Numerical integration/Gauss-Legendre Quadrature
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 is first approximated over the interval by a polynomial approximable function and a known weighting function . | |
| Those are then approximated by a sum of function values at specified points multiplied by some weights : | |
| In the case of Gauss-Legendre quadrature, the weighting function , so we can approximate an integral of with: |
![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
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 evaluation points for a n-point rule, also called "nodes", are roots of n-th order Legendre Polynomials . Legendre polynomials are defined by the following recursive rule: |
|
| There is also a recursive equation for their derivative: | |
| The roots of those polynomials are in general not analytically solvable, so they have to be approximated numerically, for example by Newton-Raphson iteration: | |
| The first guess for the -th root of a -order polynomial can be given by | |
| After we get the nodes , we compute the appropriate weights by: | |
| After we have the nodes and the weights for a n-point quadrature rule, we can approximate an integral over any interval by |
![{\displaystyle w_{i}={\frac {2}{\left(1-x_{i}^{2}\right)[P'_{n}(x_{i})]^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1e447dea4bf0b3477dc54698ec8bbd50bb62c9)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
Task description
Similar to the task Numerical Integration, the task here is to calculate the definite integral of a function , 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:
11l
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
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
n-point Gauss-Legendre quadrature rule. *)
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
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 -26C
#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 {
/*! Implementation of Gauss-Legendre quadrature
* http://en.wikipedia.org/wiki/Gaussian_quadrature
* http://rosettacode.org/wiki/Numerical_integration/Gauss-Legendre_Quadrature
*
*/
template <int N>
class GaussLegendreQuadrature {
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>
typename GaussLegendreQuadrature<N>::LegendrePolynomial GaussLegendreQuadrature<N>::s_LegendrePolynomial;
}
// This to avoid issues with exp being a templated function
double RosettaExp(double x) {
return exp(x);
}
int main() {
Rosetta::GaussLegendreQuadrature<5> gl5;
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
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));
}
- 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
EDSAC order code
The EDSAC subroutine library had two routines for Gauss-Legendre integration: Q2 for 5-point and Q3 for 6-point. The program below, however, follows the task description and calculates the roots and weights from scratch. (I can't find a copy of Q2 anyway; the code for Q3 is in Wilkes, Wheeler and Gill.) The problem has to be scaled so as to bring real numbers into EDSAC's range [-1,1). Instead of integrating exp(x) between -3 and 3, the program integrates {exp(x)/2}^5 between -3/5 and 3/5. This results in INT/160, where INT is the integral in the task description. The result is then converted to INT/100 for easy comparison with other Rosetta Code solutions.
[Gauss-Legendre quadrature for Rosetta Code.
EDSAC, initial orders 2.]
T46K P5F [EDITABLE: N parameter, order of G-L quadrature.
In the address field, maximum value 31.]
[Arrange the storage]
T48K P56F [& (delta), 36 locations: library subroutine D6 for division (*)]
T54K P92F [C, 44 locations, library subroutine T1 for cosine]
T50K P136F [X, 36 locations: library subroutine E2 for exponential.]
T52K P172F [A, 41 locations: print subroutine]
T47K P218F [M, 70 locations: main routine]
T51K P300F [G, 243 locations: subroutine for G-L quadrature]
T45K P560F [H, 16 locations: data for quadrature s/r]
T44K P576F [! (phi): function to be integrated]
[(*) Library subroutine R9, which is called at load time, is also loaded at 56.
It can be overwritten when no longer needed.]
[--------------------------------------------------------------------------
Library subroutine R9 to read integers from tape at load time.
Must be loaded at location 56]
T56K
GKT20FVDL8FA40DUDTFI40FA40FS39FG@S2FG23FA5@T5@E4@
[--------------------------------------------------------------------------
Library subroutine M3, prints header at load time and is then overwritten.]
PFGKIFAFRDLFUFOFE@A6FG@E8FEZPF
*!ROOTS!!!!!!!!!!WEIGHTS@&# [last char sets teleprinter to figures]
.. PK [after header, blank tape and PK (WWG, 1951, page 91)]
[---------------------------------------------------------------------------
! (phi) parameter: function f(x) to be integrated.
Input: 0D = x
Output: 0D = f(x)
For Rosetta Code task after scaling, f(x) = (exp(x)/2)^5]
E25K T! GK
A3F T23@ [plant return link as usual]
AD T4D [x to 4D]
[4] A4@ GX [call library subroutine E2; places exp(x) - 1 in 4D]
A4D RD A24@ [acc := exp(x)/2]
YF TD [round, and save in 0D]
HD VD YF T4D [acc := (exp(x)/2)^2, round, to 4D]
H4D V4D YF T4D [acc := (exp(x)/2)^4, round, to 4D]
H4D VD YF TD [acc := (exp(x)/2)^5, round, return in 0D]
[23] ZF [(planted) jump back to caller]
[24] IF [constant 1/2]
[-------------------------------------------------------------------------------
H parameter: Data for quadrature subroutine.
Limits of integration are c - h and c + h.]
E25K TH GK
E69K T#H [at load time, call R9 to read constants]
0F [0#H = c = 0]
10307921510F [2#H = h = 3/5]
17F [epsilon = 10^-9, criterion for convergence]
210828714# [pi/256]
T8Z [resume normal loading at 8H]
[8] PN [n = order of G-L quadrature, set by N parameter at head of program]
[9] T1024D [T order for exclusive end of EDSAC memory]
[10] P2F [add 2 to address]
[11] MF [add to T order to make A order with same address]
[The following are set up at runtime:]
[12] PF [2*n]
[13] TD [T order for first reciprocal in array]
[14] TD [T order for first root in array]
[15] TD [T order for first weight in array]
[----------------------------------------------------------------------
G parameter: subroutine for Gauss-Legendre quadrature.
Input: via H and ! (phi) parameters (see above)
Output: Estimated integral is returned in 0D.]
E25K TG GK
G22@
[1] K4096F [place this so that 35-bit values come at even addresses]
[2] PF PF [root of Legendre polynomial]
[4] PF PF [correction to root during Newton-Raphson process]
[6] PF PF [weighted sum of function values]
[8] PF PF [temporary store]
[10] PF PF [temporary store]
[The next 3 are negated values of polynomials (j = 2,...,n)]
[12] PF PF [-P_j(x)]
[14] PF PF [-P_j-1(x)]
[16] PF PF [-P_j-2(x)]
[18] Y2F [subtract to convert T to V order, with address 2 less]
[19] PF [negative count of roots]
[20] P1024F [1/32 (where max n is 31)]
[21] PF
[22] A3F T190@ [plant return link as usual]
[Set up T orders for workspace, working down from top of EDSAc memory.
Weights 1024 - 2*n, roots 1024 - 4*n, reciprocals 1024 - 6*n + 2]
A8H LD T12H [set up 2*n from n]
A9H U139@ [exclusive end of weights]
S12H U15H U131@ U127@ [start of weights = end of roots]
S12H U14H U119@ [start of roots = end of reciprocals]
S12H A10H U13H T55@ [start of reciprocals]
[Calculate the reciprocals 1/j for j := 2, ..., n]
A20@ T21@
E45@ [skip update first time]
[43] A14H T55@ [update T order]
[45] TD T4D [clear 0D and 4D for division routine]
A20@ U1F A21@ U21@ T5F
[52] A52@ G& AD [division, acc := 1/j]
[55] TD [store in array]
A55@ A10H S14H G43@
[Prepare to calculate first approximations to roots; here with acc = 0.]
A6#H TD [0D := pi/256 for division]
T4D [clear whole of 4D, including sandwich bit]
A8H LD A2F [2*n + 1 in address field i.e. *2^-15]
L32F T5F [4D := (2*n + 1)/256, extended to 35 bits]
[68] A68@ G& [call division subroutine, 0D := pi/(2*n + 1)]
AD U8#@ [store pi/(2*n + 1)]
R1F TD SD T10#@ [store -pi/4*(2*n + 1)]
[Calculate the (n div 2) positive roots; negative roots follow by symmetry.
If n is odd, the root at 0 is treated separately.]
[Given n in address field, want (n div 2) in address field,
with low bit = 0. A bit tricky.]
A8H R1F TF AF LD
TF SF [from n, get (n div 2) in the address field]
[83] T19@ [initialize negative count of roots]
A10#@ A8#@ U10#@
T4D
[88] A88@ GC [call cosine subroutine]
A4D LD [first approx to root]
E96@ [skip correction the first time]
[Inner loop; next approximation for current root]
[93] TF A2#@ A4#@ [clear acc; add correction to root]
[96] U2#@ TD [store approx to root, also pass to Legendre subroutine]
A8H T4F [pass n to Legendre subroutine]
[100] A100@ G191@ [call Legendre]
A1@ H2#@ N2#@ [acc := 1 - x^2]
T10D H10D [to mult reg.]
V8D [acc := 1/deriv]
T10D H10D [to mult reg.]
N6D U4#@ [calculate and store corection]
E115@ TD SD [acc := abs(correction)]
[115] S4#H E93@ [if not converged yet, loop back]
TF A2#@ [acc := root]
[119] TD [copy to caller's array]
A119@ A10H T119@ [post-inc forward index]
A127@ S10H T127@ [pre-dec backward index]
S2#@ [acc := -root]
[127] TD [copy to caller's array]
[Calculate weight for this root. Mult. reg. still contains 1/deriv]
V8D
LD [times 2 to conform to Rosetta code convention]
UD
[131] TD
A131@ A10H T131@ [post-inc forward index]
A139@ S10H T139@ [pre-dec backward index]
AD
[139] TD
A19@ A2F G83@
[Finished loop round roots. If n is odd, still need root at 0.]
A119@ U150@ [copy T order in case we need it]
S127@ E160@ [if T orders are equal n is even, so done]
TF A131@ T159@ [copy T order for weight]
[150] TD
TD A8H T4F [set up parameters for Legendre]
[154] A154@ G191@ [call Legendre for x = 0]
H8D V8D
LD [times 2 to conform to Rosetta code convention]
[159] TD
[Code to carry out quadrature, using roots and weights calculated above.
Based on library subroutine Q3, but simpler (doesn't exploit the symmetry)
Here with acc = 0]
[160] T6#@ [initialize weighted sum to 0]
A14H S18@ T173@ [form and plant V order for double word before roots]
S8H [initialize negative counter to -n]
[Start of loop to add weighted sums]
[165] T19@ [update negative counter]
A173@ A10H U173@ [form and plant V order for next root]
A12H T179@ [form and plant V order for next weight]
[Limits of integration are c - h and c + h]
A#H H2#H [acc := c, mult reg := h]
[173] VD [(planted) acc := c + h*root{i} (i = 1, ... n)]
YF TD [round, and pass to function to be integrated]
[176] A176@ G! [call function, returns value in 0D]
HD [mult reg := function value]
[179] VD [(planted) acc := value*weight{i} (i = 1, ... n)]
YF A6#@ T6#@ [round, update sum]
A19@ A2F G165@ [loop back till done n values]
[Now have weighted sum. Multiply by h, round, return to caller]
H6#@ V2#H YF TD
[190] ZF
[--------------]
[Subroutine to evaluate Legendre polynomial P_n(x) and (in effect) P_n'(x)]
[Input: 0D = x; 4F = n in address field, where n = 2..31 (not checked)
Output: 6D = P_n(x), 8D = 1/{(1 - x^2)*P_n'(x)}
Works with negated Legendre polynomials, because EDSAC supports -1 but not +1.
Calculates -P_2{x}, -P_3{x}, ..., -P_n{x} recursively from -P_0(x), -P_1(x)
For given j in {2..n} denote y0 = -P_j(x), y1 = -P_j-1(x), y2 = -P_j-2(x).
From the formulae in the task description we get:
y0 = (1 - 1/j)*(x*y1 - y0) + x*y1
and, with j = n, 1/{(1 - x^2)*P_n'(x)} = -(1/n)/(y1 - x*y0)]
[191] A3F T242@ [plant return link as usual]
T16#@ [clear y2 including sandwich bit]
A1@ T16#@ [y2 := -1, extended to 35 bits]
SD T14#@ [y1 := -x]
A13H A11H T219@ [form and plant A order for reciprocal]
S4F A2F T5F [negative count in 5F]
E213@ [skip update the first time]
[205] T5F [update negative count]
A14#@ T16#@ A12#@ T14#@ [y2 := y1; y1 := y0]
A219@ A10H T219@ [inc address in A order]
[213] HD V14#@ U6D [save x*y1 in 6D]
S16#@ T8D [save x*y1 - y2 in 8D]
A1@ [acc := -1]
[219] AD [(planted) acc := -1 + 1/j (j = 2,...,n)]
T10D H10D [copy to mult. reg.]
N8D [acc := (1 - 1/j) * (x*y1 - y2)]
A6D T12#@ [y0 := (2 - 1/j)*x*y1 - (1 - 1/j)*y2]
A5F A2F G205@ [iterate until have P_n(x)]
[Now want 1 / n*(y2 - x*y1)]
A219@ T230@ [plant H order for 1 - 1/n]
[230] AD HD [acc := 1/n; mult reg := x]
TD [pass 1/n as dividend to division subroutine]
A14#@ N12#@ [acc := y1 - x*y0]
T4D [pass as divisor to division s/r]
[236] A236@ G& [call division subroutine, 0D := (1/n)/(y1 - x*y0)]
[Return values to caller, remembering that local values are negated]
SD T8D S12#@ T6D
[242] ZF [(planted) return to caller]
[-------------------------------------------------------------------------------
Main routine]
E25K TM GK
E69K T#@ [call R9 at load time to read constant]
13743895347# [4/5 ]
T2Z [resume normal loading at 2@]
[2] PF PF
[4] PF
[Characters for printing]
[5] K2048F [set letters on teleprinter]
[6] #F [set figures on teleprinter]
[7] IF [letter I]
[8] NF [letter N]
[9] XF [slash]
[10] VF [equals sign]
[11] QF [figure 1]
[12] PF [figure 0]
[13] @F [carriage return]
[14] &F [line feed]
[15] !F [space]
[Enter with acc = 0]
[16] A16@ GG [do quadrature, result/160 in 0D]
HD V#@ LD YF [acc := 0D*(8/5) = result/100, and round]
T2#@ [save over printing of roots and weights]
A14H A11H U30@ [form and plant A order for first root]
A12H T36@ [form and plant A order for first weight]
S8H [acc := -n]
[29] T4@ [update negative count (also used as letter T)]
[30] AD TD [pass root to print subroutine in 0D]
[32] A32@ GA O15@ O15@ [print root and 2 spaces]
[36] AD TD [pass weight to print subroutine in 0D]
[38] A38@ GA O13@ O14@ [print weight and new line]
A30@ A10H T30@ [inc address of root]
A36@ A10H T36@ [inc address of weight]
A4@ A2F G29@ [inc negative count, loop till done]
[Print result of integration, preceded by 'INT/100 =']
O5@ O7@ O8@ O29@ O6@
O9@ O11@ O12@ O12@
O15@ O10@
A2#@ TD [pass result to print subroutine in 0D]
[64] A64@ GA O13@ O14@ [print result and new line]
[Exit]
O6@ [print dummy char to flush teleprinter buffer]
ZF [halt the machine]
[-----------------------------------------------------------------------
Library subroutine T1: calculates cos(x), where abs(x) <= pi/2
Input: 4D = x/2
Output: 4D = cos(x)/2
Requires library subroutine R9.]
E25K TC
GKT20FVDL8FA40DUDTFI40FA40FS39FG@S2FG23FA5@T5@E4@E13ZT32#@
1614F73454F243967F54539267F763549741F5726623061#
TZA3FT30@H4DV4DYFT4DH4DN32#@A34#@TDNDA36#@TDNDA38#@
TDNDA40#@TDNDA42#@TDNDYFTDNDS4DA31@YFT4DEFIFT44Z
[-----------------------------------------------------------------------
Library subroutine D6: Division, accurate, fast.
36 locations, workspace 6D and 8D.
0D := 0D/4D, where 4D <> 0, -1.]
E25K T&
GKA3FT34@S4DE13@T4DSDTDE2@T4DADLDTDA4DLDE8@RDU4DLDA35@
T6DE25@U8DN8DA6DT6DH6DS6DN4DA4DYFG21@SDVDTDEFW1526D
[--------------------------------------------------------------------------
Library subroutine E2: exponential function. 19 locations.
Input: 4D = x where -1 <= x < ln(2).
Output: 4D = exp(x) - 1]
E25K TX
GKA3FT18@YFLDT6DH4DV4DTDH6DVDRDA4DYFT4DA6DLDE4@TDEF
[-----------------------------------------------------------------------
Subroutine to print 35-bit number in 0D to 10 decimal places.
Wrapper for library subroutine P1, which prints only non-negative
numbers and does not include '0.']
E25K TA
GKA3FT17@ADE8@TDSDO@E9@O18@YFYFTDO16@O19@A14@G20@P10FZF!FMFA38@
U37@S40@T25@H39@PFT25@VDUFOFFFSFL4FTDA25@A2FG26@EFU3FJFM1F
[----------------------------------------------------------------------]
E25K TM GK [M parameter again]
E16Z [define entry point]
PF [acc = 0 on entry]- Output:
ROOTS WEIGHTS 0.9061798459 0.2369268853 0.5384693102 0.4786286710 0.0000000000 0.5688888887 -0.5384693102 0.4786286710 -0.9061798459 0.2369268853 INT/100 = 0.2003557774
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
#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
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
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
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
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
- 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]]
gaussLegendreQuadrature[Exp, {-3, 3}]
- Output:
20.0356
MATLAB
Translated from the Python solution.
%Print the result.
disp(GLGD_int(@(x) exp(x), -3, 3, 5));
%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
- 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
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
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
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
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
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
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
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)
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
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) - 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:
(Shown at about two-thirds size.)
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}
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
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))
}
- 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
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
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
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) )
})
}
fcn quadrature(n,f,a,b,nds=Void){
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 %2d-point quadrature "
"\U222B;\U208B;\U2083;\U207A;\UB3; exp(x) dx = %.13f")
.fmt(n,quadrature(n, fcn(x){ x.exp() }, -3, 3))
})- Output:
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
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