Numerical integration/Adaptive Simpson's method

From Rosetta Code
Numerical integration/Adaptive Simpson's method is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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.

Pseudocode: Simpson's method, adaptive
; 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[edit]

Translation of: zkl
#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;
}
Output:
Simpson's integration of sine from 0 to 1 = 0.459698

Factor[edit]

Translation of: Julia
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 ] [email protected] [ 4 ] [ 2 ] bi*
[ * ] [email protected] a b [ f call ] [email protected] + + + h 3 / * ; inline
 
[ sin ] 0 1 100 simps
"Simpson's rule integration of sin from 0 to 1 is: %u\n" printf
Output:
Simpson's rule integration of sin from 0 to 1 is: 0.4596976941573994

Go[edit]

Like the zkl entry, this is also a translation of the Python code in the Wikipedia article.

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)
}
Output:
Simpson's integration of sine from 0 to 1 = 0.459698

J[edit]

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
   


 
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
)
 
   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

Julia[edit]

Originally from Modesto Mas, https://mmas.github.io/simpson-integration-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))
 
Output:
Simpson's rule integration of sin from 0 to 1 is: 0.45969769415739936

Kotlin[edit]

Translation of: Go
// 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)}")
}
Output:
Simpson's integration of sine from 0.0 to 1.0 = 0.459698

Perl[edit]

Translation of: Perl 6
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
Output:
Simpson's integration of sine from 0 to 1 = 0.459697694

Perl 6[edit]

Works with: Rakudo version 2018.10

Fairly direct translation of the Python code.

sub adaptive-Simpson-quadrature(&f, $left, $right,= 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";
Output:
Simpson's integration of sine from 0 to 1 = 0.459697694

Phix[edit]

Translation of: Go
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})
Output:
Simpson's integration of sine from 0 to 1 = 0.459698

Python[edit]

 
#! 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()
 

REXX[edit]

Translation of: Go
/*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 #
output   when using the default inputs:
Simpson's integration of sine from  0  to  1  =  0.459697694131860282602

Sidef[edit]

Translation of: Perl 6
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}"
Output:
Simpson's integration of sine from 0 to 1 ≈ 0.45969769413186

zkl[edit]

Translation of: Python
# "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);
}
sinx:=quad_asr((1.0).sin.unbind(), 0.0, 1.0, 1e-09);
println("Simpson's integration of sine from 1 to 2 = ",sinx);
Output:
Simpson's integration of sine from 1 to 2 = 0.459698