Using MatLab: 1. Finish the following M-file. Run your function on the same matr

Question

Using MatLab:

1. Finish the following M-file. Run your function on the same matrix A as given in the lab.

% LU4 - The function in this M-file computes an LU factorization

% of a 4 x 4 matrix under the assumption that elimination can be

% performed without row exchanges.

% Input: 4 x 4 matrix A;

% Output: lower triangular matrix L and upper triangular matrix U.

function [L,U] = LU4(A)

L = eye(4);

U = A;

for j= ...

for i= ...

L(...) = ...; % Do not forget the semicolons here

U(...) = ...; % to suppress intermediate output!

end

end

end

2. Write a function MYLU to perform LU factorization for an arbitrary n × n matrix (under the assumption that elimination can be performed without row exchanges). Run the function on the input

T=

3 9 5 7 1   

8 8 6 0 5   

  2 2 4 2 4   

  1 7 0 4 6   

4 2 7 5 5 .

Explanation / Answer

1.

function [L, U, P] = myLU4(A)

%A must be square

%get size of input matrix
sizeofA=size(A);
n=sizeofA(1);

%instantiate L, P, U
L=eye(n);
P=eye(n);
U=A;

% process
for i=1:n
  
    %Row Reduction
    if U(i,i) == 0
        maxofU = max(abs(U(i:end,1)));
        for j=1:n
            if maxofU == abs(U(j,i))
                temp = U(1,:);
                U(1,:) = U(j,:);
                U(j,:) = temp;
              
                temp = P(:,1);
                P(1,i) = P(j,:);
                P(j,:) = temp;
            end
        end
    end
  
    if U(i,i) ~=1
        temp = eye(n);
        temp(i,i)=U(i,i);
        L=L*temp;
        U(i,:) = U(i,:)/U(i,i); %Ensure pivots are 1
    end
  
    if i~=n
      
        for k=i+1:length(U)
            temp = eye(n);
            temp(k,i) = U(k,i);
            L=L*temp;
            U(k,:)=U(k,:)-U(k,i)*U(i,:);
        end
    end
end

P = P';
end

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