How would you know how many segments to use in a Trapezoidal rule of integration to get an accurate value of the integral? This can be done by applying the Trapezoidal rule for 1 segment rule, then 2 segment rule, followed by 4 segment rule and so on. As soon as the absolute relative approximate error (page 5-6) between the consecutive answers becomes less than the pre-specified tolerance chosen by the user, you would have your integral within the accuracy you desired.

Here is a MATLAB program that does that for you. The MATLAB program that can be downloaded at http://numericalmethods.eng.usf.edu/blog/trapezoidal_rule_automatic.m (better to download it as single quotes from the web-post do not translate correctly with the MATLAB editor). The html file showing the mfile and the command window output is here: http://numericalmethods.eng.usf.edu/blog/html/trapezoidal_rule_automatic.html

% Simulation : Using Trapezoidal rule as an automatic integrator

% Language : Matlab 2007a

% Authors : Autar Kaw, http://numericalmethods.eng.usf.edu

% Mfile available at

% http://numericalmethods.eng.usf.edu/blog/trapezoidal_rule_automatic.m

% Last Revised : October 12, 2008

% Abstract: This program uses multiple-segment Trapezoidal

% rule to integrate f(x) from x=a to x=b within a pre-specified tolerance

clc

clear all

disp(‘This program uses multiple-segment Trapezoidal rule as an automatic integrator’)

disp(‘to integrate f(x) from x=a to x=b within a pre-specified tolerance’)

disp(‘ ‘)

disp(‘Author: Autar K Kaw.’)

disp(‘https://autarkaw.wordpress.com’)

disp(‘http://numericalmethods.eng.usf.edu’)

disp(‘ ‘)

%INPUTS. If you want to experiment, these are the only variables

% you should and can change.

% a = Lower limit of integration

% b = Upper limit of integration

% nmax = Maximum number of segments

% tolerance = pre-specified tolerance in percentage

% f = inline function as integrand

a=5.3;

b=10.7;

nmax=20000;

tolerance=0.005;

f=inline(‘exp(x)*sin(2*x)’);

% SIMULATION

disp(‘INPUTS’)

func=[‘ The integrand is =’ char(f)];

disp(func)

fprintf(‘ Lower limit of integration, a= %g’,a)

fprintf(‘\n Upper limit of integration, b= %g’,b)

fprintf(‘\n Maximum number of segments, nmax = %g’,nmax)

fprintf(‘\n Pre-specified percentage tolerance, eps = %g’,tolerance)

disp(‘ ‘)

disp(‘ ‘)

% Doing the automatic integration

% Calculating the integral using 1-segment rule

previous_integral=(b-a)/2*(f(a)+f(b));

% Initializing ea as greater than pre-specified tolerance for loop to work

ea=2*tolerance;

% Starting with 2-segments inside the while loop

n=2;

while (ea>tolerance) & (n<=nmax)

h=(b-a)/n;

% Keeping track of used number of segments

nused=n;

current_integral=0;

for i=1:1:n-1

current_integral=current_integral+f(a+i*h);

end

current_integral=2*current_integral+f(a)+f(b);

current_integral=(b-a)/(2*n)*current_integral;

% Calculating the absolute relative approximate error

ea = abs((current_integral-previous_integral)/current_integral)*100;

previous_integral=current_integral;

% Doubling the number of segments for next estimate of the integral

n=n*2;

end

disp(‘OUTPUTS’)

fprintf(‘ Number of segments used =%g’, nused)

fprintf(‘\n Approximate value of integral is =%g’,current_integral)

fprintf(‘\n Absolute percentage relative approximate error =%g’, ea)

if (ea>tolerance)

disp(‘ ‘)

disp(‘ ‘)

disp(‘ NOTE: The value of integral is not within the pre-specified tolerance’)

end

disp(‘ ‘)

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu.

An abridged (for low cost) book on *Numerical Methods with Applications* will be in print (includes problem sets, TOC, index) on December 10, 2008 and available at lulu storefront.

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