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## IS A GIVEN SQUARE MATRIX STRICTLY DIAGONALLY DOMINANT?

Language : Matlab 2007a Authors : Autar Kaw Last Revised : November 25, 2008 Abstract: This program shows you two ways of finding out if a square matrix is diagonally dominant. A square matrix is diagonally dominant if for all rows the absolute value of the diagonal element in a row is strictly greater than than the sum of absolute value of the rest of the elements in that row

clc clear all disp('This program shows you two ways of finding out') disp('if a square matrix is diagonally dominant. A square matrix is') disp('diagonally dominant if for all rows the absolute value of the') disp('diagonal element in a row is strictly greater than than the sum') disp('of absolute value of the rest of the elements in that row') disp(' ')

This program shows you two ways of finding out if a square matrix is diagonally dominant. A square matrix is diagonally dominant if for all rows the absolute value of the diagonal element in a row is strictly greater than than the sum of absolute value of the rest of the elements in that row

## INPUTS

The square matrix

A=[-12 1 -7 2;1 3.4 1.1 1.1; 1 0 -4.5 0;10 1 1 10]; disp ('INPUTS') disp('Here is the square matrix') A disp(' ')

INPUTS Here is the square matrix A = -12.0000 1.0000 -7.0000 2.0000 1.0000 3.4000 1.1000 1.1000 1.0000 0 -4.5000 0 10.0000 1.0000 1.0000 10.0000

## FIRST SOLUTION

This is based on finding for how many rows the condition the absolute value of the diagonal element in a row is strictly greater than than the sum of absolute value of the rest of the elements in that row.

%size gives how many rows and columns in the A matrix rowcol=size(A); n=rowcol(1); % count = for how many rows is the inequality met that % the absolute value of the diagonal element in a row is % strictly greater than than the sum of absolute value % of the rest of the elements in that row count=0; for i=1:1:n sumrow=0; for j=1:1:n if i~=j sumrow=sumrow+abs(A(i,j)); end end if abs(A(i,i))>sumrow count=count+1; end end disp('FIRST WAY') if count==n disp('Matrix is strictly diagonal dominant') else disp('Matrix is NOT strictly diagonal dominant') end

FIRST WAY Matrix is NOT strictly diagonal dominant

## SECOND SOLUTION

This is based on finding for if for any row the condition the absolute value of the diagonal element in a row is strictly greater than than the sum of absolute value of the rest of the elements in that row is NOT met

%size gives how many rows and columns in the A matrix rowcol=size(A); n=rowcol(1); % flag = keeps track if the condition is not met % flag = 1 if matrix is strictly diagonally dominant % flag = 2 if matrix is not strictly diagonally dominant % Assuming matrix is strictly diagonally dominant flag=1; for i=1:1:n sumrow=0; for j=1:1:n if i~=j sumrow=sumrow+abs(A(i,j)); end end % As soon as the condition is not met, it is not a strictly % diagonally dominant matrix if abs(A(i,i))<=sumrow flag=2; break; end end disp(' ') disp('SECOND WAY') if flag==1 disp('Matrix is strictly diagonal dominant') else disp('Matrix is NOT strictly diagonal dominant') end

SECOND WAY Matrix is NOT strictly diagonal dominant