MATLAB code for the efficient automatic integrator

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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Is a square matrix diagonal or not?

A square matrix A is diagonal if all the elements on the off-diagonal are zero. That is, A(i,j)=0 for i~=j.

In this posting, I show a MATLAB program that finds whether a square matrix is diagonal by using three different methods. These are academic ways to reinforce programming skills in a student.

The MATLAB program can be downloaded as a Mfile (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 also available.

%% IS A GIVEN SQUARE MATRIX A DIAGONAL MATRIX?
% Language : Matlab 2007a
% Authors : Autar Kaw
% Last Revised : November 15, 2008
% Abstract: This program shows you three ways of finding out
% if a square matrix is a diagonal matrix. A square matrix is
% diagonal if all the off-diagonal elements are zero, that is
% A(i,j)=0 for i~=j.
clc
clear all
disp(‘This program shows you three ways of finding out’)
disp(‘if a square matrix is a diagonal matrix.’)
disp(‘A square matrix is diagonal if all the off-diagonal’)
disp(‘elements are zero, that is A(i,j)=0 for i~=j.’)
disp(‘ ‘)
%% INPUTS
% The square matrix
A=[1 0 0 0;0 3.4 0 0; 0 0 -4.5 0;0 0 0 0];
disp (‘INPUTS’)
disp(‘Here is the square matrix’)
A
disp(‘ ‘)

%% FIRST SOLUTION
% This is based on counting the number of zeros on
% off the diagonal. If this count is n^2-n then it
% is a diagonal matrix, otherwise it is not a diagonal matrix

%size gives how many rows and columns in the A matrix
rowcol=size(A);
n=rowcol(1);
% count = how many zeros not on the diagonal
count=0;
for i=1:1:n
for j=1:1:n
if A(i,j)==0 & i~=j
count=count+1;
end
end
end
disp(‘FIRST WAY’)
if count==n^2-n
disp(‘Matrix is diagonal’)
else
disp(‘Matrix is NOT diagonal’)
end

%% SECOND SOLUTION
% This is based on finding if any of the off-diagonal elements
% are nozero. As soon as this condition is met, the matrix can be
% deemed not diagonal. If the condition is never met, the matrix is
% diagonal

%size gives how many rows and columns in the A matrix
rowcol=size(A);
n=rowcol(1);
% flag = keeps track if it is diagonal or not
% flag = 1 if matrix is diagonal
% flag = 2 if matrix is not diagonal

% Assuming matrix is diagonal
flag=1;
for i=1:1:n
for j=1:1:n
% flag=2 if off-diagonal element is nonzero.
if A(i,j)~=0 & i~=j
flag=2;
end
end
end
disp(‘ ‘)
disp(‘SECOND WAY’)
if flag==1
disp(‘Matrix is diagonal’)
else
disp(‘Matrix is NOT diagonal’)
end

%% THIRD SOLUTION
% This is based on finding if the sum of the absolute value of
% the off-diagonal elements is nonzero.
% If the sum is nonzero, the matrix is NOT diagonal.
% If the sum is zero, the matrix is diagonal

%size gives how many rows and columns in the A matrix
rowcol=size(A);
n=rowcol(1);

% sum_off_diagonal= sum of absolute value of off-diagonal elements
sum_off_diagonal=0;
for i=1:1:n
for j=1:1:n
if i~=j
sum_off_diagonal=sum_off_diagonal+abs(A(i,j));
end
end
end

disp(‘ ‘)
disp(‘THIRD WAY’)
if sum_off_diagonal==0
disp(‘Matrix is diagonal’)
else
disp(‘Matrix is NOT diagonal’)
end

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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