LU Factorization for MATLAB-CodePudding

How do you write MATLAB code for LU factorization when U is the unit matrix instead of L. The upper triangular matrix will have the diagonal of 1s instead of the lower triangular matrix.

CodePudding user response：

You are looking for Crout's Method for LU decomposition.

The Wikipedia article has the following code

function [L, U] = LUdecompCrout(A)
        
        [R, C] = size(A);
        for i = 1:R
            L(i, 1) = A(i, 1);
            U(i, i) = 1;
        end
        for j = 2:R
            U(1, j) = A(1, j) / L(1, 1);
        end
        for i = 2:R
            for j = 2:i
                L(i, j) = A(i, j) - L(i, 1:j - 1) * U(1:j - 1, j);
            end
            
            for j = i   1:R
                U(i, j) = (A(i, j) - L(i, 1:i - 1) * U(1:i - 1, j)) / L(i,i);

            end
        end
   end

CodePudding user response：

Here's one way to do this. If M is your matrix,

[P,Q] = lu(M');
L = Q'; U = P';

Alternatively, assuming that M has an LU decomposition, we could do the following:

[L,U,P,Q,D] = lu(M);
L = L*P'*D*Q';

This should yield a lower triangular L and upper triangular U where U has 1's on the diagonal whenever M has an LU decomposition.