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
% 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
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(‘Author: Autar K Kaw.’)
%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
func=[‘ The integrand is =’ char(f)];
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)
% Doing the automatic integration
% Calculating the integral using 1-segment rule
% Initializing ea as greater than pre-specified tolerance for loop to work
% Starting with 2-segments inside the while loop
while (ea>tolerance) & (n<=nmax)
% Keeping track of used number of segments
% Calculating the absolute relative approximate error
ea = abs((current_integral-previous_integral)/current_integral)*100;
% Doubling the number of segments for next estimate of the integral
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)
disp(‘ NOTE: The value of integral is not within the pre-specified tolerance’)
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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