The following program implements Gaussian elimination method with partial pivoting and scaling to solve system of linear algebraic equations. The explanations of the codes are mentioned just right of each line.
MATLAB Program for Gauss Elimination Method
% Gaussian elimination with partial pivoting and scaling
function x = gausselimination(a,b)
% a - (nxn) matrix
% b - column vector of length n
format long;
m=size(a,1); % get number of rows in matrix a
n=length(b); % get length of b
if (m ~= n)
error('a and b do not have the same number of rows')
end
a(:,n+1)=b; % Forming (n,n+1) augmented matrix
for c=1:n
for r=c:m
% Scaling of the input matrix "a"
Z(r,c) = a(r,c)/max(a(r,c:n)); % Scaling prior to pivoting
K(r:r,c:c)=Z(r,c); % Normalized coefficient parameters saved temporality
% to another matrix "K"
end
disp(K);
[~,f]=max(abs(K(1:r,c:c))); % Finding maximum absolute value from "K" matrix
% for switching rows in matrix "a"
disp(f);
a([c,f],1:n+1)=a([f,c],1:n+1); % Switching between two rows
disp(a);
for i=c
% Making diagonal element of matrix "a" into 1.0
a(i,i:n+1) = a(i,i:n+1)/a(i,i);
for j=i+1:n
% Making all elements below the diagonal element into zero
a(j,i:n+1) = a(j,i:n+1) - a(j,i)*a(i,i:n+1);
end
end
disp(a);
end
% Process of back substitution
for j=n-1:-1:1
a(j,n+1) = a(j,n+1) - a(j,j+1:n)*a(j+1:n,n+1);
end
% Returning solution
x=a(:,n+1);
end
Program Outputs:
ans =
-1.761817043997860
0.896228033874012
4.051931404116157
-1.617130802539541
2.041913538501913
0.151832487155935
Therefore, the solutions of the six linear algebraic equations are -1.76; 0.89; 4.05; -1.62; 2.04 and 0.15.
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