Rectangle method - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically in integral calculus, the rectangle method (also called the midpoint or mid-ordinate rule) computes an approximation

Approximation

An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...

to a definite integral, made by finding the area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

of a collection of rectangle

Rectangle

In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

s whose heights are determined by the values of the function.

Specifically, the interval

over which the function is to be integrated is divided into

equal subintervals of length

. The rectangles are then drawn so that either their left or right corners, or the middle of their top line lies on the graph

Graph of a function

In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

of the function, with bases running along the

-axis. The approximation to the integral is then calculated by adding up the areas (base multiplied by height) of the

rectangles, giving the formula:

where

and

.

The formula for

above gives

for the Top-left corner approximation.

As N gets larger, this approximation gets more accurate. In fact, this computation is the spirit of the definition of the Riemann integral

Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. The Riemann integral is unsuitable for many theoretical purposes...

and the limit

Limit of a sequence

The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

of this approximation as

is defined and equal to the integral of

if this Riemann integral is defined. Note that this is true regardless of which

is used, however the midpoint approximation tends to be more accurate for finite

Error

For a function

which is twice differentiable, the approximation error in each section

of the midpoint rule decays as the cube of the width of the rectangle. (For a derivation based on a Taylor approximation, see Midpoint method

Midpoint method

In numerical analysis, a branch of applied mathematics, the midpoint method is a one-step method for solving the differential equation y' = f, \quad y = y_0...

)

for some

. Summing this, the approximation error for

intervals with width

is less than or equal to

where

is the number of nodes

in terms of the total interval, we know that

so we can rewrite the expression:

for some

Matlab Example Program

function [F] = rectangle( a, b, n )

%*******************************************************

% This function computes an approximation to a definite
% integral using rectangle method. Input variables are
% a=initial value, b=end value, n=number of iterations.
% Author: User:Gulmammad
% Modified: User:fOx
% Modified: User:cholericfun

%*******************************************************

h = ( b - a ) / n ;

F = 0 ;
% x = [0 , h*[1 : n-1]]; % for the left corner: O(h)
x = [a+.5*h,a+.5*h+h*[1:n-1]]; % for the center: O(h²)
% x = [1:n]*h; % for the right corner: O(h)

for i = 1 : n

F = F + f( x(i) ) * h ;

end
function [func] = f( x )

func = 3 * x.^2 ; %specify your function here

end
end

C example program

include
include

double f(double x){
return sin(x);
}

double rectangle_integrate(double a, double b, int subintervals, double (*function)(double) ){
double result;
double interval;
int i;

interval=(b-a)/subintervals;
result=0;

for(i=1;i<=subintervals;i++){
result+=function(a+interval*(i-0.5));
}
result*=interval;

return result;
}

int main(void){
double integral;
integral=rectangle_integrate(0,2,100,f);
printf("The value of the integral is: %f \n",integral);
return 0;
}

Fortran Example Program

PROGRAM Calc
IMPLICIT NONE
DOUBLE PRECISION y, a, b, dx, xi
INTEGER n, i
DATA y /0.0d0/

PRINT*, 'Enter the start, end and numer of intervals:'
READ*, a, b, n

dx = (b-a)/n

DO i = 1, n
xi = a + (i-0.5d0)*dx
y = y + f(xi)*dx
END DO

PRINT*, 'Quadrature of f(x):', y

CONTAINS

FUNCTION f(x)
DOUBLE PRECISION f, x
f = 4.0d0/(1.0d0 + x**2)
END FUNCTION F

END PROGRAM Calc

Error

Matlab Example Program

C example program

Fortran Example Program

See also