⚠️ Warning: This is a draft ⚠️

This means it might contain formatting issues, incorrect code, conceptual problems, or other severe issues.

If you want to help to improve and eventually enable this page, please fork RosettaGit's repository and open a merge request on GitHub.

{{Task|Arithmetic operations}} [[Category:Arithmetic]] [[Category:Mathematics]]

{|border=1 cellspacing=0 cellpadding=3 |In a general [[wp:Gaussian quadrature|Gaussian quadrature]] rule, an definite integral of $f\left(x\right)$ is first approximated over the interval $\left[-1,1\right]$ by a polynomial approximable function $g\left(x\right)$ and a known weighting function $W\left(x\right)$. |$\int_\left\{-1\right\}^1 f\left(x\right) , dx = \int_\left\{-1\right\}^1 W\left(x\right) g\left(x\right) , dx$ |- |Those are then approximated by a sum of function values at specified points $x_i$ multiplied by some weights $w_i$: |$\int_\left\{-1\right\}^1 W\left(x\right) g\left(x\right) , dx \approx \sum_\left\{i=1\right\}^n w_i g\left(x_i\right)$ |- |In the case of Gauss-Legendre quadrature, the weighting function $W\left(x\right) = 1$, so we can approximate an integral of $f\left(x\right)$ with: |$\int_\left\{-1\right\}^1 f\left(x\right),dx \approx \sum_\left\{i=1\right\}^n w_i f\left(x_i\right)$ |}

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 [[Numerical Integration|simple numerical integration methods]].

{|border=1 cellspacing=0 cellpadding=3 |The $n$ evaluation points $x_i$ for a n-point rule, also called "nodes", are roots of n-th order [[wp:Legendre Polynomials|Legendre Polynomials]] $P_n\left(x\right)$. Legendre polynomials are defined by the following recursive rule: |$P_0\left(x\right) = 1$
$P_1\left(x\right) = x$
$nP_\left\{n\right\}\left(x\right) = \left(2n-1\right)xP_\left\{n-1\right\}\left(x\right)-\left(n-1\right)P_\left\{n-2\right\}\left(x\right)$ |- |There is also a recursive equation for their derivative: |$P_\left\{n\right\}\text{'}\left(x\right) = \frac\left\{n\right\}\left\{x^2-1\right\} \left\left( x P_n\left(x\right) - P_\left\{n-1\right\}\left(x\right) \right\right)$ |- |The roots of those polynomials are in general not analytically solvable, so they have to be approximated numerically, for example by [[wp:Newton's method|Newton-Raphson iteration]]: |$x_\left\{n+1\right\} = x_n - \frac\left\{f\left(x_n\right)\right\}\left\{f\text{'}\left(x_n\right)\right\}$ |- |The first guess $x_0$ for the $i$-th root of a $n$-order polynomial $P_n$ can be given by |$x_0 = \cos \left\left( \pi , \frac\left\{i - \frac\left\{1\right\}\left\{4\right\}\right\}\left\{n+\frac\left\{1\right\}\left\{2\right\}\right\} \right\right)$ |- |After we get the nodes $x_i$, we compute the appropriate weights by: |$w_i = \frac\left\{2\right\}\left\{\left\left( 1-x_i^2 \right\right) \left[P\text{'}$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 $\left[a,b\right]$ by |$\int_a^b f\left(x\right),dx \approx \frac\left\{b-a\right\}\left\{2\right\} \sum${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 $f\left(x\right)$, but by applying an n-point Gauss-Legendre quadrature rule, as described [[wp:Gaussian Quadrature|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: $\int_\left\{-3\right\}^\left\{3\right\} \exp\left(x\right) , dx \approx \sum_\left\{i=1\right\}^5 w_i ; \exp\left(x_i\right) \approx 20.036$

## Axiom

{{trans|Maxima}} Axiom provides Legendre polynomials and related solvers.

 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;
}


{{out}}

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


## 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))))))))


{{out|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


## 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.


namespace Rosetta {

/*! Implementation of Gauss-Legendre quadrature
*
*/
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 << '\n';
out << "Weights:";
for (int i = 0; i <= eDEGREE; ++i) {
out << ' ' << legpoly.weight(i);
}
out << '\n';
}
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) << '\n';
std::cout << "Actual value:                    " << RosettaExp(3) - RosettaExp(-3) << '\n';
}



{{out}}


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



## 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  ## D {{trans|C}} import std.stdio, std.math; immutable struct GaussLegendreQuadrature(size_t N, FP=double, 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() { GaussLegendreQuadrature!(5, real) glq; 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)); }  {{out}} 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  ## 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  ## 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") }  {{out}}  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  {{out}} 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:''' P =: 3 :0 NB. list of coefficients for yth Legendre polynomial if. y<:1 do. 1{.~->:y return. end. y%~ (<:(,~+:)y) -/@:* (0,P<:y),:(P y-2) ) getpoints =: 3 :0 NB. points,:weights for y points x=. 1{:: p. p=.P y w=. 2% (-.*:x)**:(p..p)p.x x,:w ) GaussLegendre =: 1 :0 NB. npoints function GaussLegendre (a,b) : 'x w'=.getpoints x -:(-~/y)* +/w* u -:((+/,-~/)y)p.x )  {{out|Example use}}  5 ^ GaussLegendre _3 3 20.0356  ## Java {{trans|C}} {{works with|Java|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  ## 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(''N''2) 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.) {{out}}  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 {{trans|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))) }  {{Out}} 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 function gaussLegendreQuadrature(f, lowerLimit, upperLimit, n) 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  {{out}} 20.035577718386  ## Mathematica 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]] gaussLegendreQuadrature[Exp, {-3, 3}]  {{out}} 20.0356  ## MATLAB Translated from the Python solution.  %Integration using Gauss-Legendre quad %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  {{out}} 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

{{trans|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()



{{out}}


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;;


{{out}}


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.

## PARI/GP

{{works with|PARI/GP|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)


{{out}}

time = 0 ms.
%1 = 20.035577718385562153928535725275093932 + 0.E-37*I


{{works with|PARI/GP|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


{{out}}

%1 = 20.035746975092343883065457558549925374
%2 = 20.035749854819803797949187238931656120


## 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
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

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)


## Pascal

See [[Numerical_integration/Gauss-Legendre_Quadrature#Delphi | Delphi]]

## Perl

{{trans|Perl 6}}

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
}

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;



{{out}}

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


## Perl 6

{{works with|rakudo|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;  {{out}} 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 {{trans|Lua}} 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] * call_func(f,{c1 * legendreRoots[i] + c2}) end for return c1 * s end function include pmaths.e -- exp() constant r_exp = routine_id("exp") string fmt = iff(machine_bits()=32?"%.13f":"%.14f") string res for i=5 to 11 by 6 do res = sprintf(fmt,{gaussLegendreQuadrature(r_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})  {{out}}  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 {{libheader|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"  {{out}}  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  ## 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)))))))))  [[File:gauss.png]] ## 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 call gaussquad 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 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 = 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 [GLQ ≡ Gauss─Legendre quadrature.] 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)-1; 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**2) * df**2) /*▓*/ 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 /*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  {{out|output|text= 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 more decimal digits of '''pi'''. It is about twice as slow as version 2, due to the increased number of decimal digits (precision). /*REXX program does numerical integration using an N-point Gauss─Legendre quadrature rule. */ pi= pi(); digs= length(pi)-1; 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**2) * df**2) /*▓*/ 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 /*stick a fork in it, we're all done. */ /*───────────────────────────────────────────────────────────────────────────────────────────*/ e: return 2.71828182845904523536028747135266249775724709369995957496696762772407663035354759, || 4571382178525166427427466391932003059921817413596629043572900334295260595630738 /*───────────────────────────────────────────────────────────────────────────────────────────*/ pi: return 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899, || 8628034825342117067982148086513282306647093844609550582231725359408128481117450 /*───────────────────────────────────────────────────────────────────────────────────────────*/ 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  {{out|output|text= when using the default inputs:}} (Shown at about two-thirds size.)  step iterate value (with 159 decimal digits being used) difference ────── ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────── ───────────── 2 17.4874646410555689643606840462449458421154284179349135091487247059537916662378882444064336021640614626063744948781912964250403870127054497392082425535068464109 -2.5483 3 19.8536919968055821921309108927158495960774667319753888929050027075848592516449832906645902758379575999249091274157148988582792112906526877518087112700785494497 -1.8206e-001 4 20.0286883952907008527738054439857661647073363250481518077257887668521514648379218096268747927750038360903142778646220077613647092768733641727539206268833693589 -7.0615e-003 5 20.0355777183855621539285357252750939315016272074471283081673242529514166130221254213250349496939691709537643294259047823350162410908440808868981982394287542091 -1.7214e-004 6 20.0357469750923438830654575585499253741529947892197512571761670590022501037527117346339483928363770582109285164930728028479549289382406446621705905363209981933 -2.8797e-006 7 20.0357498197266007755718729372891903369400657532378489130759167634362318526784010016150667027038415189719144094529764766032097831604495667799067330556673881546 -3.5093e-008 8 20.0357498544945172882260918041683132616236752579944055100693304551390338045262089091194019302017562870527315644307417688383478919210145963055448428522264642591 -3.2529e-010 9 20.0357498548174338368864419454858704839263168086955797931292590585320198342940085570553927472311015418220675609961921140415760514983040167737226050690228927443 -2.3700e-012 10 20.0357498548197898711175766908543458234008349625446568080936795730938134205900980645938318794902592556558231569959762420203929344018773329199723457149763574343 -1.3927e-014 11 20.0357498548198037305529147159697031241993516306485175808291929207610544866584568009626862857221858328844106864371425322111609007302709732793823163103980149653 -6.7396e-017 12 20.0357498548198037976759531014454017742327138984429607438017578771715767588391691509175808718708593063121709896967107496243434245185896147055314894150234262075 -2.7323e-019 13 20.0357498548198037979482458119092690701862659228785307035583081473361900008835808932495328864420024278695427964698380448330606714160259282675390182203803537594 -9.4143e-022 14 20.0357498548198037979491844483599375945130148356706886332919441446027039132743905494286471338717783707421873433644754993992655580745072286831502363474798175265 -2.7906e-024 15 20.0357498548198037979491872317401917248452734118643091749897281356338832738714150881537113815780435230011480697467170623887897830301712412973655748924184138648 -7.1915e-027 16 20.0357498548198037979491872389153958789316129464894982848020715833786709121310547889685984881568546203564135185474792767674806869872650180714616455691318778648 -1.6260e-029 17 20.0357498548198037979491872389316236038179252557440453906282250905385221873347716826354198555233437240574026019817833907372014036252533047705435353247648510898 -3.2517e-032 18 20.0357498548198037979491872389316560624360571301484111974244019477736095885421361807599231231543821951618639462965984321643251022835234451110049047608124949533 -5.7920e-035 19 20.0357498548198037979491872389316561202637283172074241556158972833578634894365092635000776399956033063018069653085902399896542171129596405210008317497301936898 -9.2480e-038 20 20.0357498548198037979491872389316561203560751340857503751994442223163866912408434007886096643419528065940077022083150476496426837665378721283432879108630468864 -1.3311e-040 21 20.0357498548198037979491872389316561203562080727616463861143647576984994047530870779393715057751591887673397688454357985082021265151278191050057935329724907161 -1.7360e-043 22 20.0357498548198037979491872389316561203562082461596244537077863602238433892612703628843743785373313737563806457244053157873973239947461987202443878362979981218 -2.0610e-046 23 20.0357498548198037979491872389316561203562082463655032534484950691669880046406047078766996078695370527223056578914332723730363863326194707715142045831095820995 -2.2368e-049 24 20.0357498548198037979491872389316561203562082463657267060509015976314523758814742624773428457390528961843568960502876896215809857825164102337905868347725395891 -2.2276e-052 25 20.0357498548198037979491872389316561203562082463657269286070017882824923688080311511389836619043005851350331110867389220628954338053656628671036072512306223102 -2.0430e-055 26 20.0357498548198037979491872389316561203562082463657269288111333795426189423729667519158562143832977811003145168351321839626313132075697513253761673496828204601 -1.7312e-058 27 20.0357498548198037979491872389316561203562082463657269288113063661454822050198926197665008333893008724687497228278730367375441075263700413282548634210907331356 -1.3595e-061 28 20.0357498548198037979491872389316561203562082463657269288113065019935786483820352375621786828318969009163053743757325024448325026804644277866300802833611297358 -9.9207e-065 29 20.0357498548198037979491872389316561203562082463657269288113065020927177593233999249852447888627901300469719564790181325442944469692690797774430312247159512798 -6.7451e-068 30 20.0357498548198037979491872389316561203562082463657269288113065020927851675301934062025341716601075750412806887227020916063849030412480955063639628314338766826 -4.2832e-071 31 20.0357498548198037979491872389316561203562082463657269288113065020927852103363148863217394106431702791915956948972366384835732103508918001327415359847661271744 -2.5459e-074 32 20.0357498548198037979491872389316561203562082463657269288113065020927852103617599854934274435013875248206413049448382025586066461615726348079942124364780837615 -1.4196e-077 33 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741736109635347323907131494641377410353985987829217992622815976248321170584831199 -7.4395e-081 34 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810467715704772209566910717933633388969835872983190631663850670877759345234946 -3.6713e-084 35 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504412069378446854036859408497315019337333762510854198446941961793973563786 -1.7091e-087 36 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429152646383719980280460795167918691617029439367737607466797014691070736 -7.5175e-091 37 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160160714391043273984198489693834991216803247954607301723271542659342 -3.1292e-094 38 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163842341789925349746540298990681930753381942866562579949258588756 -1.2345e-097 39 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843575809851614709658383098559963930599249691243554488598937473 -4.6221e-101 40 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576271898405984614568086424291240202255560215705599799392784 -1.6447e-104 41 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062816325200556739918251890227721352129415324255316078 -5.5685e-108 42 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062871992591098295378332977741979460337064627095382229 -1.7962e-111 43 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010547683152388008632372342584043989711962229120 -5.5262e-115 44 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553207686109280576604524821212627658539678173 -1.6234e-118 45 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309007883789778553235095713392096309190 -4.5581e-122 46 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463568475856093621233977451367882545 -1.2245e-125 47 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690894669420459178304255584843389 -3.1504e-129 48 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926165554457658247246849536195 -7.7695e-133 49 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173322092766217806542725001 -1.8383e-136 50 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323930695861756195366068 -3.9547e-140 51 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323931067751563248835441 -2.3582e-141 52 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323931067290550361259682 -2.4043e-141 53 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323931250155044682676454 1.5882e-140 ────── ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────── ───────────── ↑ 20.0357498548198037979491872389316561203562082463657269288113065020927852103617741810504429160163843576272062872010553209309463690926173323931091333618506603714 {exact value} Using 159 digit precision, the N-point Gauss─Legendre quadrature (GLQ) had an accuracy of 141 digits.  ## Scala {{Out}}Best seen in running your browser either by [https://scalafiddle.io/sf/rrvzhH1/0 ScalaFiddle (ES aka JavaScript, non JVM)] or [https://scastie.scala-lang.org/yYqRqizfSZip2DhYbdfZ2w Scastie (remote JVM)]. Scala import scala.math.{Pi, cos, exp} object GaussLegendreQuadrature extends App { 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 {{trans|Perl 6}} ruby 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", i, quadrature(i, {.exp}, -3, +3)) }  {{out}} txt 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 {{trans|Common Lisp}} {{tcllib|math::constants}} {{tcllib|math::polynomials}} {{tcllib|math::special}} tcl 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:

tcl
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]"


{{out}}

txt

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

Ursala
#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(
@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:
Ursala
#show+

demo =

~&lNrCT (
^|lNrCT(:/'nodes:',:/'weights:')@lSrSX ..mp2str~~* nodes/160 5,
:/'integral:' ~&iNC ..mp2str integral(160,5) (mp..exp,-3E0,3E0))


{{out}}

txt

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


## zkl

{{trans|Perl 6}}

zkl
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
}


zkl
[5..10].walk().append(20).pump(Console.println,fcn(n){
("Gauss-Legendre %2d-point quadrature "
"\U222B;\U208B;\U2083;\U207A;\UB3; exp(x) dx = %.13f")
.fmt(n,quadrature(n, fcn(x){ x.exp() }, -3, 3))
})


{{out}}

txt

Gauss-Legendre  5-point quadrature ∫₋₃⁺³ exp(x) dx = 20.0355777183856
Gauss-Legendre  6-point quadrature ∫₋₃⁺³ exp(x) dx = 20.0357469750924
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



{{omit from|GUISS}}

`