Question

Using MATLAB, develop an M-file to determine LU factorization of a square matrix with partial pivoting. That is, develop a function called mylu that is passed the square matrix [A] and returns the triangular matrices [L] and [U] and the permutation P. You are not to use MATLAB built-in function lu in your codes. Test your function by using it to solve a system of equations listed below in part 3. Confirm that your function is working properly by verifying that [L][U]=P[A] and by using the MATLAB built-in function lu.

Using MATLAB, develop an M-file to determine matrix inverse based on the LU factorization method above. That is, develop a function called myinv that is passed the square matrix [A] and utilizing codes of part 1 above to return the inversed matrix. You are not to use MATLAB built-in function inv or left-division in your codes. Test your function by using it to solve a system of equations listed below in part 3. Confirm that your function is working properly by verifying that [A]ሾܣሿିଵ=[I] and by using the MATLAB built-in function inv.

Using MATLAB, develop an M-file for the Gauss-Seidel Method to solve the system of equations listed below until the percent relative error falls below ߝ௦ = 5%.

Please solve all parts.

1. Using MATLAB, develop an M-file to determine LU factorization of a square matrix with partial pivoting. That is, develop a function called mylu that is passed the square matrix [A] and returns the triangular matrices [L] and [U] and the permutation P. You are not to use MATLAB built-in function lu in your codes. Test your function by using it to solve a system of equations listed below in part 3. Confirm that your function is working properly by verifying that [L]lU] P[A] and by using the MATLAB built-in function lu 2. Using MATLAB, develop an M-file to determine matrix inverse based on the LU factorization method above. That is, develop a function called myinv that is passed the square matrix [A] and utilizing codes of part 1 above to return the inversed matrix. You are not to use MATLAB built-in function inv or left-division in your codes. Test your function by using it to solve a system of equations listed below in part 3. Confirm that your function is working properly by verifying that [A][Al- and by using the MATLAB built-in function inv. 3. Using MATLAB, develop an M-file for the Gauss-Seidel Method to solve the system of equations listed below until the percent relative error falls below 5% X1 + X2 + 5X3 -21.5 3X1-6X2 + 2X3 =-61.5 10X1 + 2X2-X3-27 Hints: Your M-file codes for all 3 problems must be able to handle generally a square matrix of general size n x n (i.e. not just limited to the 3 x 3 matrix in the example above)

Answer 1

Part 1: Program Screen Shot:

Sample Output:

Part 1 Program Code to Copy :

% create a function mylu
% it is passed the square matrix A
% returns to the triangular matrix
% functio [L], [U] and permutation p.
function [L,U,P] = myLu(A)

% store the length of square matrix A
s=length(A);

% A matrix is assigned to U
U=A;

L=zeros(s,s);
PivVot=(0:s-1)';
for j=1:s,

% Check out the pivot voting
% The maximum numeric value in the column first.
[~,inditi]=max(abs(U(j:s,j)));
inditi=inditi+(j-1);
tri=PivVot(j); PivVot(j)=PivVot(inditi); PivVot(inditi)=tri;
tri=L(j,1:j-1); L(j,1:j-1)=L(inditi,1:j-1); L(inditi,1:j-1)=tri;
tri=U(j,j:end); U(j,j:end)=U(inditi,j:end); U(inditi,j:end)=tri;

% Check out the LU factorization
L(j,j)=1;

% loop to check out the L and U
for i=(1+j):size(U,1)
c= U(i,j)/U(j,j);
U(i,j:s)=U(i,j:s)-U(j,j:s)*c;
L(i,j)=c;
end
end

% Pivot is the matrix
P=zeros(s,s);
P(PivVot(:)*s+(1:s)')=1;
end

Part 2: Program Screen Shot:

Sample output:

octave:5 A*Ainv ans[1.0000 -0.0000 0 0.0000 1.0000 0.0000 0.0000 0.0000 1.0000] 1.0001e+00 -1.1736e-04 1.1980e-05 6.7930e-05 9.9992e-01 9.7400e-06 1.4960e-04 1.5075e-04 9.9991e-01 ans -0 1

Part 2: Program code to copy:

% create the Ainv inverse function
function Ainv=myinv(A)

% used to utilizing the part 1 function to find the LU decomposition
[L,U,P] = myLu(A);

% Check out the identity matrix
I =eye(size(A));
s=size(A,1);
Ainv=zeros(size(A));
for i=1:s
b=I(:,i);

% A inverse function of triangular forward and backward sub matrix.
Ainv(:,i)=TriBackS(U,TriForS(L,P*b));
end

% create a function of Triangular for submatrix.
function C=TriForS(L,b)
s=length(b);
C=zeros(s,1);
C(1)=b(1)/L(1,1);
for j=2:s
C(j)=(b(j) -sum(L(j,1:j-1)'.*C(1:j-1)))/L(j,j);
% end of the funtion
end

% Create a function of triangular backward Submatrix
function C=TriBackS(U,b)
s=length(b);
C=zeros(s,1);
C(s)=b(s)/U(s,s);

% loop to enter the triangual backward Submatrix element
for j=(s-1):-1:1
C(j)=(b(j) -sum(U(j,j+1:end)'.*C(j+1:end)))/U(j,j);

% end of the function
end

Part 3: Program code to Screen Shot:

Sample Output:

Part 3: Program Code to Copy:

% clear all the memory

clear all

% close all the previous function.

close all

% clear all the screen

clc

% Input the any A square matrix

A = [1 1 5;

-3 -6 2;

10 2 -1];

% The constant vector of coefecient matrix M

M = [-21.5;-61.5;27];

n = length(M);

X = zeros(n,1);

Error = ones(n,1);

% loop to verify the matrix is diagnol

for i = 1:n

j = 1:n;

j(i) = [];

B = abs(A(i,j));

% Verify the diagnol value of the matrix and check is greater that or not

% with remaining row values.

Check(i) = abs(A(i,i)) - sum(B);

if Check(i) < 0

fprintf(' Matrix is diagonally strict %2i\n\n',i)

end

end

% Initialize iteration of the method

iteration = 0;

% Check the relative error falls below the 5%

while max(Error) > 0.05

% goes to the etration step by step

iteration = iteration + 1;

Z = X;

% loop to enter the lement of Gauess_seidel Method

for i = 1:n

j = 1:n;

% Unknown remaining coefiecient is eliminate

j(i) = [];

Xtemp = X;

Xtemp(i) = [];

% coefiecient matrix store

X(i) = (M(i) - sum(A(i,j) * Xtemp)) / A(i,i);

end

% evluauvate the eroor from the iteration

Xsolution(:,iteration) = X;

Error= sqrt((X - Z).^2);

end

% The Gauss_Seidel Method

GaussSeidelTable = [1:iteration;Xsolution]';

MatrixSquare = [A X M]