Numerical integration/Adaptive Simpson's method
Lychee (1969)'s Modified Adaptive Simpson's method (doi:10.1145/321526.321537) is a numerical quadrature method that recursively bisects the interval until the precision is high enough.
; Lychee's ASR, Modifications 1, 2, 3
procedure _quad_asr_simpsons(f, a, fa, b, fb)
m := (a + b) / 2
fm := f(m)
h := b - a
return multiple [m, fm, (h / 6) * (f(a) + f(b) + 4*sum1 + 2*sum2)]
procedure _quad_asr(f, a, fa, b, fb, tol, whole, m, fm, depth)
lm, flm, left := _quad_asr_simpsons(f, a, fa, m, fm)
rm, frm, right := _quad_asr_simpsons(f, m, fm, b, fb)
delta := left + right - whole
tol' := tol / 2
if depth <= 0 or tol' == tol or abs(delta) <= 15 * tol:
return left + right + delta / 15
else:
return _quad_asr(f, a, fa, m, fm, tol', left , lm, flm, depth - 1) +
_quad_asr(f, m, fm, b, fb, tol', right, rm, frm, depth - 1)
procedure quad_asr(f, a, b, tol, depth)
fa := f(a)
fb := f(b)
m, fm, whole := _quad_asr_simpsons(f, a, fa, b, fb)
return _quad_asr(f, a, fa, b, fb, tol, whole, m, fm, depth)
|
C
<lang c>#include <stdio.h>
- include <math.h>
typedef struct { double m; double fm; double simp; } triple;
/* "structured" adaptive version, translated from Racket */ triple _quad_simpsons_mem(double (*f)(double), double a, double fa, double b, double fb) {
// Evaluates Simpson's Rule, also returning m and f(m) to reuse.
double m = (a + b) / 2;
double fm = f(m);
double simp = fabs(b - a) / 6 * (fa + 4*fm + fb);
triple t = {m, fm, simp};
return t;
}
double _quad_asr(double (*f)(double), double a, double fa, double b, double fb, double eps, double whole, double m, double fm) {
// Efficient recursive implementation of adaptive Simpson's rule.
// Function values at the start, middle, end of the intervals are retained.
triple lt = _quad_simpsons_mem(f, a, fa, m, fm);
triple rt = _quad_simpsons_mem(f, m, fm, b, fb);
double delta = lt.simp + rt.simp - whole;
if (fabs(delta) <= eps * 15) return lt.simp + rt.simp + delta/15;
return _quad_asr(f, a, fa, m, fm, eps/2, lt.simp, lt.m, lt.fm) +
_quad_asr(f, m, fm, b, fb, eps/2, rt.simp, rt.m, rt.fm);
}
double quad_asr(double (*f)(double), double a, double b, double eps) {
// Integrate f from a to b using ASR with max error of eps. double fa = f(a); double fb = f(b); triple t = _quad_simpsons_mem(f, a, fa, b, fb); return _quad_asr(f, a, fa, b, fb, eps, t.simp, t.m, t.fm);
}
int main(){
double a = 0.0, b = 1.0;
double sinx = quad_asr(sin, a, b, 1e-09);
printf("Simpson's integration of sine from %g to %g = %f\n", a, b, sinx);
return 0;
}</lang>
- Output:
Simpson's integration of sine from 0 to 1 = 0.459698
Factor
<lang factor>USING: formatting kernel locals math math.functions math.ranges sequences ; IN: rosetta-code.simpsons
- simps ( f a b n -- x )
n even? [ n "n must be even; %d was given" sprintf throw ] unless b a - n / :> h 1 n 2 <range> 2 n 1 - 2 <range> [ [ a + h * f call ] map-sum ] bi@ [ 4 ] [ 2 ] bi* [ * ] 2bi@ a b [ f call ] bi@ + + + h 3 / * ; inline
[ sin ] 0 1 100 simps "Simpson's rule integration of sin from 0 to 1 is: %u\n" printf</lang>
- Output:
Simpson's rule integration of sin from 0 to 1 is: 0.4596976941573994
Go
Like the zkl entry, this is also a translation of the Python code in the Wikipedia article. <lang go>package main
import (
"fmt" "math"
)
type F = func(float64) float64
/* "structured" adaptive version, translated from Racket */ func quadSimpsonsMem(f F, a, fa, b, fb float64) (m, fm, simp float64) {
// Evaluates Simpson's Rule, also returning m and f(m) to reuse. m = (a + b) / 2 fm = f(m) simp = math.Abs(b-a) / 6 * (fa + 4*fm + fb) return
}
func quadAsrRec(f F, a, fa, b, fb, eps, whole, m, fm float64) float64 {
// Efficient recursive implementation of adaptive Simpson's rule.
// Function values at the start, middle, end of the intervals are retained.
lm, flm, left := quadSimpsonsMem(f, a, fa, m, fm)
rm, frm, right := quadSimpsonsMem(f, m, fm, b, fb)
delta := left + right - whole
if math.Abs(delta) <= eps*15 {
return left + right + delta/15
}
return quadAsrRec(f, a, fa, m, fm, eps/2, left, lm, flm) +
quadAsrRec(f, m, fm, b, fb, eps/2, right, rm, frm)
}
func quadAsr(f F, a, b, eps float64) float64 {
// Integrate f from a to b using ASR with max error of eps. fa, fb := f(a), f(b) m, fm, whole := quadSimpsonsMem(f, a, fa, b, fb) return quadAsrRec(f, a, fa, b, fb, eps, whole, m, fm)
}
func main() {
a, b := 0.0, 1.0
sinx := quadAsr(math.Sin, a, b, 1e-09)
fmt.Printf("Simpson's integration of sine from %g to %g = %f\n", a, b, sinx)
}</lang>
- Output:
Simpson's integration of sine from 0 to 1 = 0.459698
J
Typically one would choose the library implementation:
load'~addons/math/misc/integrat.ijs' NB. integrate returns definite integral and estimated digits of accuracy 1&o. integrate 0 1 0.459698 9 NB. adapt implements adaptive Simpson's rule, however recomputes some integrands 1&o. adapt 0 1 1e_9 0.459698
<lang J>
Note'expected answer computed by j www.jsoftware.com'
1-&:(1&o.d._1)0 0.459698
translated from c
)
mp=: +/ .* NB. matrix product
NB. Evaluates Simpson's Rule, also returning m and f(m) to reuse. uquad_simpsons_mem=: adverb define
'a fa b fb'=. y em=. a ([ + [: -: -~) b fm=. u em simp=. ((| b - a) % 6) * 1 4 1 mp fa , fm , fb em, fm, simp
)
Simp=: 1 :'2{m' Fm=: 1 :'1{m' M=: 1 :'0{m'
NB. Efficient recursive implementation of adaptive Simpson's rule. NB. Function values at the start, middle, end of the intervals are retained. uquad_asr=: adverb define
'a fa b fb eps whole em fm'=. y lt=. u uquad_simpsons_mem(a, fa, em, fm) rt=. u uquad_simpsons_mem(em, fm, b, fb) delta=. lt Simp + rt Simp - whole if. (| delta) <: eps * 15 do. lt Simp + rt Simp + delta % 15 else. (a, fa, em, fm, (-: eps), lt Simp, lt M, lt Fm) +&(u uquad_asr) (em, fm, b, fb, (-: eps), rt Simp, rt M, rt Fm) end.
)
NB. Integrate u from a to b using ASR with max error of eps. quad_asr=: adverb define
'a b eps'=. y fa=. u a fb=. u b t=. u uquad_simpsons_mem a, fa, b, fb u uquad_asr a, fa, b, fb, eps, t Simp, t M, t Fm
) </lang>
echo 'Simpson''s integration of sine from 0 to 1 = ' , ": 1&o. quad_asr 0 1 1e_9 Simpson's integration of sine from 0 to 1 = 0.459698
Java
<lang Java> import java.util.function.Function;
public class NumericalIntegrationAdaptiveSimpsons {
public static void main(String[] args) {
Function<Double,Double> f = x -> sin(x);
System.out.printf("integrate sin(x), x = 0 .. Pi = %2.12f. Function calls = %d%n", quadratureAdaptiveSimpsons(f, 0, Math.PI, 1e-8), functionCount);
functionCount = 0;
System.out.printf("integrate sin(x), x = 0 .. 1 = %2.12f. Function calls = %d%n", quadratureAdaptiveSimpsons(f, 0, 1, 1e-8), functionCount);
}
private static double quadratureAdaptiveSimpsons(Function<Double,Double> function, double a, double b, double error) {
double fa = function.apply(a);
double fb = function.apply(b);
Triple t = quadratureAdaptiveSimpsonsOne(function, a, fa, b ,fb);
return quadratureAdaptiveSimpsonsRecursive(function, a, fa, b, fb, error, t.s, t.x, t.fx);
}
private static double quadratureAdaptiveSimpsonsRecursive(Function<Double,Double> function, double a, double fa, double b, double fb, double error, double whole, double m, double fm) {
Triple left = quadratureAdaptiveSimpsonsOne(function, a, fa, m, fm);
Triple right = quadratureAdaptiveSimpsonsOne(function, m, fm, b, fb);
double delta = left.s + right.s - whole;
if ( Math.abs(delta) <= 15*error ) {
return left.s + right.s + delta / 15;
}
return quadratureAdaptiveSimpsonsRecursive(function, a, fa, m, fm, error/2, left.s, left.x, left.fx) +
quadratureAdaptiveSimpsonsRecursive(function, m, fm, b, fb, error/2, right.s, right.x, right.fx);
}
private static Triple quadratureAdaptiveSimpsonsOne(Function<Double,Double> function, double a, double fa, double b, double fb) {
double m = (a + b) / 2;
double fm = function.apply(m);
return new Triple(m, fm, Math.abs(b-a) / 6 * (fa + 4*fm + fb));
}
private static class Triple {
double x, fx, s;
private Triple(double m, double fm, double s) {
this.x = m;
this.fx = fm;
this.s = s;
}
}
private static int functionCount = 0;
private static double sin(double x) {
functionCount++;
return Math.sin(x);
}
} </lang>
- Output:
integrate sin(x), x = 0 .. Pi = 1.999999999998. Function calls = 121 integrate sin(x), x = 0 .. 1 = 0.459697694131. Function calls = 33
Julia
Originally from Modesto Mas, https://mmas.github.io/simpson-integration-julia <lang julia>function simps(f::Function, a::Number, b::Number, n::Number)
iseven(n) || throw("n must be even, and $n was given")
h = (b-a)/n
s = f(a) + f(b)
s += 4 * sum(f.(a .+ collect(1:2:n) .* h))
s += 2 * sum(f.(a .+ collect(2:2:n-1) .* h))
h/3 * s
end
println("Simpson's rule integration of sin from 0 to 1 is: ", simps(sin, 0.0, 1.0, 100))
</lang>
- Output:
Simpson's rule integration of sin from 0 to 1 is: 0.45969769415739936
Kotlin
<lang scala>// Version 1.2.71
import kotlin.math.abs import kotlin.math.sin
typealias F = (Double) -> Double typealias T = Triple<Double, Double, Double>
/* "structured" adaptive version, translated from Racket */ fun quadSimpsonsMem(f: F, a: Double, fa: Double, b: Double, fb: Double): T {
// Evaluates Simpson's Rule, also returning m and f(m) to reuse val m = (a + b) / 2 val fm = f(m) val simp = abs(b - a) / 6 * (fa + 4 * fm + fb) return T(m, fm, simp)
}
fun quadAsrRec(f: F, a: Double, fa: Double, b: Double, fb: Double,
eps: Double, whole: Double, m: Double, fm: Double): Double {
// Efficient recursive implementation of adaptive Simpson's rule.
// Function values at the start, middle, end of the intervals are retained.
val (lm, flm, left) = quadSimpsonsMem(f, a, fa, m, fm)
val (rm, frm, right) = quadSimpsonsMem(f, m, fm, b, fb)
val delta = left + right - whole
if (abs(delta) <= eps * 15) return left + right + delta / 15
return quadAsrRec(f, a, fa, m, fm, eps / 2, left, lm, flm) +
quadAsrRec(f, m, fm, b, fb, eps / 2, right, rm, frm)
}
fun quadAsr(f: F, a: Double, b: Double, eps: Double): Double {
// Integrate f from a to b using ASR with max error of eps. val fa = f(a) val fb = f(b) val (m, fm, whole) = quadSimpsonsMem(f, a, fa, b, fb) return quadAsrRec(f, a, fa, b, fb, eps, whole, m, fm)
}
fun main(args: Array<String>) {
val a = 0.0
val b = 1.0
val sinx = quadAsr(::sin, a, b, 1.0e-09)
println("Simpson's integration of sine from $a to $b = ${"%6f".format(sinx)}")
}</lang>
- Output:
Simpson's integration of sine from 0.0 to 1.0 = 0.459698
Perl
<lang perl>use strict; use warnings;
sub adaptive_Simpson_quadrature {
my($f, $left, $right, $eps) = @_; my $lf = eval "$f($left)"; my $rf = eval "$f($right)"; my ($mid, $midf, $whole) = Simpson_quadrature_mid($f, $left, $lf, $right, $rf); return recursive_Simpsons_asr($f, $left, $lf, $right, $rf, $eps, $whole, $mid, $midf);
sub Simpson_quadrature_mid {
my($g, $l, $lf, $r, $rf) = @_;
my $mid = ($l + $r) / 2;
my $midf = eval "$g($mid)";
($mid, $midf, abs($r - $l) / 6 * ($lf + 4 * $midf + $rf))
}
sub recursive_Simpsons_asr {
my($h, $a, $fa, $b, $fb, $eps, $whole, $m, $fm) = @_;
my ($lm, $flm, $left) = Simpson_quadrature_mid($h, $a, $fa, $m, $fm);
my ($rm, $frm, $right) = Simpson_quadrature_mid($h, $m, $fm, $b, $fb);
my $delta = $left + $right - $whole;
abs($delta) <= 15 * $eps
? $left + $right + $delta / 15
: recursive_Simpsons_asr($h, $a, $fa, $m, $fm, $eps/2, $left, $lm, $flm) +
recursive_Simpsons_asr($h, $m, $fm, $b, $fb, $eps/2, $right, $rm, $frm)
}
}
my ($a, $b) = (0, 1); my $sin = adaptive_Simpson_quadrature('sin', $a, $b, 1e-9); printf "Simpson's integration of sine from $a to $b = %.9f", $sin</lang>
- Output:
Simpson's integration of sine from 0 to 1 = 0.459697694
Phix
<lang Phix>function quadSimpsonsMem(integer f, atom a, fa, b, fb)
-- Evaluates Simpson's Rule, also returning m and f(m) to reuse.
atom m = (a + b) / 2,
fm = call_func(f,{m}),
simp = abs(b-a) / 6 * (fa + 4*fm + fb)
return {m, fm, simp}
end function
function quadAsrRec(integer f, atom a, fa, b, fb, eps, whole, m, fm)
-- Efficient recursive implementation of adaptive Simpson's rule.
-- Function values at the start, middle, end of the intervals are retained.
atom {lm, flm, left} := quadSimpsonsMem(f, a, fa, m, fm),
{rm, frm, rght} := quadSimpsonsMem(f, m, fm, b, fb),
delta := left + rght - whole
if abs(delta) <= eps*15 then
return left + rght + delta/15
end if
return quadAsrRec(f, a, fa, m, fm, eps/2, left, lm, flm) +
quadAsrRec(f, m, fm, b, fb, eps/2, rght, rm, frm)
end function
function quadAsr(integer f, atom a, b, eps)
-- Integrate f from a to b using ASR with max error of eps.
atom fa := call_func(f,{a}),
fb := call_func(f,{b}),
{m, fm, whole} := quadSimpsonsMem(f, a, fa, b, fb)
return quadAsrRec(f, a, fa, b, fb, eps, whole, m, fm)
end function
-- we need a mini wrapper to get a routine_id for sin() -- (because sin() is implemented in low-level assembly) function _sin(atom a)
return sin(a)
end function
atom a := 0.0, b := 1.0,
sinx := quadAsr(routine_id("_sin"), a, b, 1e-09)
printf(1,"Simpson's integration of sine from %g to %g = %f\n", {a, b, sinx})</lang>
- Output:
Simpson's integration of sine from 0 to 1 = 0.459698
Python
<lang Python>
- ! python3
example
$ python3 /tmp/integrate.py Simpson's integration of sine from 0.0 to 1.0 = 0.4596976941317858
expected answer computed by j www.jsoftware.com
1-&:(1&o.d._1)0 0.459698
translated from c
import math
import collections triple = collections.namedtuple('triple', 'm fm simp')
def _quad_simpsons_mem(f: callable, a: float , fa: float, b: float, fb: float)->tuple:
Evaluates Simpson's Rule, also returning m and f(m) to reuse.
m = a + (b - a) / 2
fm = f(m)
simp = abs(b - a) / 6 * (fa + 4*fm + fb)
return triple(m, fm, simp,)
def _quad_asr(f: callable, a: float, fa: float, b: float, fb: float, eps: float, whole: float, m: float, fm: float)->float:
Efficient recursive implementation of adaptive Simpson's rule.
Function values at the start, middle, end of the intervals are retained.
lt = _quad_simpsons_mem(f, a, fa, m, fm)
rt = _quad_simpsons_mem(f, m, fm, b, fb)
delta = lt.simp + rt.simp - whole
return (lt.simp + rt.simp + delta/15
if (abs(delta) <= eps * 15) else
_quad_asr(f, a, fa, m, fm, eps/2, lt.simp, lt.m, lt.fm) +
_quad_asr(f, m, fm, b, fb, eps/2, rt.simp, rt.m, rt.fm)
)
def quad_asr(f: callable, a: float, b: float, eps: float)->float:
Integrate f from a to b using ASR with max error of eps.
fa = f(a)
fb = f(b)
t = _quad_simpsons_mem(f, a, fa, b, fb)
return _quad_asr(f, a, fa, b, fb, eps, t.simp, t.m, t.fm)
def main():
(a, b,) = (0.0, 1.0,)
sinx = quad_asr(math.sin, a, b, 1e-09);
print("Simpson's integration of sine from {} to {} = {}\n".format(a, b, sinx))
main() </lang>
Raku
(formerly Perl 6)
Fairly direct translation of the Python code.
<lang perl6>sub adaptive-Simpson-quadrature(&f, $left, $right, \ε = 1e-9) {
my $lf = f($left); my $rf = f($right); my ($mid, $midf, $whole) = Simpson-quadrature-mid(&f, $left, $lf, $right, $rf); return recursive-Simpsons-asr(&f, $left, $lf, $right, $rf, ε, $whole, $mid, $midf);
sub Simpson-quadrature-mid(&g, $l, $lf, $r, $rf){
my $mid = ($l + $r) / 2;
my $midf = g($mid);
($mid, $midf, ($r - $l).abs / 6 * ($lf + 4 * $midf + $rf))
}
sub recursive-Simpsons-asr(&h, $a, $fa, $b, $fb, $eps, $whole, $m, $fm){
my ($lm, $flm, $left) = Simpson-quadrature-mid(&h, $a, $fa, $m, $fm);
my ($rm, $frm, $right) = Simpson-quadrature-mid(&h, $m, $fm, $b, $fb);
my $delta = $left + $right - $whole;
$delta.abs <= 15 * $eps
?? $left + $right + $delta / 15
!! recursive-Simpsons-asr(&h, $a, $fa, $m, $fm, $eps/2, $left, $lm, $flm) +
recursive-Simpsons-asr(&h, $m, $fm, $b, $fb, $eps/2, $right, $rm, $frm)
}
}
my ($a, $b) = 0e0, 1e0; my $sin = adaptive-Simpson-quadrature(&sin, $a, $b, 1e-9).round(10**-9);; say "Simpson's integration of sine from $a to $b = $sin";</lang>
- Output:
Simpson's integration of sine from 0 to 1 = 0.459697694
REXX
<lang rexx>/*REXX program performs numerical integration using adaptive Simpson's method. */ numeric digits length( pi() ) - length(.) /*use # of digits in pi for precision. */ a= 0; b= 1; f= 'SIN' /*define values for A, B, and F. */ sinx= quadAsr('SIN',a,b,"1e" || (-digits() + 1) ) say "Simpson's integration of sine from " a ' to ' b ' = ' sinx exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ pi: pi= 3.14159265358979323846; return pi /*pi has twenty-one decimal digits. */ r2r: return arg(1) // (pi() *2) /*normalize radians ──► a unit circle, */ /*──────────────────────────────────────────────────────────────────────────────────────*/ quadSimp: procedure; parse arg f,a,fa,b,fb; m= (a+b) / 2; interpret 'fm=' f"(m)"
simp= abs(b-a) / 6 * (fa + 4*fm + fb); return m fm simp
/*──────────────────────────────────────────────────────────────────────────────────────*/ quadAsr: procedure; parse arg f,a,b,eps; interpret 'fa=' f"(a)"
interpret 'fb=' f"(b)"
parse value quadSimp(f,a,fa,b,fb) with m fm whole
return quadAsrR(f,a,fa,b,fb,eps,whole,m,fm)
/*──────────────────────────────────────────────────────────────────────────────────────*/ quadAsrR: procedure; parse arg f,a,fa,b,fb,eps,whole,m,fm; frac= digits() * 3 / 4
parse value quadSimp(f,a,fa,m,fm) with lm flm left
parse value quadSimp(f,m,fm,b,fb) with rm frm right
$= left + right - whole /*calculate delta.*/
if abs($)<=eps*frac then return left + right + $/frac
return quadAsrR(f,a,fa,m,fm,eps/2, left,lm,flm) + ,
quadAsrR(f,m,fm,b,fb,eps/2,right,rm,frm)
/*──────────────────────────────────────────────────────────────────────────────────────*/ sin: procedure; parse arg x; x= r2r(x); numeric fuzz min(5, max(1, digits() -3) )
if x=pi*.5 then return 1; if x==pi * 1.5 then return -1
if abs(x)=pi | x=0 then return 0
#= x; _= x; q= x*x
do k=2 by 2 until p=#; p= #; _= - _ * q / (k * (k+1) ); #= # + _
end /*k*/
return #</lang>
- output when using the default inputs:
Simpson's integration of sine from 0 to 1 = 0.459697694131860282602
Sidef
<lang ruby>func adaptive_Simpson_quadrature(f, left, right, ε = 1e-9) {
func quadrature_mid(l, lf, r, rf) {
var mid = (l+r)/2
var midf = f(mid)
(mid, midf, abs(r-l)/6 * (lf + 4*midf + rf))
}
func recursive_asr(a, fa, b, fb, ε, whole, m, fm) {
var (lm, flm, left) = quadrature_mid(a, fa, m, fm)
var (rm, frm, right) = quadrature_mid(m, fm, b, fb)
var Δ = (left + right - whole)
abs(Δ) <= 15*ε
? (left + right + Δ/15)
: (__FUNC__(a, fa, m, fm, ε/2, left, lm, flm) +
__FUNC__(m, fm, b, fb, ε/2, right, rm, frm))
}
var (lf = f(left), rf = f(right)) var (mid, midf, whole) = quadrature_mid(left, lf, right, rf) recursive_asr(left, lf, right, rf, ε, whole, mid, midf)
}
var (a = 0, b = 1) var area = adaptive_Simpson_quadrature({ .sin }, a, b, 1e-15).round(-15) say "Simpson's integration of sine from #{a} to #{b} ≈ #{area}"</lang>
- Output:
Simpson's integration of sine from 0 to 1 ≈ 0.45969769413186
Wren
<lang ecmascript>import "/fmt" for Fmt
/* "structured" adaptive version, translated from Racket */ var quadSimpsonMem = Fn.new { |f, a, fa, b, fb|
// Evaluates Simpson's Rule, also returning m and f.call(m) to reuse. var m = (a + b) / 2 var fm = f.call(m) var simp = (b - a).abs / 6 * (fa + 4*fm + fb) return [m, fm, simp]
}
var quadAsrRec // recursive quadAsrRec = Fn.new { |f, a, fa, b, fb, eps, whole, m, fm|
// Efficient recursive implementation of adaptive Simpson's rule.
// Function values at the start, middle, end of the intervals are retained.
var r1 = quadSimpsonMem.call(f, a, fa, m, fm)
var r2 = quadSimpsonMem.call(f, m, fm, b, fb)
var lm = r1[0]
var flm = r1[1]
var left = r1[2]
var rm = r2[0]
var frm = r2[1]
var right = r2[2]
var delta = left + right - whole
if (delta.abs < eps * 15) return left + right + delta/15
return quadAsrRec.call(f, a, fa, m, fm, eps/2, left, lm, flm) +
quadAsrRec.call(f, m, fm, b, fb, eps/2, right, rm, frm)
}
var quadAsr = Fn.new { |f, a, b, eps|
// Integrate f from a to b using ASR with max error of eps. var fa = f.call(a) var fb = f.call(b) var r = quadSimpsonMem.call(f, a, fa, b, fb) var m = r[0] var fm = r[1] var whole = r[2] return quadAsrRec.call(f, a, fa, b, fb, eps, whole, m, fm)
}
var a = 0 var b = 1 var sinx = quadAsr.call(Fn.new { |x| x.sin }, a, b, 1e-09) Fmt.print("Simpson's integration of sine from $d to $d = $f", a, b, sinx)</lang>
- Output:
Simpson's integration of sine from 0 to 1 = 0.459698
zkl
<lang zkl># "structured" adaptive version, translated from Racket fcn _quad_simpsons_mem(f, a,fa, b,fb){
#Evaluates the Simpson's Rule, also returning m and f(m) to reuse""" m,fm := (a + b)/2, f(m); return(m,fm, (b - a).abs()/6*(fa + fm*4 + fb));
} fcn _quad_asr(f, a,fa, b,fb, eps, whole, m,fm){
# Efficient recursive implementation of adaptive Simpson's rule. # Function values at the start, middle, end of the intervals are retained.
lm,flm,left := _quad_simpsons_mem(f, a,fa, m,fm); rm,frm,right := _quad_simpsons_mem(f, m,fm, b,fb); delta:=left + right - whole; if(delta.abs() <= eps*15) return(left + right + delta/15); _quad_asr(f, a,fa, m,fm, eps/2, left , lm,flm) + _quad_asr(f, m,fm, b,fb, eps/2, right, rm,frm)
} fcn quad_asr(f,a,b,eps){
#Integrate f from a to b using Adaptive Simpson's Rule with max error of eps fa,fb := f(a),f(b); m,fm,whole := _quad_simpsons_mem(f, a,fa, b,fb); _quad_asr(f, a,fa, b,fb, eps,whole,m,fm);
}</lang> <lang zkl>sinx:=quad_asr((1.0).sin.unbind(), 0.0, 1.0, 1e-09); println("Simpson's integration of sine from 1 to 2 = ",sinx);</lang>
- Output:
Simpson's integration of sine from 1 to 2 = 0.459698