Numerical Integration
From Rosetta Code
Programming Task
This is a programming task. It lays out a problem which Rosetta Code users are encouraged to solve, using languages they know.
Write functions to calculate the definite integral of a function (f(x)) using rectangular (left, right, and midpoint), trapezium, and Simpson's methods. Your functions should take in the upper and lower bounds (a and b) and the number of approximations to make in that range (n). Assume that your example already has a function that gives values for f(x).
Simpson's method is defined by the following pseudocode:
h = (b - a) / n sum1 = sum2 = 0 loop on i from 0 to (n - 1) sum1 = sum1 + f(a + h * i + h / 2) loop on i from 1 to (n - 1) sum2 = sum2 + f(a + h * i) answer = (h / 6) * (f(a) + f(b) + 4 * sum1 + 2 * sum2)
What it will be doing is computing integrals for multiple quadratics (like the one shown in Wikipedia) and summing them.
Contents |
[edit] Ada
This solution creates a generic package into which the function F(X) is passed during generic instantiation. The first part is the package specification. The second part is the package body.
generic with function F(X : Long_Float) return Long_Float; package Integrate is function Left_Rect(A, B, N : Long_Float) return Long_Float; function Right_Rect(A, B, N : Long_Float) return Long_Float; function Mid_Rect(A, B, N : Long_Float) return Long_Float; function Trapezium(A, B, N : Long_Float) return Long_Float; function Simpson(A, B, N : Long_Float) return Long_Float; end Integrate;
package body Integrate is
---------------
-- Left_Rect --
---------------
function Left_Rect (A, B, N : Long_Float) return Long_Float is
H : constant Long_Float := (B - A) / N;
Sum : Long_Float := 0.0;
X : Long_Float := A;
begin
while X <= B - H loop
Sum := Sum + (H * F(X));
X := X + H;
end loop;
return Sum;
end Left_Rect;
----------------
-- Right_Rect --
----------------
function Right_Rect (A, B, N : Long_Float) return Long_Float is
H : constant Long_Float := (B - A) / N;
Sum : Long_Float := 0.0;
X : Long_Float := A + H;
begin
while X <= B - H loop
Sum := Sum + (H * F(X));
X := X + H;
end loop;
return Sum;
end Right_Rect;
--------------
-- Mid_Rect --
--------------
function Mid_Rect (A, B, N : Long_Float) return Long_Float is
H : constant Long_Float := (B - A) / N;
Sum : Long_Float := 0.0;
X : Long_Float := A;
begin
while X <= B - H loop
Sum := Sum + (H / 2.0) * (F(X) + F(X + H));
X := X + H;
end loop;
return Sum;
end Mid_Rect;
---------------
-- Trapezium --
---------------
function Trapezium (A, B, N : Long_Float) return Long_Float is
H : constant Long_Float := (B - A) / N;
Sum : Long_Float := F(A) + F(B);
X : Long_Float := 1.0;
begin
while X <= N - 1.0 loop
Sum := Sum + 2.0 * F(A + X * (B - A) / N);
X := X + 1.0;
end loop;
return (B - A) / (2.0 * N) * Sum;
end Trapezium;
-------------
-- Simpson --
-------------
function Simpson (A, B, N : Long_Float) return Long_Float is
H : constant Long_Float := (B - A) / N;
Sum1 : Long_Float := 0.0;
Sum2 : Long_Float := 0.0;
Limit : Integer := Integer(N) - 1;
begin
for I in 0..Limit loop
Sum1 := Sum1 + F(A + H * Long_Float(I) + H / 2.0);
end loop;
for I in 1..Limit loop
Sum2 := Sum2 + F(A + H * Long_Float(I));
end loop;
return H / 6.0 * (F(A) + F(B) + 4.0 * Sum1 + 2.0 * Sum2);
end Simpson;
end Integrate;
[edit] ALGOL 68
MODE F = PROC(LONG REAL)LONG REAL;
###############
## left rect ##
###############
PROC left rect = (F f, LONG REAL a, b, INT n) LONG REAL:
BEGIN
LONG REAL h= (b - a) / n;
LONG REAL sum:= 0;
LONG REAL x:= a;
WHILE x <= b - h DO
sum := sum + (h * f(x));
x +:= h
OD;
sum
END # left rect #;
#################
## right rect ##
#################
PROC right rect = (F f, LONG REAL a, b, INT n) LONG REAL:
BEGIN
LONG REAL h= (b - a) / n;
LONG REAL sum:= 0;
LONG REAL x:= a + h;
WHILE x <= b - h DO
sum := sum + (h * f(x));
x +:= h
OD;
sum
END # right rect #;
###############
## mid rect ##
###############
PROC mid rect = (F f, LONG REAL a, b, INT n) LONG REAL:
BEGIN
LONG REAL h= (b - a) / n;
LONG REAL sum:= 0;
LONG REAL x:= a;
WHILE x <= b - h DO
sum := sum + (h / 2) * (f(x) + f(x + h));
x +:= h
OD;
sum
END # mid rect #;
###############
## trapezium ##
###############
PROC trapezium = (F f, LONG REAL a, b, INT n) LONG REAL:
BEGIN
LONG REAL h= (b - a) / n;
LONG REAL sum:= f(a) + f(b);
LONG REAL x:= 1;
WHILE x <= n - 1 DO
sum := sum + 2 * f(a + x * h );
x +:= 1
OD;
(b - a) / (2 * n) * sum
END # trapezium #;
#############
## simpson ##
#############
PROC simpson = (F f, LONG REAL a, b, INT n) LONG REAL:
BEGIN
LONG REAL h= (b - a) / n;
LONG REAL sum1:= 0;
LONG REAL sum2:= 0;
INT limit:= n - 1;
FOR i FROM 0 TO limit DO
sum1 := sum1 + f(a + h * LONG REAL(i) + h / 2)
OD;
FOR i FROM 1 TO limit DO
sum2 +:= f(a + h * LONG REAL(i))
OD;
h / 6 * (f(a) + f(b) + 4 * sum1 + 2 * sum2)
END # simpson #;
SKIP
[edit] BASIC
Works with: QuickBasic version 4.5
Translation of: Java
FUNCTION leftRect(a, b, n)
h = (b - a) / n
sum = 0
FOR x = a TO b - h STEP h
sum = sum + h * (f(x))
NEXT x
leftRect = sum
END FUNCTION
FUNCTION rightRect(a, b, n)
h = (b - a) / n
sum = 0
FOR x = a + h TO b - h STEP h
sum = sum + h * (f(x))
NEXT x
rightRect = sum
END FUNCTION
FUNCTION midRect(a, b, n)
h = (b - a) / n
sum = 0
FOR x = a TO b - h STEP h
sum = sum + (h / 2) * (f(x) + f(x + h))
NEXT x
midRect = sum
END FUNCTION
FUNCTION trap(a, b, n)
h = (b - a) / n
sum = f(a) + f(b)
FOR i = 1 TO n-1
sum = sum + 2 * f((a + i * h))
NEXT i
trap = h / 2 * sum
END FUNCTION
FUNCTION simpson(a, b, n)
h = (b - a) / n
sum1 = 0
sum2 = 0
FOR i = 0 TO n-1
sum1 = sum + f(a + h * i + h / 2)
NEXT i
FOR i = 1 TO n - 1
sum2 = sum2 + f(a + h * i)
NEXT i
simpson = h / 6 * (f(a) + f(b) + 4 * sum1 + 2 * sum2)
END FUNCTION
[edit] C++
Due to their similarity, it makes sense to make the integration method a policy.
// the integration routine
template<typename Method, typename F, typename Float>
double integrate(F f, Float a, Float b, int steps, Method m)
{
double s = 0;
double h = (b-a)/steps;
for (int i = 0; i < steps; ++i)
s += m(f, a + h*i, h);
return h*s;
}
// methods
class rectangular
{
public:
enum position_type { left, middle, right };
rectangular(position_type pos): position(pos) {}
template<typename F, typename Float>
double operator()(F f, Float x, Float h)
{
switch(position)
{
case left:
return f(x);
case middle:
return f(x+h/2);
case right:
return f(x+h);
}
}
private:
position_type position;
};
class trapezium
{
public:
template<typename F, typename Float>
double operator()(F f, Float x, Float h)
{
return (f(x) + f(x+h))/2;
}
};
class simpson
{
public:
template<typename F, typename Float>
double operator()(F f, Float x, Float h)
{
return (f(x) + 4*f(x+h/2) + f(x+h))/6;
}
};
// sample usage
double f(double x) { return x*x; )
// inside a function somewhere:
double rl = integrate(f, 0.0, 1.0, 10, rectangular(rectangular::left));
double rm = integrate(f, 0.0, 1.0, 10, rectangular(rectangular::middle));
double rr = integrate(f, 0.0, 1.0, 10, rectangular(rectangular::right));
double t = integrate(f, 0.0, 1.0, 10, trapezium());
double s = integrate(f, 0.0, 1.0, 10, simpson());
[edit] D
The function f(x) is Func1/Func2 in this program.
module integrate ;
import std.stdio ;
import std.math ;
alias real function(real) realFn ;
class Sigma{
int n ;
real a, b , h ;
realFn fn ;
string desc ;
enum Method {LEFT = 0, RGHT, MIDD, TRAP, SIMP} ;
static string[5] methodName =
["LeftRect ", "RightRect", "MidRect ", "Trapezium", "Simpson "] ;
static Sigma opCall() {
Sigma s = new Sigma() ;
return s ;
}
static Sigma opCall(realFn f, int n, real a, real b) {
Sigma s = new Sigma(f, n, a, b) ;
return s ;
}
static real opCall(realFn f, int n, real a, real b, Method m) {
return Sigma(f,n,a,b).getSum(m) ;
}
private this() {} ;
this(realFn f, int n, real a, real b) {
setFunction(f) ;
setStep(n) ;
setRange(a,b) ;
}
Sigma opCall(Method m) {
return doSum(m) ;
}
Sigma setFunction(realFn f) {
this.fn = f ;
return this ;
}
Sigma setRange(real a, real b) {
this.a = a ; this.b = b ;
setInterval() ;
return this ;
}
Sigma setStep(int n) {
this.n = n ;
setInterval() ;
return this ;
}
Sigma setDesc(string d) {
this.desc = d ;
return this ;
}
private void setInterval() {
this.h = (b - a) / n ;
}
private real partSum(int i, Method m) {
real x = a + h * i ;
switch(m) {
case Method.LEFT:
return fn(x) ;
case Method.RGHT:
return fn(x + h) ;
case Method.MIDD:
return fn(x + h/2) ;
case Method.TRAP:
return (fn(x) + fn(x + h))/2 ;
default:
}
//case SIMPSON:
return (fn(x) + 4 * fn(x + h/2) + fn(x + h))/6 ;
}
real getSum(Method m) {
real sum = 0 ;
for(int i = 0; i < n ; i++)
sum += partSum(i, m) ;
return sum * h ;
}
Sigma doSum(Method m) {
writefln("%10s = %9.6f", methodName[m], getSum(m)) ;
return this ;
}
Sigma showSetting() {
writefln("\n%s\nA = %9.6f, B = %9.6f, N = %s, h = %s", desc, a, b, n, h) ;
return this ;
}
Sigma doLeft() { return doSum(Method.LEFT) ; }
Sigma doRight() { return doSum(Method.RGHT) ; }
Sigma doMid() { return doSum(Method.MIDD) ; }
Sigma doTrap() { return doSum(Method.TRAP) ; }
Sigma doSimp() { return doSum(Method.SIMP) ; }
Sigma doAll() {
showSetting() ;
doLeft() ; doRight() ; doMid() ; doTrap() ; doSimp() ;
return this ;
}
}
real Func1(real x) {
return cos(x) + sin(x) ;
}
real Func2(real x) {
return 2.0L/(1 + 4*x*x) ;
}
void main() {
// use as a re-usable and manageable object
Sigma s = Sigma(&Func1, 10, -PI/2, PI/2).showSetting()
.doLeft().doRight().doMid().doTrap()(Sigma.Method.SIMP) ;
s.setFunction(&Func2).setStep(4).setRange(-1.0L,2.0L) ;
s.setDesc("Function(x) = 2 / (1 + 4x^2)").doAll() ;
// use as a single function call
writefln("\nLeftRect Integral of FUNC2 =") ;
writefln("%12.9f (%3dstep)\n%12.9f (%3dstep)\n%12.9f (%3dstep).",
Sigma(&Func2, 8, -1.0L, 2.0L, Sigma.Method.LEFT), 8,
Sigma(&Func2, 64, -1.0L, 2.0L, Sigma.Method.LEFT), 64,
Sigma(&Func2, 512, -1.0L, 2.0L, Sigma.Method.LEFT),512) ;
writefln("\nSimpson Integral of FUNC2 =") ;
writefln("%12.9f (%3dstep).",
Sigma(&Func2, 512, -1.0L, 2.0L, Sigma.Method.SIMP),512) ;
}
Parts of the output:
Function(x) = 2 / (1 + 4x^2) A = -1.000000, B = 2.000000, N = 4, h = 0.75 LeftRect = 2.456897 RightRect = 2.245132 MidRect = 2.496091 Trapezium = 2.351014 Simpson = 2.447732
[edit] Forth
fvariable step
defer method ( fn F: x -- fn[x] )
: left execute ;
: right step f@ f+ execute ;
: mid step f@ 2e f/ f+ execute ;
: trap
dup fdup left
fswap right f+ 2e f/ ;
: simpson
dup fdup left
dup fover mid 4e f* f+
fswap right f+ 6e f/ ;
: set-step ( n F: a b -- n F: a )
fover f- dup 0 d>f f/ step f! ;
: integrate ( xt n F: a b -- F: sigma )
set-step
0e
0 do
dup fover method f+
fswap step f@ f+ fswap
loop
drop fnip
step f@ f* ;
\ testing similar to the D example : test ' is method ' 4 -1e 2e integrate f. ; : fn1 fsincos f+ ; : fn2 fdup f* 4e f* 1e f+ 2e fswap f/ ; 7 set-precision test left fn2 \ 2.456897 test right fn2 \ 2.245132 test mid fn2 \ 2.496091 test trap fn2 \ 2.351014 test simpson fn2 \ 2.447732
[edit] Fortran
In ISO Fortran 95 and later if function f() is not already defined to be "elemental", define an elemental wrapper function around it to allow for array-based initialization:
elemental function elemf(x)
real :: elemf, x
elemf = f(x)
end function elemf
Use Array Initializers, Pointers, Array invocation of Elemental functions, Elemental array-array and array-scalar arithmetic, and the SUM intrinsic function
program integrate
integer, parameter :: n = 20 ! or whatever
real, parameter :: a = 0.0, b = 15.0 ! or whatever
real, parameter :: h = (b - a) / n
real, parameter, dimension(0:2*n) :: xpoints = (/ (a + h*i/(n*2), i = 0, 2*n) /)
real, dimension(0:2*n), target :: fpoints ! gather up all the f(x) values needed for all methods once and distribute via pointers
real, dimension(:), pointer :: fleft, fmid, fright
real :: leftrect, midrect, rightrect, trapezoid, simpson
fpoints = elemf(xpoints) ! elemental function wrapper for F runs once for each element of XPOINTS
fleft => fpoints(0 : 2*n-2 : 2)
fmid => fpoints(1 : 2*n-1 : 2)
fright => fpoints(2 : 2*n : 2)
! Left rectangular rule integral
leftrect = h * sum(fleft)
! Middle rectangular rule integral
midrect = h * sum(fmid)
! Right rectangular rule integral
rightrect = h * sum(fright)
! Trapezoid rule integral
trapezoid = h / 2 * sum(fleft + fright)
! Simpson's rule integral
simpson = h / 6 * sum(fleft + fright + 4*fmid)
end program integrate
[edit] Haskell
Different approach from most of the other examples: First, the function f might be expensive to calculate, and so it should not be evaluated several times. So, ideally, we want to have positions x and weights w for each method and then just calculate the approximation of the integral by
approx f xs ws = sum [w * f x | (x,w) <- zip xs ws]
Second, let's to generalize all integration methods into one scheme. The methods can all be characterized by the coefficients vs they use in a particular interval. These will be fractions, and for terseness, we extract the denominator as an extra argument v.
Now there are the closed formulas (which include the endpoints) and the open formulas (which exclude them). Let's do the open formulas first, because then the coefficients don't overlap:
integrateOpen :: Fractional a => a -> [a] -> (a -> a) -> a -> a -> Int -> a integrateOpen v vs f a b n = approx f xs ws * h / v where m = fromIntegral (length vs) * n h = (b-a) / fromIntegral m ws = concat $ replicate n vs c = a + h/2 xs = [c + h * fromIntegral i | i <- [0..m-1]]
Similarly for the closed formulas, but we need an additional function overlap which sums the coefficients overlapping at the interior interval boundaries:
integrateClosed :: Fractional a => a -> [a] -> (a -> a) -> a -> a -> Int -> a integrateClosed v vs f a b n = approx f xs ws * h / v where m = fromIntegral (length vs - 1) * n h = (b-a) / fromIntegral m ws = overlap n vs xs = [a + h * fromIntegral i | i <- [0..m]] overlap :: Num a => Int -> [a] -> [a] overlap n [] = [] overlap n (x:xs) = x : inter n xs where inter 1 ys = ys inter n [] = x : inter (n-1) xs inter n [y] = (x+y) : inter (n-1) xs inter n (y:ys) = y : inter n ys
And now we can just define
intLeftRect = integrateClosed 1 [1,0] intRightRect = integrateClosed 1 [0,1] intMidRect = integrateOpen 1 [1] intTrapezium = integrateClosed 2 [1,1] intSimpson = integrateClosed 3 [1,4,1]
or, as easily, some additional schemes:
intMilne = integrateClosed 45 [14,64,24,64,14] intOpen1 = integrateOpen 2 [3,3] intOpen2 = integrateOpen 3 [8,-4,8]
Some examples:
*Main> intLeftRect (\x -> x*x) 0 1 10 0.2850000000000001 *Main> intRightRect (\x -> x*x) 0 1 10 0.38500000000000006 *Main> intMidRect (\x -> x*x) 0 1 10 0.3325 *Main> intTrapezium (\x -> x*x) 0 1 10 0.3350000000000001 *Main> intSimpson (\x -> x*x) 0 1 10 0.3333333333333334
[edit] Java
The function in this example is assumed to be f(double x).
public class Integrate{
public static double leftRect(double a, double b, double n){
double h = (b - a) / n;
double sum = 0;
for(double x = a;x <= b - h;x += h)
sum += f(x);
return h * sum;
}
public static double rightRect(double a, double b, double n){
double h = (b - a) / n;
double sum = 0;
for(double x = a + h;x <= b - h;x += h)
sum += f(x);
return h * sum;
}
public static double midRect(double a, double b, double n){
double h = (b - a) / n;
double sum = 0;
for(double x = a;x <= b - h;x += h)
sum += (f(x) + f(x + h));
return (h / 2) * sum;
}
public static double trap(double a, double b, double n){
double h = (b - a) / n;
double sum = f(a) + f(b);
for(int i = 1;i < n;i++)
sum += f(a + i * h);
return h * sum;
}
public static double simpson(double a, double b, double n){
double h = (b - a) / n;
double sum1 = 0;
double sum2 = 0;
for(int i = 0;i < n;i++)
sum1 += f(a + h * i + h / 2);
for(int i = 1;i < n;i++)
sum2 += f(a + h * i);
return h / 6 * (f(a) + f(b) + 4 * sum1 + 2 * sum2);
}
//assume f(double x) elsewhere in the class
}
[edit] Logo
to i.left :fn :x :step
output invoke :fn :x
end
to i.right :fn :x :step
output invoke :fn :x + :step
end
to i.mid :fn :x :step
output invoke :fn :x + :step/2
end
to i.trapezium :fn :x :step
output ((i.left :fn :x :step) + (i.right :fn :x :step)) / 2
end
to i.simpsons :fn :x :step
output ( (i.left :fn :x :step)
+ (i.mid :fn :x :step) * 4
+ (i.right :fn :x :step) ) / 6
end
to integrate :method :fn :steps :a :b
localmake "step (:b - :a) / :steps
localmake "sigma 0
; for [x :a :b-:step :step] [make "sigma :sigma + apply :method (list :fn :x :step)]
repeat :steps [
make "sigma :sigma + (invoke :method :fn :a :step)
make "a :a + :step ]
output :sigma * :step
end
to fn2 :x
output 2 / (1 + 4 * :x * :x)
end
print integrate "i.left "fn2 4 -1 2 ; 2.456897
print integrate "i.right "fn2 4 -1 2 ; 2.245132
print integrate "i.mid "fn2 4 -1 2 ; 2.496091
print integrate "i.trapezium "fn2 4 -1 2 ; 2.351014
print integrate "i.simpsons "fn2 4 -1 2 ; 2.447732
[edit] OCaml
let integrate f a b steps meth =
let h = (b -. a) /. float_of_int steps in
let rec helper i s =
if i >= steps then s
else helper (succ i) (s +. meth f (a +. h *. float_of_int i) h)
in
h *. helper 0 0.
let left_rect f x _ =
f x
let mid_rect f x h =
f (x +. h /. 2.)
let right_rect f x h =
f (x +. h)
let trapezium f x h =
(f x +. f (x +. h)) /. 2.
let simpson f x h =
(f x +. 4. *. f (x +. h /. 2.) +. f (x +. h)) /. 6.
let square x = x *. x
let rl = integrate square 0. 1. 10 left_rect let rm = integrate square 0. 1. 10 mid_rect let rr = integrate square 0. 1. 10 right_rect let t = integrate square 0. 1. 10 trapezium let s = integrate square 0. 1. 10 simpson
[edit] Pascal
function RectLeft(function f(x: real): real; xl, xr: real): real; begin RectLeft := f(xl) end; function RectMid(function f(x: real): real; xl, xr: real) : real; begin RectMid := f((xl+xr)/2) end; function RectRight(function f(x: real): real; xl, xr: real): real; begin RectRight := f(xr) end; function Trapezium(function f(x: real): real; xl, xr: real): real; begin Trapezium := (f(xl) + f(xr))/2 end; function Simpson(function f(x: real): real; xl, xr: real): real; begin Simpson := (f(xl) + 4*f((xl+xr)/2) + f(xr))/6 end; function integrate(function method(function f(x: real): real; xl, xr: real): real; function f(x: real): real; a, b: real; n: integer); var integral, h: real; k: integer; begin integral := 0; h := (b-a)/n; for k := 0 to n-1 do begin integral := integral + method(f, a + k*h, a + (k+1)*h) end; integrate := integral end;
[edit] Python
def left_rect(f,x,h): return f(x) def mid_rect(f,x,h): return f(x + h/2) def right_rect(f,x,h): return f(x+h) def trapezium(f,x,h): return (f(x) + f(x+h))/2.0 def simpson(f,x,h): return (f(x) + 4*f(x + h/2) + f(x+h))/6.0 def square(x): return x*x def integrate( f, a, b, steps, meth): h= (b-a)/steps ival = h*sum( (meth(f, a+i*h, h) for i in range(0,steps))) return ival t = integrate( square, 3.0, 7.0, 30, simpson )
A faster Simpson's rule integrator is
def faster_simpson(f, a, b, steps): h = (b-a)/steps a1 = a+h/2 s1 = sum( f(a1+i*h) for i in range(0,steps)) s2 = sum( f(a+i*h) for i in range(1,steps)) return (h/6.0)*(f(a)+f(b)+4.0*s1+2.0*s2)
[edit] Scheme
(define (integrate f a b steps meth)
(define h (/ (- b a) steps))
(* h
(let loop ((i 0) (s 0))
(if (>= i steps)
s
(loop (+ i 1) (+ s (meth f (+ a (* h i)) h)))))))
(define (left-rect f x h) (f x))
(define (mid-rect f x h) (f (+ x (/ h 2))))
(define (right-rect f x h) (f (+ x h)))
(define (trapezium f x h) (/ (+ (f x) (f (+ x h))) 2))
(define (simpson f x h) (/ (+ (f x) (* 4 (f (+ x (/ h 2)))) (f (+ x h))) 6))
(define (square x) (* x x))
(define rl (integrate square 0 1 10 left-rect))
(define rm (integrate square 0 1 10 mid-rect))
(define rr (integrate square 0 1 10 right-rect))
(define t (integrate square 0 1 10 trapezium))
(define s (integrate square 0 1 10 simpson))
