Numerical integration/Gauss-Legendre Quadrature: Difference between revisions
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<br>visual indication of what's what, but it's the dickens to pay when updating the source code. |
<br>visual indication of what's what, but it's the dickens to pay when updating the source code. |
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<lang rexx>/*REXX program does numerical integration using an N-point Gauss─Legendre quadrature rule. */ |
<lang rexx>/*REXX program does numerical integration using an N-point Gauss─Legendre quadrature rule. */ |
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pi=pi(); digs=length(pi)-1; numeric digits digs; reps=digs%2 |
pi= pi(); digs= length(pi)-1; numeric digits digs; reps= digs % 2 |
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!.=.; b=3; |
!.= .; b= 3; a= -b; bma= b - a; bmaH= bma / 2; tiny= '1e-'digs |
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trueV=exp(b)-exp(a); bpa=b+a; bpaH=bpa/2 |
trueV= exp(b)-exp(a); bpa= b + a; bpaH= bpa / 2 |
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say ' step ' center("iterative value",digs+3) ' difference' |
say ' step ' center("iterative value", digs+3) ' difference' /*show hdr*/ |
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sep='──────' copies("─" ,digs+3) '─────────────'; |
sep='──────' copies("─" , digs+3) '─────────────'; say sep /* " sep*/ |
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do #=1 until dif>0; p0z=1; p0.1=1; p1z=2; p1.1=1; p1.2=0; |
do #=1 until dif>0; p0z= 1; p0.1= 1; p1z= 2; p1.1= 1; p1.2= 0; ##= # + .5; r.= 0 |
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/*█*/ |
/*█*/ do k=2 to #; km= k - 1; do y=1 for p1z; T.y= p1.y; end /*y*/ |
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/*█*/ |
/*█*/ T.y= 0; TT.= 0; do L=1 for p0z; _= L + 2; TT._= p0.L; end /*L*/ |
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/*█*/ |
/*█*/ |
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/*█*/ |
/*█*/ kkm= k + km; do j=1 for p1z +1; T.j= (kkm*T.j -km*TT.j)/k; end /*j*/ |
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/*█*/ |
/*█*/ p0z= p1z; do n=1 for p0z; p0.n= p1.n ; end /*n*/ |
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/*█*/ |
/*█*/ p1z= p1z + 1; do p=1 for p1z; p1.p= T.p ; end /*p*/ |
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/*█*/ |
/*█*/ end /*k*/ |
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/*▓*/ do !=1 for #; x= cos( pi * (! - .25) / ## ) |
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/*▓*/ |
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/*▓*/ /*░*/ do reps until abs(dx) <= tiny |
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/*▓*/ /*░*/ f= p1.1; df= 0; do u=2 to p1z; df= f + x*df |
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/*▓*/ /*░*/ f= p1.u +x*f |
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/*▓*/ /*░*/ end /*u*/ |
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/*▓*/ /*░*/ dx= f / df; x= x - dx |
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/*▓*/ /*░*/ end /*reps ···*/ |
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/*▓*/ r.1.!= x |
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/*▓*/ r.2.!= 2 / ( (1 - x**2) * df**2) |
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/*▓*/ end /*!*/ |
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⚫ | |||
/*▓*/ end /*!*/ |
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/*▒*/ do m=1 for #; $=$ + r.2.m * exp(bpaH + r.1.m*bmaH); end /*m*/ |
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⚫ | |||
z= bmaH * $ /*calculate target value (Z)*/ |
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dif= z - trueV; z= format(z, 3, digs - 2) /* " difference. */ |
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Ndif= translate( format(dif, 3, 4, 2, 0), 'e', "E") |
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⚫ | |||
Ndif=translate( format(dif, 3, 4, 2, 0), 'e', "E") |
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⚫ | |||
end /*#*/ |
end /*#*/ |
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say sep; xdif=compare(strip(z), trueV); say right("↑", 6+1+xdif) |
say sep; xdif= compare( strip(z), trueV); say right("↑", 6 + 1 + xdif) |
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say left('', 6+1) |
say left('', 6 + 1) trueV " {exact value}"; say |
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say 'Using ' digs " digit precision, the" , |
say 'Using ' digs " digit precision, the" , |
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'N-point Gauss─Legendre quadrature (GLQ) had an accuracy of ' xdif-2 " digits." |
'N-point Gauss─Legendre quadrature (GLQ) had an accuracy of ' xdif-2 " digits." |
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pi: return 3.141592653589793238462643383279502884197169399375105820974944592307816406286286209 |
pi: return 3.141592653589793238462643383279502884197169399375105820974944592307816406286286209 |
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/*───────────────────────────────────────────────────────────────────────────────────────────*/ |
/*───────────────────────────────────────────────────────────────────────────────────────────*/ |
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cos: procedure expose !.; parse arg x; if !.x\==. then return !.x; _=1; z=1; y=x*x |
cos: procedure expose !.; parse arg x; if !.x\==. then return !.x; _=1; z=1; y=x*x |
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do k=2 by 2 until p==z; p=z; _=-_*y/(k*(k-1)); z=z+_; end; |
do k=2 by 2 until p==z; p=z; _= -_*y/(k*(k-1)); z=z+_; end; !.x=z; return z |
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/*───────────────────────────────────────────────────────────────────────────────────────────*/ |
/*───────────────────────────────────────────────────────────────────────────────────────────*/ |
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exp: procedure; parse arg x; ix=x % 1; if abs(x-ix) > .5 then ix=ix + sign(x) |
exp: procedure; parse arg x; ix= x % 1; if abs(x-ix) > .5 then ix= ix + sign(x) |
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x=x-ix; z=1; _=1; |
x= x-ix; z=1; _=1; do j=1 until p==z; p=z; _= _*x/j; z= z+_; end |
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return z * e()**ix</lang> |
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{{out|output|text= when using the default inputs:}} |
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'''output''' |
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<pre> |
<pre> |
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step iterative value difference |
step iterative value difference |
Revision as of 02:28, 24 August 2018
![Task](http://static.miraheze.org/rosettacodewiki/thumb/b/ba/Rcode-button-task-crushed.png/64px-Rcode-button-task-crushed.png)
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: |
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: |
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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 |
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:
Axiom
Axiom provides Legendre polynomials and related solvers.<lang Axiom>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])</lang>Example:<lang Axiom>digits(50)
gaussIntegrate(4/(1+x^2), x=0..1, 20)
(1) 3.1415926535_8979323846_2643379815_9534002592_872901276 Type: Float
% - %pi
(2) - 0.3463549483_9378821092_475 E -26</lang>
C
<lang C>#include <stdio.h>
- include <math.h>
- define N 5
double Pi; double lroots[N]; double weight[N]; double lcoef[N + 1][N + 1] = Template: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; }</lang>
- 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
Common Lisp
<lang 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))))))))</lang>
- Example:
<lang lisp>(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</lang> Comparison of the 5-point rule with simpler, but more costly methods from the task Numerical Integration: <lang lisp>(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</lang>
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. <lang cpp> 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 << '\n'; out << "Weights:"; for (int i = 0; i <= eDEGREE; ++i) { out << ' ' << legpoly.weight(i); } out << '\n'; } private: /*! Implementation of the Legendre polynomials that form * the basis of this quadrature */ class LegendrePolynomial { public: LegendrePolynomial () { // Solve roots and weights for (int i = 0; i <= eDEGREE; ++i) { double dr = 1;
// Find zero Evaluation eval(cos(M_PI * (i - 0.25) / (eDEGREE + 0.5))); do { dr = eval.v() / eval.d(); eval.evaluate(eval.x() - dr); } while (fabs (dr) > 2e-16);
this->_r[i] = eval.x(); this->_w[i] = 2 / ((1 - eval.x() * eval.x()) * eval.d() * eval.d()); } }
double root(int i) const { return this->_r[i]; } double weight(int i) const { return this->_w[i]; } private: double _r[eDEGREE + 1]; double _w[eDEGREE + 1];
/*! Evaluate the value *and* derivative of the * Legendre polynomial */ class Evaluation { public: explicit Evaluation (double x) : _x(x), _v(1), _d(0) { this->evaluate(x); }
void evaluate(double x) { this->_x = x;
double vsub1 = x; double vsub2 = 1; double f = 1 / (x * x - 1); for (int i = 2; i <= eDEGREE; ++i) { this->_v = ((2 * i - 1) * x * vsub1 - (i - 1) * vsub2) / i; this->_d = i * f * (x * this->_v - vsub1);
vsub2 = vsub1; vsub1 = this->_v; } }
double v() const { return this->_v; } double d() const { return this->_d; } double x() const { return this->_x; }
private: double _x; double _v; double _d; }; };
/*! Pre-compute the weights and abscissae of the Legendre polynomials */ static LegendrePolynomial s_LegendrePolynomial; };
template <int N> 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) << '\n'; std::cout << "Actual value: " << RosettaExp(3) - RosettaExp(-3) << '\n';
} </lang>
- 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
Delphi
<lang 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.</lang>
Roots: 0.9061798459 0.5384693101 0.0000000000 -0.5384693101 -0.9061798459 Weight: 0.2369268851 0.4786286705 0.5688888889 0.4786286705 0.2369268851 Integrating Exp(X) over [-3, 3]: 20.0355777184 Actual value: 20.0357498548
D
<lang 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));
}</lang>
- 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
Fortran
<lang 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 </lang>
n numerical integral error -------------------------------------------------- 1 6.00000000000000000000000000000000 -14. 2 17.4874646410555689643606840462449 -2.5 3 19.8536919968055821921309108927158 -.18 4 20.0286883952907008527738054439858 -.71E-02 5 20.0355777183855621539285357252751 -.17E-03 6 20.0357469750923438830654575585499 -.29E-05 7 20.0357498197266007755718729372892 -.35E-07 8 20.0357498544945172882260918041684 -.33E-09 9 20.0357498548174338368864419454859 -.24E-11 10 20.0357498548197898711175766908548 -.14E-13 11 20.0357498548198037305529147159695 -.67E-16 12 20.0357498548198037976759531014464 -.27E-18 13 20.0357498548198037979482458119095 -.94E-21 14 20.0357498548198037979491844483597 -.28E-23 15 20.0357498548198037979491872317190 -.72E-26 16 20.0357498548198037979491872388913 -.40E-28 17 20.0357498548198037979491872389166 -.15E-28 18 20.0357498548198037979491872389259 -.58E-29 19 20.0357498548198037979491872388910 -.41E-28 20 20.0357498548198037979491872388495 -.82E-28
Go
Implementation pretty much by the methods given in the task description. <lang go>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")
}</lang>
- 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 <lang haskell>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))</lang>
Calculation of Legendre polynomials <lang haskell>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)</lang>
Universal auxilary functions <lang haskell>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</lang>
Integration on a given mesh using Gauss-Legendre quadrature: <lang haskell>integrate _ [] = 0 integrate f (m:ms) = sum $ zipWith (gaussLegendre 5 f) (m:ms) ms</lang>
- 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: <lang j>P =: 3 :0 NB. list of coefficients for yth Legendre polynomial
if. y<:1 do. 1{.~->:y return. end. y%~ (<:(,~+:)y) -/@:* (0,P<:y),:(P y-2)
)
getpoints =: 3 :0 NB. points,:weights for y points
x=. 1{:: p. p=.P y w=. 2% (-.*:x)**:(p..p)p.x x,:w
)
GaussLegendre =: 1 :0 NB. npoints function GaussLegendre (a,b)
'x w'=.getpoints x -:(-~/y)* +/w* u -:((+/,-~/)y)p.x
)</lang>
- Example use:
<lang j> 5 ^ GaussLegendre _3 3 20.0356</lang>
Java
<lang 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)); }
}</lang>
Roots: 0,906180 0,538469 0,000000 -0,538469 -0,906180 Weight: 0,236927 0,478629 0,568889 0,478629 0,236927 integrating Exp(x) over [-3, 3]: 20,03557772, compared to actual 20,03574985
Julia
This function computes the points and weights of an N-point Gauss–Legendre quadrature rule on the interval (a,b). It uses the O(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: <lang julia>function gauss(a,b, N)
λ, Q = eig(SymTridiagonal(zeros(N), [ n / sqrt(4n^2 - 1) for n = 1:N-1 ])) return (λ + 1) * (b-a)/2 + a, [ 2*Q[1,i]^2 for i = 1:N ] * (b-a)/2
end</lang>
(This code is a simplified version of the Base.gauss
subroutine in the Julia standard library.)
- Output:
julia> x,w = gauss(-3,3,5) ([-2.718539537815986,-1.6154079303170459,1.3322676295501878e-15,1.6154079303170512,2.7185395378159916],[0.7107806551685667,1.4358860114981036,1.706666666666666,1.435886011498098,0.7107806551685655]) julia> sum(exp(x) .* w) 20.03557771838554
Kotlin
<lang scala>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)))
}</lang>
- 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
<lang 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</lang>
- Output:
20.035577718386
Mathematica
code assumes function to be integrated has attribute Listable which is true of most built in Mathematica functions <lang Mathematica>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}]</lang>
- Output:
20.0356
MATLAB
Translated from the Python solution. <lang MATLAB> %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 </lang>
- Output:
20.0356
Maxima
<lang 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 */</lang>
OCaml
<lang 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));;</lang>
which can be used in: <lang OCaml>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;;</lang>
- 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 <lang Ocaml>Printf.printf "%.16f\n" ((exp 3.0) -.(exp (-3.0)));; 20.0357498548198052</lang> although going beyond 20 points starts reducing the accuracy, due to accumulated rounding errors.
PARI/GP
This task is easy in GP thanks to built-in support for Legendre polynomials and efficient (Schonhage-Gourdon) polynomial root finding. <lang parigp>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)</lang>
- Output:
time = 0 ms. %1 = 20.035577718385562153928535725275093932 + 0.E-37*I
Gauss-Legendre quadrature is built-in from 2.9 forward. <lang parigp>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</lang>
- Output:
%1 = 20.035746975092343883065457558549925374 %2 = 20.035749854819803797949187238931656120
ooRexx
<lang 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</lang>
Output:
1 6.0000000000000000000000000000000000000000 -1.4036E+1 2 17.4874646410555686000000000000000000000000 -2.5483 3 19.8536919968055914500000000000000000000000 -1.8206E-1 4 20.0286883952907032246391703165575495371776 -7.0615E-3 5 20.0355777183855623345965085871972344078167 -1.7214E-4 6 20.0357469750923433031000982816859525440756 -2.8797E-6 7 20.0357498197266007450081506439422093510041 -3.5093E-8 8 20.0357498544945192648654062025059252571210 -3.2529E-10 9 20.0357498548174362426073138353882519240177 -2.3698E-12 10 20.0357498548197905075149387536361754813374 -1.5552E-14 11 20.0357498548198049052166074059523608613749 -1.1548E-15 12 20.0357498548198068119347633275378821700762 7.5193E-16 13 20.0357498548198063256375618073806663013152 2.6564E-16 14 20.0357498548198035202546245888922276792447 -2.5397E-15 15 20.0357498548198027919824444452012138941729 -3.2680E-15 16 20.0357498548198037471314715729442546019171 -2.3129E-15 17 20.0357498548198067452377635761033686644343 6.8524E-16 18 20.0357498548198042026084719530842757694873 -1.8574E-15 19 20.0357498548198042304714191024916472961732 -1.8295E-15 20 20.0357498548198034525095801113268011014944 -2.6075E-15 20.03574985481980606 (exact)
Pascal
See Delphi
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.
<lang perl6>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;</lang>
- 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
PL/I
Translated from Fortran. <lang PL/I>(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; </lang>
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
<lang 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"</lang>
- 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
Racket
Computation of the Legendre polynomials and derivatives:
<lang racket> (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))))
</lang>
Computation of the Legendre polynomial roots:
<lang racket> (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′)))))
</lang>
Computation of Gauss-Legendre nodes and weights
<lang racket> (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))))
</lang>
Integration using Gauss-Legendre quadratures:
<lang racket> (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))))))
</lang>
Usage:
<lang racket> > (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 </lang>
Accuracy of the method:
<lang racket> > (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)))))))))
REXX
version 1
<lang rexx>/*---------------------------------------------------------------------
- 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</lang>
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.
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.
<lang rexx>/*REXX program does numerical integration using an N-point Gauss─Legendre quadrature rule. */
pi= pi(); digs= length(pi)-1; numeric digits digs; reps= digs % 2
!.= .; b= 3; a= -b; bma= b - a; bmaH= bma / 2; tiny= '1e-'digs
trueV= exp(b)-exp(a); bpa= b + a; bpaH= bpa / 2
say ' step ' center("iterative value", digs+3) ' difference' /*show hdr*/
sep='──────' copies("─" , digs+3) '─────────────'; say sep /* " sep*/
do #=1 until dif>0; p0z= 1; p0.1= 1; p1z= 2; p1.1= 1; p1.2= 0; ##= # + .5; r.= 0
/*█*/ do k=2 to #; km= k - 1; do y=1 for p1z; T.y= p1.y; end /*y*/ /*█*/ T.y= 0; TT.= 0; do L=1 for p0z; _= L + 2; TT._= p0.L; end /*L*/ /*█*/ /*█*/ kkm= k + km; do j=1 for p1z +1; T.j= (kkm*T.j -km*TT.j)/k; end /*j*/ /*█*/ p0z= p1z; do n=1 for p0z; p0.n= p1.n ; end /*n*/ /*█*/ p1z= p1z + 1; do p=1 for p1z; p1.p= T.p ; end /*p*/ /*█*/ end /*k*/
/*▓*/ do !=1 for #; x= cos( pi * (! - .25) / ## ) /*▓*/ /*▓*/ /*░*/ do reps until abs(dx) <= tiny /*▓*/ /*░*/ f= p1.1; df= 0; do u=2 to p1z; df= f + x*df /*▓*/ /*░*/ f= p1.u +x*f /*▓*/ /*░*/ end /*u*/ /*▓*/ /*░*/ dx= f / df; x= x - dx /*▓*/ /*░*/ end /*reps ···*/ /*▓*/ r.1.!= x /*▓*/ r.2.!= 2 / ( (1 - x**2) * df**2) /*▓*/ end /*!*/ $= 0 /*▒*/ do m=1 for #; $=$ + r.2.m * exp(bpaH + r.1.m*bmaH); end /*m*/ z= bmaH * $ /*calculate target value (Z)*/ dif= z - trueV; z= format(z, 3, digs - 2) /* " difference. */ Ndif= translate( format(dif, 3, 4, 2, 0), 'e', "E") if #\==1 then say center(#, 6) z' ' Ndif /*no display if not computed*/ end /*#*/
say sep; xdif= compare( strip(z), trueV); say right("↑", 6 + 1 + xdif) say left(, 6 + 1) trueV " {exact value}"; say say 'Using ' digs " digit precision, the" ,
'N-point Gauss─Legendre quadrature (GLQ) had an accuracy of ' xdif-2 " digits."
exit /*stick a fork in it, we're all done. */ /*───────────────────────────────────────────────────────────────────────────────────────────*/ e: return 2.718281828459045235360287471352662497757247093699959574966967627724076630353547595 pi: return 3.141592653589793238462643383279502884197169399375105820974944592307816406286286209 /*───────────────────────────────────────────────────────────────────────────────────────────*/ cos: procedure expose !.; parse arg x; if !.x\==. then return !.x; _=1; z=1; y=x*x
do k=2 by 2 until p==z; p=z; _= -_*y/(k*(k-1)); z=z+_; end; !.x=z; return z
/*───────────────────────────────────────────────────────────────────────────────────────────*/ exp: procedure; parse arg x; ix= x % 1; if abs(x-ix) > .5 then ix= ix + sign(x)
x= x-ix; z=1; _=1; do j=1 until p==z; p=z; _= _*x/j; z= z+_; end return z * e()**ix</lang>
- output when using the default inputs:
step iterative value 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.
Sidef
<lang ruby>func legendre_pair((1), x) { (x, 1) } func legendre_pair( n, x) {
var (m1, m2) = legendre_pair(n - 1, x) var u = (1 - 1/n) ((1 + u)*x*m1 - u*m2, m1)
}
func legendre((0), _) { 1 } func legendre( n, x) { [legendre_pair(n, x)][0] }
func legendre_prime({ .is_zero }, _) { 0 } func legendre_prime({ .is_one }, _) { 1 }
func legendre_prime(n, x) {
var (m0, m1) = legendre_pair(n, x) (m1 - x*m0) * n / (1 - x**2)
}
func approximate_legendre_root(n, k) {
# Approximation due to Francesco Tricomi var t = ((4*k - 1) / (4*n + 2)) (1 - ((n - 1)/(8 * n**3))) * cos(Num.pi * t)
}
func newton_raphson(f, f_prime, r, eps = 2e-16) {
loop { var dr = (-f(r) / f_prime(r)) dr.abs >= eps || break r += dr } return r
}
func legendre_root(n, k) {
newton_raphson(legendre.method(:call, n), legendre_prime.method(:call, n), approximate_legendre_root(n, k))
}
func weight(n, r) { 2 / ((1 - r**2) * legendre_prime(n, r)**2) }
func nodes(n) {
gather { take(Pair(0, weight(n, 0))) if n.is_odd { |i| var r = legendre_root(n, i) var w = weight(n, r) take(Pair(r, w), Pair(-r, w)) }.each(1 .. (n >> 1)) }
}
func quadrature(n, f, a, b, nds = nodes(n)) {
func scale(x) { (x*(b - a) + a + b) / 2 } (b - a) / 2 * nds.sum { .second * f(scale(.first)) }
}
[(5..10)..., 20].each { |i|
printf("Gauss-Legendre %2d-point quadrature ∫₋₃⁺³ exp(x) dx ≈ %.15f\n", i, quadrature(i, {.exp}, -3, +3))
}</lang>
- 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
<lang 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 * $rangesize2}
}</lang> Demonstrating: <lang tcl>puts "nodes(5) = [nodes 5]" puts "weights(5) = [weights [nodes 5]]" set exp {x {expr {exp($x)}}} puts "int(exp,-3,3) = [gausslegendreintegrate $exp 5 -3 3]"</lang>
- 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 <lang Ursala>#import std
- import nat
legendre = # takes n to the pair of functions (P_n,P'_n), where P_n is the Legendre polynomial of order n
~&?\(1E0!,0E0!)! -+
^|/~& //mp..vid^ mp..sub\1E0+ mp..sqr, ~~ "c". ~&\1E0; ~&\"c"; ~&ar^?\0E0! mp..add^/mp..mul@alrPrhPX ^|R/~& ^|\~&t ^/~&l mp..mul, @iiXNX ~&rZ->r @l ^/^|(~&tt+ sum@NNiCCiX+ successor,~&) both~&g&&~&+ -+ ~* mp..zero_p?/~& (&&~&r ~&EZ+ ~~ mp..prec)^/~& ^(~&,..shr\8); mp..equ^|(~&,..gro\8)->l @r ^/~& ..shr\8, ^(~&rl,mp..mul*lrrPD)^/..nat2mp@r -+ ^(~&l,mp..sub*+ zipp0E0^|\~& :/0E0)+ ~&rrt->lhthPX ^( ^lrNCC\~&lh mp..vid^*D/..nat2mp@rl -+ mp..sub*+ zipp0E0^|\~& :/0E0, mp..mul~*brlD^|bbI/~&hthPX @l ..nat2mp~~+ predecessor~~NiCiX+-, @r ^|/successor predecessor), ^|(mp..grow/1E0; @iNC ^lrNCC\~& :/0E0,~&/2)+-+-+-
nodes = # takes precision and order (p,n) to a list of nodes and weights <(x_1,w_1)..(x_n,w_n)>
-+
^H( @lrr *+ ^/~&+ mp..div/( ..nat2mp 2)++ mp..mul^/(mp..sqr; //mp..sub ..nat2mp 1)+ mp..sqr+, mp..shr^*DrlXS/~&ll ^|H\~& *+ @NiX+ ->l^|(~&lZ!|+ not+ //mp..eq,@r+ ^/~&+ mp..sub^/~&+ mp..div^)), ^/^|(~&,legendre) mp..cos*+ mp..mul^*D( 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+-</lang>
demonstration program:<lang Ursala>#show+
demo =
~&lNrCT (
^|lNrCT(:/'nodes:',:/'weights:')@lSrSX ..mp2str~~* nodes/160 5, :/'integral:' ~&iNC ..mp2str integral(160,5) (mp..exp,-3E0,3E0))</lang>
- 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
zkl
<lang 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
}</lang> <lang zkl>[5..10].walk().append(20).pump(Console.println,fcn(n){
("Gauss-Legendre %2d-point quadrature " "\U222B;\U208B;\U2083;\U207A;\UB3; exp(x) dx = %.13f") .fmt(n,quadrature(n, fcn(x){ x.exp() }, -3, 3))
})</lang>
- 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