Gaussian Elimination
code -- partial pivoting
Usual headers
Script
I have left the lines below uncommented so that your output is written to the command window. If you comment
anything out for testing, please uncomment it before submitting the file
% Make sure the first four cases work correctly before attempting the
% last case
%load the provided .mat file first
load('HW8testdata.mat')
% load('HW8testdat.mat')
% x1
x2 =
x3 =
x4 =
= gauss_elim(A1, b1)
gauss_elim(A2, b2)
gauss_elim(A3, b3)
gauss_elim(A4, b4)
function [Solutions, ResiduumNorm, resCorrected] = gauss_elim(A,b)
%check whether the matrix is square, and vector is of proper dimension
%creating temporary variables to store inputs size to check them
size1 = size(A);
size2 = size(b);
%checking if the matrix is square
if(size1(1) ~= size1(2))
error("Given matrix is not square one.")
end
%checking if the vector has proper dimension
if (size2(2) ~= 1)
error("Given vector has wrong dimensions.")
end
%checking vector and matrix dimensions to augment them
if(size1(1) ~= size2(1))
error("Vector and matrix dimension don't match.")
end
%deleting the temporary variables
clear size1 size2;
%-----------------------------------------
1
Gaussian Elimination
code -- partial pivoting
%input is correct, proceed to function
%----------------------------------------%create augmented matrix with given matrix A and vector b
augmentedMatrix = [A,b];
% 1x1 matrix with size of given square matrix
sizeOfA = size(A,1);
for count = 1:sizeOfA
%----------------------------------------% partial pivoting
%----------------------------------------%additional variable not to
i = count;
for j = count+1 : sizeOfA
if (augmentedMatrix(j,
%index of row with
i = j;
end
end
%swap rows if necessary
if( count~=i )
augmentedMatrix([count
end
modify the 'count' from main loop
count) > augmentedMatrix(i, count) )
the biggest value
i], :) = augmentedMatrix([i count], :);
%----------------------------------------% gaussian elimination
%----------------------------------------%transforming matrix to echelon form
for j = count+1 : sizeOfA
L = augmentedMatrix(j, count) / augmentedMatrix(count , count);
augmentedMatrix(j,:) = augmentedMatrix(j,:) - augmentedMatrix(count,:)
* L;
end
end
%----------------------------------------% solving the linear equations
%-----------------------------------------
%setting up solutions vector for keeping the answers
Solutions = eye(sizeOfA);
solutionsVector = zeros(1, sizeOfA);
%separating augmented matrix for easier calculations
Aechelon = augmentedMatrix(:, 1:end-1);
2
Gaussian Elimination
code -- partial pivoting
bechelon = augmentedMatrix(:, end);
for count = 1: sizeOfA
k = sizeOfA - count + 1;
sum = 0;
for j = k+1 : sizeOfA
sum = sum + (augmentedMatrix(k,j) * solutionsVector(j));
end
solutionsVector(k) = (bechelon(k) - sum)/Aechelon(k,k);
end
solutionsVector = solutionsVector';
Solutions = [Solutions, solutionsVector];
%----------------------------------------% solution error calculation
%-----------------------------------------
%calculating the residuum
residuum = A * solutionsVector - b;
%calculating the Euclidan norm of residuum
norm = 0;
for count = 1 : size(residuum)
norm = norm + (residuum(count) * residuum(count));
end
%outputting normalized residuum
ResiduumNorm = sqrt(norm);
%residuum correction
deltaX = residuum' * inv(A);
deltaX = deltaX';
x2 = solutionsVector - deltaX;
res2 = A * x2 - b;
%calculating the Euclidan norm of corrected residuum
norm2 = 0;
for count = 1 : size(res2)
norm2 = norm2 + (res2(count) * res2(count));
end
resCorrected = sqrt(norm2);
end
x2 =
1.0000
0
0
1.0000
0
0
0
0
3
0
0
-1.1289
3.2454
Gaussian Elimination
code -- partial pivoting
0
0
0
0
0
0
1.0000
0
0
0
1.0000
0
0
0
1.0000
0.2531
0.3573
-1.2804
1.0000
0
0
0
0
0
1.0000
0
0
0
0
0
1.0000
0
0
0
0
0
1.0000
0
0
0
0
0
1.0000
2.4308
-7.9864
0.0391
-0.3953
2.1780
x3 =
Warning: Matrix is singular to working precision.
x4 =
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
NaN
NaN
-Inf
-Inf
Inf
Published with MATLAB® R2022a
4