQuadrature(exp, -3, 3, i)}) if i=5 then puts(1,"roots:") ?legendreRoots puts(1,"weights:") ?legendreWeights end if printf(1,"Gauss-Legendre %2d-point quadrature for exp over [-3..3] = %s\n",{order,res}) end for res = sprintf(fmt,{exp(3)-exp(-3)}) printf(1," compared to actual = %s\n",{res})  Output: roots:{0.9061798459,0.5384693101,0,-0.5384693101,-0.9061798459} weights:{0.2369268851,0.4786286705,0.5688888889,0.4786286705,0.2369268851} Gauss-Legendre 5-point quadrature for exp over [-3..3] = 20.0355777183856 Gauss-Legendre 11-point quadrature for exp over [-3..3] = 20.0357498548198 compared to actual = 20.0357498548198  Tests showed the result appeared to be accurate to 13 decimal places (15 significant figures) for order 10 to 30 on 32-bit, and one more for order 11+ on 64-bit. ## PL/I Translated from Fortran. (subscriptrange, size, fofl): Integration_Gauss: procedure options (main); declare (n, k) fixed binary; declare r(*,*) float (18) controlled; declare (z, a, b, exact) float (18); do n = 1 to 20; a = -3; b = 3; if allocation(r) > 0 then free r; allocate r(2, n); r = 0; call gaussquad(n, r); z = (b-a)/2 * sum(r(2,*) * exp((a+b)/2+r(1,*)*(b-a)/2)); exact = exp(3.0q0)-exp(-3.0q0); put skip edit (n, z, z-exact) (f(5), f(25,16), e(15,2)); end; gaussquad: procedure(n, r); /*declare n fixed binary, r(2, n) float (18);*/ declare n fixed binary, r(2, *) float (18);/* corrected */ declare pi float (18) value (4*atan(1.0q0)); declare (x, f, df, dx) float (18); declare (i, iter, L) fixed binary; declare (p0(*), p1(*), tmp(*), tmp2(*)) float (18) controlled; allocate p0(1) initial (1); allocate p1(2) initial (1, 0); do k = 2 to n; allocate tmp(hbound(p1)+1); do L = 1 to hbound(p1); tmp(L) = p1(L); end; tmp(L) = 0; allocate tmp2(hbound(p0)+2); tmp2(1), tmp2(2) = 0; do L = 1 to hbound(p0); tmp2(L+2) = p0(L); end; tmp = ((2*k-1)*tmp - (k-1)*tmp2)/k; free p0; allocate p0(hbound(p1)); p0 = p1; free p1; allocate p1(hbound(tmp)); p1 = tmp; free tmp, tmp2; end; do i = 1 to n; x = cos(pi*(i-0.25q0)/(n+0.5q0)); do iter = 1 to 10; f = p1(1); df = 0; do k = 2 to hbound(p1); df = f + x*df; f = p1(k) + x * f; end; dx = f / df; x = x - dx; if abs(dx) < 10*epsilon(dx) then leave; end; r(1,i) = x; r(2,i) = 2/((1-x**2)*df**2); end; end gaussquad; end Integration_Gauss;  1 6.0000000000000000 -1.40E+0001 2 17.4874646410555690 -2.55E+0000 3 19.8536919968055822 -1.82E-0001 4 20.0286883952907009 -7.06E-0003 5 20.0355777183855621 -1.72E-0004 6 20.0357469750923439 -2.88E-0006 7 20.0357498197266008 -3.51E-0008 8 20.0357498544945173 -3.25E-0010 9 20.0357498548174338 -2.37E-0012 10 20.0357498548197897 -1.41E-0014 11 20.0357498548198037 -6.94E-0017 12 20.0357498548198037 -6.25E-0017 13 20.0357498548198037 -1.25E-0016 14 20.0357498548198026 -1.16E-0015 15 20.0357498548198144 1.06E-0014 16 20.0357498548198021 -1.74E-0015 17 20.0357498548198359 3.21E-0014 18 20.0357498548198473 4.35E-0014 19 20.0357498548198848 8.10E-0014 20 20.0357498548200728 2.69E-0013  program gave me an error message: D:\ig.pli(19:2) : IBM1937I S Extents for parameters must be asterisks or restricted expressions with computational type. I tried to correct that. ok?  ## Python Library: NumPy from numpy import * ################################################################## # Recursive generation of the Legendre polynomial of order n def Legendre(n,x): x=array(x) if (n==0): return x*0+1.0 elif (n==1): return x else: return ((2.0*n-1.0)*x*Legendre(n-1,x)-(n-1)*Legendre(n-2,x))/n ################################################################## # Derivative of the Legendre polynomials def DLegendre(n,x): x=array(x) if (n==0): return x*0 elif (n==1): return x*0+1.0 else: return (n/(x**2-1.0))*(x*Legendre(n,x)-Legendre(n-1,x)) ################################################################## # Roots of the polynomial obtained using Newton-Raphson method def LegendreRoots(polyorder,tolerance=1e-20): if polyorder<2: err=1 # bad polyorder no roots can be found else: roots=[] # The polynomials are alternately even and odd functions. So we evaluate only half the number of roots. for i in range(1,int(polyorder)/2 +1): x=cos(pi*(i-0.25)/(polyorder+0.5)) error=10*tolerance iters=0 while (error>tolerance) and (iters<1000): dx=-Legendre(polyorder,x)/DLegendre(polyorder,x) x=x+dx iters=iters+1 error=abs(dx) roots.append(x) # Use symmetry to get the other roots roots=array(roots) if polyorder%2==0: roots=concatenate( (-1.0*roots, roots[::-1]) ) else: roots=concatenate( (-1.0*roots, [0.0], roots[::-1]) ) err=0 # successfully determined roots return [roots, err] ################################################################## # Weight coefficients def GaussLegendreWeights(polyorder): W=[] [xis,err]=LegendreRoots(polyorder) if err==0: W=2.0/( (1.0-xis**2)*(DLegendre(polyorder,xis)**2) ) err=0 else: err=1 # could not determine roots - so no weights return [W, xis, err] ################################################################## # The integral value # func : the integrand # a, b : lower and upper limits of the integral # polyorder : order of the Legendre polynomial to be used # def GaussLegendreQuadrature(func, polyorder, a, b): [Ws,xs, err]= GaussLegendreWeights(polyorder) if err==0: ans=(b-a)*0.5*sum( Ws*func( (b-a)*0.5*xs+ (b+a)*0.5 ) ) else: # (in case of error) err=1 ans=None return [ans,err] ################################################################## # The integrand - change as required def func(x): return exp(x) ################################################################## # order=5 [Ws,xs,err]=GaussLegendreWeights(order) if err==0: print "Order : ", order print "Roots : ", xs print "Weights : ", Ws else: print "Roots/Weights evaluation failed" # Integrating the function [ans,err]=GaussLegendreQuadrature(func , order, -3,3) if err==0: print "Integral : ", ans else: print "Integral evaluation failed"  Output: Order : 5 Roots : [-0.90617985 -0.53846931 0. 0.53846931 0.90617985] Weights : [ 0.23692689 0.47862867 0.56888889 0.47862867 0.23692689] Integral : 20.0355777184  ### With library routine One can also use the already invented wheel in NumPy: import numpy as np # func is a function that takes a list-like input values def gauss_legendre_integrate(func, domain, deg): x, w = np.polynomial.legendre.leggauss(deg) s = (domain[1] - domain[0])/2 a = (domain[1] + domain[0])/2 return np.sum(s*w*func(s*x + a)) for d in range(3, 10): print(d, gauss_legendre_integrate(np.exp, [-3, 3], d))  Output: 3 19.853691996805587 4 20.028688395290693 5 20.035577718385575 6 20.035746975092323 7 20.03574981972664 8 20.035749854494522 9 20.03574985481744 ## Racket Computation of the Legendre polynomials and derivatives: (define (LegendreP n x) (let compute ([n n] [Pn-1 x] [Pn-2 1]) (case n [(0) Pn-2] [(1) Pn-1] [else (compute (- n 1) (/ (- (* (- (* 2 n) 1) x Pn-1) (* (- n 1) Pn-2)) n) Pn-1)]))) (define (LegendreP′ n x) (* (/ n (- (* x x) 1)) (- (* x (LegendreP n x)) (LegendreP (- n 1) x))))  Computation of the Legendre polynomial roots: (define (LegendreP-root n i) ; newton-raphson step (define (newton-step x) (- x (/ (LegendreP n x) (LegendreP′ n x)))) ; initial guess (define x0 (cos (* pi (/ (- i 1/4) (+ n 1/2))))) ; computation of a root with relative accuracy 1e-15 (if (< (abs x0) 1e-15) 0 (let next ([x′ (newton-step x0)] [x x0]) (if (< (abs (/ (- x′ x) (+ x′ x))) 1e-15) x′ (next (newton-step x′) x′)))))  Computation of Gauss-Legendre nodes and weights (define (Gauss-Legendre-quadrature n) ;; positive roots (define roots (for/list ([i (in-range (floor (/ n 2)))]) (LegendreP-root n (+ i 1)))) ;; weights for positive roots (define weights (for/list ([x (in-list roots)]) (/ 2 (- 1 (sqr x)) (sqr (LegendreP′ n x))))) ;; all roots and weights (values (append (map - roots) (if (odd? n) (list 0) '()) (reverse roots)) (append weights (if (odd? n) (list (/ 2 (sqr (LegendreP′ n 0)))) '()) (reverse weights))))  Integration using Gauss-Legendre quadratures: (define (integrate f a b #:nodes (n 5)) (define m (/ (+ a b) 2)) (define d (/ (- b a) 2)) (define-values [x w] (Gauss-Legendre-quadrature n)) (define (g x) (f (+ m (* d x)))) (* d (+ (apply + (map * w (map g x))))))  Usage: > (Gauss-Legendre-quadrature 5) '(-0.906179845938664 -0.5384693101056831 0 0.5384693101056831 0.906179845938664) '(0.23692688505618875 0.47862867049936625 128/225 0.47862867049936625 0.23692688505618875) > (integrate exp -3 3) 20.035577718385547 > (- (exp 3) (exp -3) 20.035749854819805  Accuracy of the method: > (require plot) > (parameterize ([plot-x-label "Number of Gaussian nodes"] [plot-y-label "Integration error"] [plot-y-transform log-transform] [plot-y-ticks (log-ticks #:base 10)]) (plot (points (for/list ([n (in-range 2 11)]) (list n (abs (- (integrate exp -3 3 #:nodes n) (- (exp 3) (exp -3)))))))))  ## Raku (formerly Perl 6) Works with: rakudo version 2015-09-24 A free translation of the OCaml solution. We save half the effort to calculate the nodes by exploiting the (skew-)symmetry of the Legendre Polynomials. The evaluation of Pn(x) is kept linear in n by also passing Pn-1(x) in the recursion. The quadrature function allows passing in a precalculated list of nodes for repeated integrations. Note: The calculations of Pn(x) and P'n(x) could be combined to further reduce duplicated effort. We also could cache P'n(x) from the last Newton-Raphson step for the weight calculation. multi legendre-pair( 1 ,$x) { $x, 1 } multi legendre-pair(Int$n, $x) { my ($m1, $m2) = legendre-pair($n - 1, $x); my \u = 1 - 1 /$n;
(1 + u) * $x *$m1 - u * $m2,$m1;
}

multi legendre(    0 , $) { 1 } multi legendre(Int$n, $x) { legendre-pair($n, $x)[0] } multi legendre-prime( 0 ,$ ) { 0 }
multi legendre-prime(    1 , $) { 1 } multi legendre-prime(Int$n, $x) { my ($m0, $m1) = legendre-pair($n, $x); ($m1 - $x *$m0) * $n / (1 -$x**2);
}

sub approximate-legendre-root(Int $n, Int$k) {
# Approximation due to Francesco Tricomi
my \t = (4*$k - 1) / (4*$n + 2);
(1 - ($n - 1) / (8 *$n**3)) * cos(pi * t);
}

sub newton-raphson(&f, &f-prime, $r is copy, :$eps = 2e-16) {
while abs(my \dr = - f($r) / f-prime($r)) >= $eps {$r += dr;
}
$r; } sub legendre-root(Int$n, Int $k) { newton-raphson(&legendre.assuming($n), &legendre-prime.assuming($n), approximate-legendre-root($n, $k)); } sub weight(Int$n, $r) { 2 / ((1 -$r**2) * legendre-prime($n,$r)**2) }

sub nodes(Int $n) { flat gather { take 0 => weight($n, 0) if $n !%% 2; for 1 ..$n div 2 {
my $r = legendre-root($n, $_); my$w = weight($n,$r);
take $r =>$w, -$r =>$w;
}
}
}

sub quadrature(Int $n, &f,$a, $b, :@nodes = nodes($n)) {
sub scale($x) { ($x * ($b -$a) + $a +$b) / 2 }
($b -$a) / 2 * [+] @nodes.map: { .value * f(scale(.key)) }
}

say "Gauss-Legendre $_.fmt('%2d')-point quadrature ∫₋₃⁺³ exp(x) dx ≈ ", quadrature($_, &exp, -3, +3) for flat 5 .. 10, 20;

Output:
Gauss-Legendre  5-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0355777183856
Gauss-Legendre  6-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357469750923
Gauss-Legendre  7-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357498197266
Gauss-Legendre  8-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357498544945
Gauss-Legendre  9-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357498548174
Gauss-Legendre 10-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357498548198
Gauss-Legendre 20-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.0357498548198

## REXX

### version 1

/*---------------------------------------------------------------------
* 31.10.2013 Walter Pachl  Translation from PL/I
* 01.11.2014 -"- see Version 2 for improvements
*--------------------------------------------------------------------*/
Call time 'R'
prec=60
Numeric Digits prec
epsilon=1/10**prec
pi=3.141592653589793238462643383279502884197169399375105820974944592307
exact = exp(3,prec)-exp(-3,prec)
Do n = 1 To 20
a = -3; b = 3
r.=0
sum=0
Do j=1 To n
sum=sum + r.2.j * exp((a+b)/2+r.1.j*(b-a)/2,prec)
End
z = (b-a)/2 * sum
Say right(n,2) format(z,2,40) format(z-exact,2,4,,0)
End
Say  '  ' exact '(exact)'
say '... and took' format(time('E'),,2) "seconds"
Exit

p0.0=1; p0.1=1
p1.0=2; p1.1=1; p1.2=0
Do k = 2 To n
tmp.0=p1.0+1
Do L = 1 To p1.0
tmp.l = p1.l
End
tmp.l=0
tmp2.0=p0.0+2
tmp2.1=0
tmp2.2=0
Do L = 1 To p0.0
l2=l+2
tmp2.l2=p0.l
End
Do j=1 To tmp.0
tmp.j = ((2*k-1)*tmp.j - (k-1)*tmp2.j)/k
End
p0.0=p1.0
Do j=1 To p0.0
p0.j = p1.j
End
p1.0=tmp.0
Do j=1 To p1.0
p1.j=tmp.j
End
End
Do i = 1 To n
x = cos(pi*(i-0.25)/(n+0.5),prec)
Do iter = 1 To 10
f = p1.1; df = 0
Do k = 2 To p1.0
df = f + x*df
f  = p1.k + x * f
End
dx =  f / df
x = x - dx
If abs(dx) < epsilon then leave
End
r.1.i = x
r.2.i = 2/((1-x**2)*df**2)
End
Return

cos: Procedure
/* REXX ****************************************************************
* Return cos(x) -- with specified precision
* cos(x) = 1-(x**2/2!)+(x**4/4!)-(x**6/6!)+-...
* 920903 Walter Pachl
***********************************************************************/
Parse Arg x,prec
If prec='' Then prec=9
Numeric Digits (2*prec)
Numeric Fuzz 3
o=1
u=1
r=1
Do i=1 By 2
ra=r
o=-o*x*x
u=u*i*(i+1)
r=r+(o/u)
If r=ra Then Leave
End
Numeric Digits prec
Return r+0

exp: Procedure
/***********************************************************************
* Return exp(x) -- with reasonable precision
* 920903 Walter Pachl
***********************************************************************/
Parse Arg x,prec
If prec<9 Then prec=9
Numeric Digits (2*prec)
Numeric Fuzz   3
o=1
u=1
r=1
Do i=1 By 1
ra=r
o=o*x
u=u*i
r=r+(o/u)
If r=ra Then Leave
End
Numeric Digits (prec)
Return r+0


Output:

 1  6.0000000000000000000000000000000000000000 -1.4036E+1
2 17.4874646410555689643606840462449458421154 -2.5483
3 19.8536919968055821921309108927158495960775 -1.8206E-1
4 20.0286883952907008527738054439857661647073 -7.0615E-3
5 20.0355777183855621539285357252750939315016 -1.7214E-4
6 20.0357469750923438830654575585499253741530 -2.8797E-6
7 20.0357498197266007755718729372891903369401 -3.5093E-8
8 20.0357498544945172882260918041683132616237 -3.2529E-10
9 20.0357498548174338368864419454858704839263 -2.3700E-12
10 20.0357498548197898711175766908543458234008 -1.3927E-14
11 20.0357498548198037305529147159697031241994 -6.7396E-17
12 20.0357498548198037976759531014454017742327 -2.7323E-19
13 20.0357498548198037979482458119092690701863 -9.4143E-22
14 20.0357498548198037979491844483599375945130 -2.7906E-24
15 20.0357498548198037979491872317401917248453 -7.1915E-27
16 20.0357498548198037979491872389153958789316 -1.6260E-29
17 20.0357498548198037979491872389316236038179 -3.2517E-32
18 20.0357498548198037979491872389316560624361 -5.7920E-35
19 20.0357498548198037979491872389316561202637 -9.2480E-38
20 20.0357498548198037979491872389316561203561 -1.3311E-40
20.0357498548198037979491872389316561203562082463657269288113 (exact)
... and took 4.97 seconds

### version 2

This REXX version (an optimized version of version 1)   and uses:

•   a faster   cos   function   (with full precision)
•   a faster   exp   function   (with full precision)
•   some simple variables instead of stemmed arrays
•   some static variables instead of repeated expressions
•   calculations using full (specified) precision (numeric digits)
•   multiplication using   [··· *.5]   instead of division using   [··· /2]
•   a generic approach for setting the   numeric digits
•   a better test for earlier termination (stopping) of calculations
•   a more precise value for   pi
•   shows an arrow that points where the GLQ number matches the exact value
•   displays the number of decimal digits that match the exact value

The execution speed of this REXX program is largely dependent on the number of decimal digits in   pi.
If faster speed is desired,   the number of the decimal digits of   pi   can be reduced.

Each iteration yields around three more (fractional) decimal digits   (past the decimal point).

The use of "vertical bars" is one of the very few times to use leading comments, as there isn't that many
situations where there exists nested     do     loops with different (grouped) sizable indentations,   and
where there's practically no space on the right side of the REXX source statements.   It presents a good
visual indication of what's what,   but it's the dickens to pay when updating the source code.

/*REXX program does numerical integration using an N─point Gauss─Legendre quadrature rule.   */
pi= pi();     digs= length(pi) - length(.);          numeric digits digs;       reps= digs % 2

!.= .;        b= 3;        a= -b;       bma= b - a;          bmaH= bma / 2;     tiny= '1e-'digs
trueV= exp(b)-exp(a);                   bpa= b + a;          bpaH= bpa / 2
hdr= 'iterate value       (with '   digs   " decimal digits being used)"
say ' step '  center(hdr, digs+3)    '  difference'                  /*show hdr*/
sep='──────'  copies("─", digs+3)    '─────────────';      say sep   /*  "  sep*/

do #=1  until dif>0;  p0z= 1;  p0.1= 1;  p1z= 2;  p1.1= 1;  p1.2= 0;  ##= # + .5;  r.= 0
/*█*/   do k=2  to #;  km= k - 1
/*█*/                     do y=1  for p1z;   T.y= p1.y;                           end  /*y*/
/*█*/   T.y= 0;  TT.= 0;          do L=1  for p0z;   _= L + 2;   TT._= p0.L;      end  /*L*/
/*█*/   kkm= k + km;      do j=1  for p1z  +1;       T.j= (kkm*T.j - km*TT.j)/k;  end  /*j*/
/*█*/   p0z= p1z;         do n=1  for p0z;           p0.n= p1.n                ;  end  /*n*/
/*█*/   p1z= p1z + 1;     do p=1  for p1z;           p1.p= T.p                 ;  end  /*p*/
/*█*/   end   /*k*/
/*▓*/       do !=1  for #;             x= cos( pi * (! - .25)  /  ## )
/*▓*/           /*░*/   do reps  until abs(dx) <= tiny
/*▓*/           /*░*/   f= p1.1;  df= 0;   do u=2  to p1z; df= f +  x*df
/*▓*/           /*░*/                                       f= p1.u +x*f
/*▓*/           /*░*/                      end   /*u*/
/*▓*/           /*░*/   dx= f / df;   x= x - dx
/*▓*/           /*░*/   end             /*reps ···*/
/*▓*/       r.1.!= x
/*▓*/       r.2.!= 2 / ( (1 - x*x) * df*df)
/*▓*/       end   /*!*/
$= 0 /*▒*/ do m=1 for #;$=$+ r.2.m * exp(bpaH + r.1.m*bmaH); end /*m*/ z= bmaH *$                                                    /*calculate target value (Z)*/
dif= z - trueV;            z= format(z, 3, digs - 2)           /*    "     difference.     */
Ndif= translate( format(dif, 3, 4, 2, 0),  'e',  "E")
if #\==1  then  say center(#, 6)      z' '      Ndif           /*no display if not computed*/
end   /*#*/

say sep;  xdif= compare( strip(z), trueV);                       say right("↑", 6 + 1 + xdif)
say  left('', 6 + 1)      trueV         " {exact value}";        say
say 'Using '      digs      " digit precision, the" ,
'N-point Gauss─Legendre quadrature (GLQ) had an accuracy of '      xdif-2       " digits."
exit 0                                                /*stick a fork in it,  we're all done. */
/*───────────────────────────────────────────────────────────────────────────────────────────*/
e:   return 2.718281828459045235360287471352662497757247093699959574966967627724076630353547595
pi:  return 3.141592653589793238462643383279502884197169399375105820974944592307816406286286209
/*───────────────────────────────────────────────────────────────────────────────────────────*/
cos: procedure  expose !.; parse arg x;   if !.x\==.  then return !.x;   _= 1;   z=1;  y= x*x
do k=2  by 2  until p==z; p=z; _= -_*y/(k*(k-1)); z=z+_; end;  !.x=z;  return z
/*───────────────────────────────────────────────────────────────────────────────────────────*/
exp: procedure; parse arg x; ix= x % 1;  if abs(x-ix)>.5  then ix= ix + sign(x); x= x-ix;  z= 1
_=1;  do j=1  until p==z; p=z;  _= _*x/j;  z= z+_;   end;    return z * e()**ix

output   when using the default inputs:
 step                iterate value       (with  82  decimal digits being used)                 difference
────── ───────────────────────────────────────────────────────────────────────────────────── ─────────────
2     17.48746464105556896436068404624494584211542841793491350914872470595379166623788825   -2.5483
3     19.85369199680558219213091089271584959607746673197538889290500270758485925164498330   -1.8206e-01
4     20.02868839529070085277380544398576616470733632504815180772578876685215146483792186   -7.0615e-03
5     20.03557771838556215392853572527509393150162720744712830816732425295141661302212542   -1.7214e-04
6     20.03574697509234388306545755854992537415299478921975125717616705900225010375271175   -2.8797e-06
7     20.03574981972660077557187293728919033694006575323784891307591676343623185267840087   -3.5093e-08
8     20.03574985449451728822609180416831326162367525799440551006933045513903380452620872   -3.2529e-10
9     20.03574985481743383688644194548587048392631680869557979312925905853201983429400861   -2.3700e-12
10    20.03574985481978987111757669085434582340083496254465680809367957309381342059009668   -1.3927e-14
11    20.03574985481980373055291471596970312419935163064851758082919292076105448665845694   -6.7396e-17
12    20.03574985481980379767595310144540177423271389844296074380175787717157675883917151   -2.7323e-19
13    20.03574985481980379794824581190926907018626592287853070355830814733619000088357912   -9.4143e-22
14    20.03574985481980379794918444835993759451301483567068863329194414460270391327442654   -2.7906e-24
15    20.03574985481980379794918723174019172484527341186430917498972813563388327387142320   -7.1915e-27
16    20.03574985481980379794918723891539587893161294648949828480207158337867091213105210   -1.6260e-29
17    20.03574985481980379794918723893162360381792525574404539062822509053852218733547782   -3.2517e-32
18    20.03574985481980379794918723893165606243605713014841119742440194777360958854209572   -5.7920e-35
19    20.03574985481980379794918723893165612026372831720742415561589728335786348943623570   -9.2480e-38
20    20.03574985481980379794918723893165612035607513408575037519944422231638669124167990   -1.3311e-40
21    20.03574985481980379794918723893165612035620807276164638611436475769849940475037458   -1.7360e-43
22    20.03574985481980379794918723893165612035620824615962445370778636022384338924992003   -2.0610e-46
23    20.03574985481980379794918723893165612035620824636550325344849506916698800464997617   -2.2368e-49
24    20.03574985481980379794918723893165612035620824636572670605090159763145237587025264   -2.2276e-52
25    20.03574985481980379794918723893165612035620824636572692860700178828249236875179273   -2.0430e-55
26    20.03574985481980379794918723893165612035620824636572692881113337954261894220969394   -1.7312e-58
27    20.03574985481980379794918723893165612035620824636572692881130636614548220525870297   -1.3595e-61
28    20.03574985481980379794918723893165612035620824636572692881130650199357864896908624   -9.9207e-65
29    20.03574985481980379794918723893165612035620824636572692881130650209271775421848621   -6.7456e-68
30    20.03574985481980379794918723893165612035620824636572692881130650209278516823348154   -4.2128e-71
31    20.03574985481980379794918723893165612035620824636572692881130650209278518859457416   -2.1767e-71
32    20.03574985481980379794918723893165612035620824636572692881130650209278521040018937    3.8415e-74
────── ───────────────────────────────────────────────────────────────────────────────────── ─────────────
↑
20.03574985481980379794918723893165612035620824636572692881130650209278521036177419  {exact value}

Using  82  digit precision,  the N-point Gauss─Legendre quadrature (GLQ) had an accuracy of  74  digits.


### version 3, more precision

This REXX version is almost an exact copy of REXX version 2,   but with about twice as the number of decimal digits of   pi   and   e.

It is about twice as slow as version 2,   due to the doubling of the number of decimal digits   (precision).

/*REXX program does numerical integration using an N─point Gauss─Legendre quadrature rule.   */
pi= pi();     digs= length(pi) - length(.);          numeric digits digs;       reps= digs % 2
!.= .;        b= 3;        a= -b;       bma= b - a;          bmaH= bma / 2;     tiny= '1e-'digs
trueV= exp(b)-exp(a);                   bpa= b + a;          bpaH= bpa / 2
hdr= 'iterate value       (with '   digs   " decimal digits being used)"
say ' step '  center(hdr, digs+3)    '  difference'                  /*show hdr*/
sep='──────'  copies("─", digs+3)    '─────────────';      say sep   /*  "  sep*/

do #=1  until dif>0;  p0z= 1;  p0.1= 1;  p1z= 2;  p1.1= 1;  p1.2= 0;  ##= # + .5;  r.= 0
/*█*/   do k=2  to #;  km= k - 1
/*█*/                     do y=1  for p1z;   T.y= p1.y;                           end  /*y*/
/*█*/   T.y= 0;  TT.= 0;          do L=1  for p0z;   _= L + 2;   TT._= p0.L;      end  /*L*/
/*█*/   kkm= k + km;      do j=1  for p1z  +1;       T.j= (kkm*T.j - km*TT.j)/k;  end  /*j*/
/*█*/   p0z= p1z;         do n=1  for p0z;           p0.n= p1.n                ;  end  /*n*/
/*█*/   p1z= p1z + 1;     do p=1  for p1z;           p1.p= T.p                 ;  end  /*p*/
/*█*/   end   /*k*/
/*▓*/       do !=1  for #;             x= cos( pi * (! - .25)  /  ## )
/*▓*/           /*░*/   do reps  until abs(dx) <= tiny
/*▓*/           /*░*/   f= p1.1;  df= 0;   do u=2  to p1z; df= f +  x*df
/*▓*/           /*░*/                                       f= p1.u +x*f
/*▓*/           /*░*/                      end   /*u*/
/*▓*/           /*░*/   dx= f / df;   x= x - dx
/*▓*/           /*░*/   end             /*reps ···*/
/*▓*/       r.1.!= x
/*▓*/       r.2.!= 2 / ( (1 - x*x) * df*df)
/*▓*/       end   /*!*/
$= 0 /*▒*/ do m=1 for #;$=$+ r.2.m * exp(bpaH + r.1.m*bmaH); end /*m*/ z= bmaH *$                                                    /*calculate target value (Z)*/
dif= z - trueV;            z= format(z, 3, digs - 2)           /*    "     difference.     */
Ndif= translate( format(dif, 3, 4, 3, 0),  'e',  "E")
if #\==1  then  say center(#, 6)      z' '      Ndif           /*no display if not computed*/
end   /*#*/

say sep;  xdif= compare( strip(z), trueV);                       say right("↑", 6 + 1 + xdif)
say  left('', 6 + 1)      trueV         " {exact value}";        say
say 'Using '      digs      " digit precision, the" ,
'N-point Gauss─Legendre quadrature (GLQ) had an accuracy of '      xdif-2       " digits."
exit 0                                                /*stick a fork in it,  we're all done. */
/*───────────────────────────────────────────────────────────────────────────────────────────*/
e:   return 2.71828182845904523536028747135266249775724709369995957496696762772407663035354759,
||457138217852516642742746639193200305992181741359662904357290033429526059563073813232862794
/*───────────────────────────────────────────────────────────────────────────────────────────*/
pi:  return 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899,
||862803482534211706798214808651328230664709384460955058223172535940812848111745028410270194
/*───────────────────────────────────────────────────────────────────────────────────────────*/
cos: procedure  expose !.; parse arg x;   if !.x\==.  then return !.x;   _= 1;   z=1;  y= x*x
do k=2  by 2  until p==z; p=z; _= -_*y/(k*(k-1)); z=z+_; end;  !.x=z;  return z
/*───────────────────────────────────────────────────────────────────────────────────────────*/
exp: procedure; parse arg x; ix= x % 1;  if abs(x-ix)>.5  then ix= ix + sign(x); x= x-ix;  z= 1
_=1;  do j=1  until p==z; p=z;  _= _*x/j;  z= z+_;   end;    return z * e()**ix

output   when using the default inputs:

 step                                                            iterate value       (with  171  decimal digits being used)                                                             difference
────── ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────── ─────────────
2     17.4874646410555689643606840462449458421154284179349135091487247059537916662378882444064336021640614626063744948781912964250403870127054497392082425535068464109311173377377   -2.5483
3     19.8536919968055821921309108927158495960774667319753888929050027075848592516449832906645902758379575999249091274157148988582792112906526877518087112700785494497813902725450   -1.8206e-001
4     20.0286883952907008527738054439857661647073363250481518077257887668521514648379218096268747927750038360903142778646220077613647092768733641727539206268833693587721944236294   -7.0615e-003
5     20.0355777183855621539285357252750939315016272074471283081673242529514166130221254213250349496939691709537643294259047823350162410908440808868981982394287542087129417151006   -1.7214e-004
6     20.0357469750923438830654575585499253741529947892197512571761670590022501037527117346339483928363770582109285164930728028479549289382406446621705905363209981936742762651248   -2.8797e-006
7     20.0357498197266007755718729372891903369400657532378489130759167634362318526784010016150667027038415189719144094529764766032097831604495667799067330556673881537789420232152   -3.5093e-008
8     20.0357498544945172882260918041683132616236752579944055100693304551390338045262089091194019302017562870527315644307417688383478919210145963055448428522264642589709805903057   -3.2529e-010
9     20.0357498548174338368864419454858704839263168086955797931292590585320198342940085570553927472311015418220675609961921140415760514983040167737226050690228927266115828876520   -2.3700e-012
10    20.0357498548197898711175766908543458234008349625446568080936795730938134205900980645938318794902592556558231569959762420203929344018773329199723457149763574278017459859529   -1.3927e-014
11    20.0357498548198037305529147159697031241993516306485175808291929207610544866584568009626862857221858328844106864371425322111609007302709732793823163103980149601875492907998   -6.7396e-017
12    20.0357498548198037976759531014454017742327138984429607438017578771715767588391691509175808718708593063121709896967107496243434245185896147055314894150234262032514577087792   -2.7323e-019
13    20.0357498548198037979482458119092690701862659228785307035583081473361900008835808932495328864420024278695427964698380448330606714160259282675390182203803538192726572599929   -9.4143e-022
14    20.0357498548198037979491844483599375945130148356706886332919441446027039132743905494286471338717783707421873433644754993992655580745072286831502363474798170771121237677390   -2.7906e-024
15    20.0357498548198037979491872317401917248452734118643091749897281356338832738714150881537113815780435230011480697467170623887897830301712412973655748924184136940242004265158   -7.1915e-027
16    20.0357498548198037979491872389153958789316129464894982848020715833786709121310547889685984881568546203564135185474792767674806869872650180714616455691318785641503320488704   -1.6260e-029
17    20.0357498548198037979491872389316236038179252557440453906282250905385221873347716826354198555233437240574026019817833907372014036252533047705435353247648512336234642790641   -3.2517e-032
18    20.0357498548198037979491872389316560624360571301484111974244019477736095885421361807599231231543821951618639462965984321643251022835234451110049047608124964855646728491571   -5.7920e-035
19    20.0357498548198037979491872389316561202637283172074241556158972833578634894365092635000776399956033063018069653085902399896542171129596405210008317497301938111107401607602   -9.2480e-038
20    20.0357498548198037979491872389316561203560751340857503751994442223163866912408434007886096643419528065940077022083150476496426837665378721283432879108630829513249759484353   -1.3311e-040
21    20.0357498548198037979491872389316561203562080727616463861143647576984994047530870779393715057751591887673397688454357985082021265151278191050057935329724914648356586984041   -1.7360e-043
22    20.0357498548198037979491872389316561203562082461596244537077863602238433892612703628843743785373313737563806457244053157873973239947461987202443878362980281616080907191625   -2.0610e-046
23    20.0357498548198037979491872389316561203562082463655032534484950691669880046406047078766996078695370527223056578914332723730363863326194707715142045831095238426102807682133   -2.2368e-049
24    20.0357498548198037979491872389316561203562082463657267060509015976314523758814742624773428457390528961843568960502876896215809857825164102337905868347722728364661655423691   -2.2276e-052
25    20.0357498548198037979491872389316561203562082463657269286070017882824923688080311511389836619043005851350331110867389220628954338053656628671036072512304656757933297348289   -2.0430e-055
26    20.0357498548198037979491872389316561203562082463657269288111333795426189423729667519158562143832977811003145168351321839626313132075697513253761673496847193697358302206599   -1.7312e-058
27    20.0357498548198037979491872389316561203562082463657269288113063661454822050198926197665008333893008724687497228278730367375441075263700413282548634210893951621431572014401   -1.3595e-061
28    20.0357498548198037979491872389316561203562082463657269288113065019935786483820352375621786828318969009163053743757325024448325026804644277866300802833735429200407643132066   -9.9207e-065
29    20.0357498548198037979491872389316561203562082463657269288113065020927177593233999249852447888627901300469719564790181325442944469692690797774430312247184030485560959159838   -6.7451e-068
30    20.0357498548198037979491872389316561203562082463657269288113065020927851675301934062025341716601075750412806887227020916063849030412480955063639628314158527843447097540104   -4.2832e-071
31    20.0357498548198037979491872389316561203562082463657269288113065020927852103363148863217394106431702791915956948972366384835732103508918001327415359845732098066185095970907   -2.5459e-074
32    20.0357498548198037979491872389316561203562082463657269288113065020927852103617599854934274435013875248206413049448382025586066461615726348079942124358556139880490254984356   -1.4196e-077
33    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741736109635347323907131494641377410353985987829217992622815976248321175867964752131506800051   -7.4395e-081
34    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810467715704772209566910717933633388969835872983190631663850670877761750073036465167190394   -3.6713e-084
35    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504412069378446854036859408497315019337333762510854198446941961781825098514532469683329   -1.7091e-087
36    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429152646383719980280460795167918691617029439367737607466797188696985999193933984760   -7.5175e-091
37    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160160714391043273984198489693834991216803247954607301723484371150995472545047773   -3.1292e-094
38    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163842341789925349746540298990681930753381942866562579916746319742876113289347   -1.2345e-097
39    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843575809851614709658383098559963930599249691243551258257666303808499450058   -4.6221e-101
40    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576271898405984614568086424291240202255560215708382127745410021921505433   -1.6447e-104
41    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062816325200556739918251890227721352129417700490117766374046259608   -5.5685e-108
42    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062871992591098295378332977741979460337046289653292852558991470138   -1.7962e-111
43    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010547683152388008632372342584044010171728180146818963258851   -5.5262e-115
44    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553207686109280576604524821212783897594720674601807871384   -1.6234e-118
45    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309007883789778553235095566868388697120439100484887   -4.5581e-122
46    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463568475856093621234235103013989004662708710524   -1.2245e-125
47    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690894669420459180906124959094321731281026582   -3.1504e-129
48    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926165554457660180369582395139692931382774   -7.7695e-133
49    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173322092766618204788515060681700867922   -1.8383e-136
50    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323930673674948485214173181944866992   -4.1766e-140
51    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323931091242430000931491744551038314   -9.1189e-144
52    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323931091333599358956493201909187002   -1.9148e-147
53    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323931091333618502674999936415396203   -3.9287e-151
54    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323931091333618506574837300809273433   -2.8877e-153
55    20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323931091333618507014523492331476840    4.1081e-152
────── ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────── ─────────────
↑
20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323931091333618506603713959668429768  {exact value}

Using  171  digit precision, the N-point Gauss─Legendre quadrature (GLQ) had an accuracy of  152  digits.


## Scala

Output:

Best seen in running your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).

import scala.math.{Pi, cos, exp}

private val N = 5

private def legeInte(a: Double, b: Double): Double = {
val (c1, c2) = ((b - a) / 2, (b + a) / 2)
val tuples: IndexedSeq[(Double, Double)] = {
val lcoef = {
val lcoef = Array.ofDim[Double](N + 1, N + 1)

lcoef(0)(0) = 1
lcoef(1)(1) = 1
for (i <- 2 to N) {
lcoef(i)(0) = -(i - 1) * lcoef(i - 2)(0) / i
for (j <- 1 to i) lcoef(i)(j) =
((2 * i - 1) * lcoef(i - 1)(j - 1) - (i - 1) * lcoef(i - 2)(j)) / i
}
lcoef
}

def legeEval(n: Int, x: Double): Double =
lcoef(n).take(n).foldRight(lcoef(n)(n))((o, s) => s * x + o)

def legeDiff(n: Int, x: Double): Double =
n * (x * legeEval(n, x) - legeEval(n - 1, x)) / (x * x - 1)

@scala.annotation.tailrec
def convergention(x0: Double, x1: Double): Double = {
if (x0 == x1) x1
else convergention(x1, x1 - legeEval(N, x1) / legeDiff(N, x1))
}

for {i <- 0 until 5
x = convergention(0.0, cos(Pi * (i + 1 - 0.25) / (N + 0.5)))
x1 = legeDiff(N, x)
} yield (x, 2 / ((1 - x * x) * x1 * x1))
}

println(s"Roots: ${tuples.map(el => f"${el._1}%10.6f").mkString}")
println(s"Weight:${tuples.map(el => f"${el._2}%10.6f").mkString}")

c1 * tuples.map { case (lroot, weight) => weight * exp(c1 * lroot + c2) }.sum
}

println(f"Integrating exp(x) over [-3, 3]:\n\t${legeInte(-3, 3)}%10.8f,") println(f"compared to actual%n\t${exp(3) - exp(-3)}%10.8f")

}


## Sidef

Translation of: Raku
func legendre_pair((1), x) { (x, 1) }
func legendre_pair( n,  x) {
var (m1, m2) = legendre_pair(n - 1, x)
var u = (1 - 1/n)
((1 + u)*x*m1 - u*m2, m1)
}

func legendre((0), _) { 1 }
func legendre( n,  x) { [legendre_pair(n, x)][0] }

func legendre_prime({ .is_zero }, _) { 0 }
func legendre_prime({ .is_one  }, _) { 1 }

func legendre_prime(n, x) {
var (m0, m1) = legendre_pair(n, x)
(m1 - x*m0) * n / (1 - x**2)
}

func approximate_legendre_root(n, k) {
# Approximation due to Francesco Tricomi
var t = ((4*k - 1) / (4*n + 2))
(1 - ((n - 1)/(8 * n**3))) * cos(Num.pi * t)
}

func newton_raphson(f, f_prime, r, eps = 2e-16) {
loop {
var dr = (-f(r) / f_prime(r))
dr.abs >= eps || break
r += dr
}
return r
}

func legendre_root(n, k) {
newton_raphson(legendre.method(:call, n), legendre_prime.method(:call, n),
approximate_legendre_root(n, k))
}

func weight(n, r) { 2 / ((1 - r**2) * legendre_prime(n, r)**2) }

func nodes(n) {
gather {
take(Pair(0, weight(n, 0))) if n.is_odd
{ |i|
var r = legendre_root(n, i)
var w = weight(n, r)
take(Pair(r, w), Pair(-r, w))
}.each(1 .. (n >> 1))
}
}

func quadrature(n, f, a, b, nds = nodes(n)) {
func scale(x) { (x*(b - a) + a + b) / 2 }
(b - a) / 2 * nds.sum { .second * f(scale(.first)) }
}

[(5..10)..., 20].each { |i|
printf("Gauss-Legendre %2d-point quadrature ∫₋₃⁺³ exp(x) dx ≈ %.15f\n",
}

Output:
Gauss-Legendre  5-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035577718385561
Gauss-Legendre  6-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035746975092344
Gauss-Legendre  7-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035749819726600
Gauss-Legendre  8-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035749854494515
Gauss-Legendre  9-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035749854817432
Gauss-Legendre 10-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035749854819791
Gauss-Legendre 20-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035749854819805

## Tcl

Translation of: Common Lisp
Library: Tcllib (Package: math::constants)
Library: Tcllib (Package: math::polynomials)
Library: Tcllib (Package: math::special)
package require Tcl 8.5
package require math::special
package require math::polynomials
package require math::constants
math::constants::constants pi

# Computes the initial guess for the root i of a n-order Legendre polynomial
proc guess {n i} {
global pi
expr { cos($pi * ($i - 0.25) / ($n + 0.5)) } } # Computes and evaluates the n-order Legendre polynomial at the point x proc legpoly {n x} { math::polynomials::evalPolyn [math::special::legendre$n] $x } # Computes and evaluates the derivative of an n-order Legendre polynomial at point x proc legdiff {n x} { expr {$n / ($x**2 - 1) * ($x * [legpoly $n$x] - [legpoly [incr n -1] $x])} } # Computes the n nodes for an n-point quadrature rule. (i.e. n roots of a n-order polynomial) proc nodes n { set x [lrepeat$n 0.0]
for {set i 0} {$i <$n} {incr i} {
set val [guess $n [expr {$i + 1}]]
foreach . {1 2 3 4 5} {
set val [expr {$val - [legpoly$n $val] / [legdiff$n $val]}] } lset x$i $val } return$x
}

# Computes the weight for an n-order polynomial at the point (node) x
proc legwts {n x} {
expr {2.0 / (1 - $x**2) / [legdiff$n $x]**2} } # Takes a array of nodes x and computes an array of corresponding weights w proc weights x { set n [llength$x]
set w {}
foreach xi $x { lappend w [legwts$n $xi] } return$w
}

# Integrates a lambda term f with a n-point Gauss-Legendre quadrature rule over the interval [a,b]
proc gausslegendreintegrate {f n a b} {
set x [nodes $n] set w [weights$x]
set rangesize2 [expr {($b -$a)/2}]
set rangesum2 [expr {($a +$b)/2}]
set sum 0.0
foreach xi $x wi$w {
set y [expr {$rangesize2*$xi + $rangesum2}] set sum [expr {$sum + $wi*[apply$f $y]}] } expr {$sum * $rangesize2} }  Demonstrating: puts "nodes(5) = [nodes 5]" puts "weights(5) = [weights [nodes 5]]" set exp {x {expr {exp($x)}}}
puts "int(exp,-3,3) = [gausslegendreintegrate $exp 5 -3 3]"  Output: nodes(5) = 0.906179845938664 0.5384693101056831 -1.198509146801203e-94 -0.5384693101056831 -0.906179845938664 weights(5) = 0.2369268850561896 0.4786286704993664 0.5688888888888889 0.4786286704993664 0.2369268850561896 int(exp,-3,3) = 20.03557771838559  ## Ursala using arbitrary precision arithmetic #import std #import nat legendre = # takes n to the pair of functions (P_n,P'_n), where P_n is the Legendre polynomial of order n ~&?\(1E0!,0E0!)! -+ ^|/~& //mp..vid^ mp..sub\1E0+ mp..sqr, ~~ "c". ~&\1E0; ~&\"c"; ~&ar^?\0E0! mp..add^/mp..mul@alrPrhPX ^|R/~& ^|\~&t ^/~&l mp..mul, @iiXNX ~&rZ->r @l ^/^|(~&tt+ sum@NNiCCiX+ successor,~&) both~&g&&~&+ -+ ~* mp..zero_p?/~& (&&~&r ~&EZ+ ~~ mp..prec)^/~& ^(~&,..shr\8); mp..equ^|(~&,..gro\8)->l @r ^/~& ..shr\8, ^(~&rl,mp..mul*lrrPD)^/..nat2mp@r -+ ^(~&l,mp..sub*+ zipp0E0^|\~& :/0E0)+ ~&rrt->lhthPX ^( ^lrNCC\~&lh mp..vid^*D/..nat2mp@rl -+ mp..sub*+ zipp0E0^|\~& :/0E0, mp..mul~*brlD^|bbI/~&hthPX @l ..nat2mp~~+ predecessor~~NiCiX+-, @r ^|/successor predecessor), ^|(mp..grow/1E0; @iNC ^lrNCC\~& :/0E0,~&/2)+-+-+- nodes = # takes precision and order (p,n) to a list of nodes and weights <(x_1,w_1)..(x_n,w_n)> -+ ^H( @lrr *+ ^/~&+ mp..div/( ..nat2mp 2)++ mp..mul^/(mp..sqr; //mp..sub ..nat2mp 1)+ mp..sqr+, mp..shr^*DrlXS/~&ll ^|H\~& *+ @NiX+ ->l^|(~&lZ!|+ not+ //mp..eq,@r+ ^/~&+ mp..sub^/~&+ mp..div^)), ^/^|(~&,legendre) mp..cos*+ mp..mul^*D( mp..div^|/mp..pi@NiC mp..add/5E-1+ ..nat2mp, @r mp..bus/*2.5E-1+ ..nat2mp*+ nrange/1)+- integral = # takes precision and order (p,n) to a function taking a function and interval (f,(a,b)) ("p","n"). -+ mp..shrink^/~& difference\"p"+ mp..prec, mp..mul^|/~& mp..add:-0E0+ * mp..mul^/~&rr ^H/~&ll mp..add^\~&lrr mp..mul@lrPrXl, ^(~&rl,-*nodes("p","n"))^|/~& mp..vid~~G/2E0+ ^/mp..bus mp..add+- demonstration program: #show+ demo = ~&lNrCT ( ^|lNrCT(:/'nodes:',:/'weights:')@lSrSX ..mp2str~~* nodes/160 5, :/'integral:' ~&iNC ..mp2str integral(160,5) (mp..exp,-3E0,3E0)) Output: nodes: 9.0617984593866399279762687829939296512565191076233E-01 5.3846931010568309103631442070020880496728660690555E-01 0.0000000000000000000000000000000000000000000000000E+00 -5.3846931010568309103631442070020880496728660690555E-01 -9.0617984593866399279762687829939296512565191076233E-01 weights: 2.3692688505618908751426404071991736264326000220463E-01 4.7862867049936646804129151483563819291229555334456E-01 5.6888888888888888888888888888888888888888888888896E-01 4.7862867049936646804129151483563819291229555334456E-01 2.3692688505618908751426404071991736264326000220463E-01 integral: 2.0035577718385562153928535725275093931501627207110E+01 ## Wren Translation of: C Library: Wren-fmt import "./fmt" for Fmt var N = 5 var lroots = List.filled(N, 0) var weight = List.filled(N, 0) var lcoef = List.filled(N+1, null) for (i in 0..N) lcoef[i] = List.filled(N + 1, 0) var legeCoef = Fn.new { lcoef[0][0] = lcoef[1][1] = 1 for (n in 2..N) { lcoef[n][0] = -(n-1) * lcoef[n -2][0] / n for (i in 1..n) { lcoef[n][i] = ((2*n - 1) * lcoef[n-1][i-1] - (n - 1) * lcoef[n-2][i]) / n } } } var legeEval = Fn.new { |n, x| (n..1).reduce(lcoef[n][n]) { |s, i| s*x + lcoef[n][i-1] } } var legeDiff = Fn.new { |n, x| return n * (x * legeEval.call(n, x) - legeEval.call(n-1, x)) / (x*x - 1) } var legeRoots = Fn.new { var x = 0 var x1 = 0 for (i in 1..N) { x = (Num.pi * (i - 0.25) / (N + 0.5)).cos while (true) { x1 = x x = x - legeEval.call(N, x) / legeDiff.call(N, x) if (x == x1) break } lroots[i-1] = x x1 = legeDiff.call(N, x) weight[i-1] = 2 / ((1 - x*x) * x1 * x1) } } var legeIntegrate = Fn.new { |f, a, b| var c1 = (b - a) / 2 var c2 = (b + a) / 2 var sum = 0 for (i in 0...N) sum = sum + weight[i] * f.call(c1*lroots[i] + c2) return c1 * sum } legeCoef.call() legeRoots.call() System.write("Roots: ") for (i in 0...N) Fmt.write("$f", lroots[i])
System.write("\nWeight:")
for (i in 0...N) Fmt.write(" $f", weight[i]) var f = Fn.new { |x| x.exp } var actual = 3.exp - (-3).exp Fmt.print("\nIntegrating exp(x) over [-3, 3]:\n\t$10.8f,\n" +
"compared to actual\n\t\$10.8f", legeIntegrate.call(f, -3, 3), actual)

Output:
Roots:  0.906180 0.538469 0.000000 -0.538469 -0.906180
Weight: 0.236927 0.478629 0.568889 0.478629 0.236927
Integrating exp(x) over [-3, 3]:
20.03557772,
compared to actual
20.03574985


## zkl

Translation of: Raku
fcn legendrePair(n,x){ //-->(float,float)
if(n==1) return(x,1.0);
m1,m2:=legendrePair(n-1,x);
u:=1.0 - 1.0/n;
return( (u + 1)*x*m1 - u*m2, m1);
}
fcn legendre(n,x){ //-->float
if(n==0) return(0.0);
legendrePair(n,x)[0]
}
fcn legendrePrime(n,x){ //-->float
if(n==0) return(0.0);
if(n==1) return(1.0);
m0,m1:=legendrePair(n,x);
(m1 - m0*x)*n/(1.0 - x*x);
}
fcn approximateLegendreRoot(n,k){ # Approximation due to Francesco Tricomi
t:=(4.0*k - 1)/(4.0*n + 2);
(1.0 - (n - 1)/(8*n*n*n))*((0.0).pi*t).cos();
}
fcn newtonRaphson(f,fPrime,r,eps=2.0e-16){
while(not (dr:=-f(r)/fPrime(r)).closeTo(0.0,eps)){ r+=dr }
r;
}
fcn legendreRoot(n,k){
newtonRaphson(legendre.fp(n),legendrePrime.fp(n),
approximateLegendreRoot(n,k));
}
fcn weight(n,r){
lp:=legendrePrime(n,r);
2.0/((1.0 - r*r)*lp*lp)
}
fcn nodes(n){ //-->( (r,weight), (r,w), ...) length n
sink:=n.isOdd and L(T(0.0,weight(n,0))) or List;
(1).pump(n/2,sink,'wrap(m){
r:=legendreRoot(n,m);
w:=weight(n,r);
return( Void.Write,T(r,w),T(-r,w) )
})
}
if(not nds) nds=nodes(n);
scale:='wrap(x){ (x*(b - a) + a + b) / 2 };
nds.reduce('wrap(p,[(r,w)]){ p + w*f(scale(r)) },0.0) * (b - a)/2
}
[5..10].walk().append(20).pump(Console.println,fcn(n){
})
Gauss-Legendre  5-point quadrature ∫₋₃⁺³ exp(x) dx = 20.0355777183856