In the previous post, we discussed why doubling the number of segments in the automatic integrator based on multiple-segment trapezoidal rule is more efficient than increasing the number of segments one at a time. But this advantage involves having to store the individual function values from previous calculations and then having to retrieve them properly. This drawback was circumvented very efficiently by using the formula derived in another previous post where there is no need to store individual function values.

The matlab file for finding a definite integral by directly using the multiple segment trapezoidal rule from this post is given here (matlab file, html file), while the matlab file that uses the more efficient formula from this post is given here (matlab file, html file). Here are the inputs to the programs.

% 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=200000;

tolerance=0.000005;

f=inline(‘exp(x)*sin(2*x)’)

We ran both the program on a PC and found that the more efficient algorithm (51 seconds) ran in half the time as the other one (82 seconds). This is expected, as only n function evaluations are made for 2n-segments rule with the efficient formula, while 2n+1 functions evaluations are made for the original formula.

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu, the textbook on Numerical Methods with Applications available from the lulu storefront, and the YouTube video lectures available at http://www.youtube.com/numericalmethodsguy.

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