Write a function named ubi Zr.axe qet LU () that takes (as input) any size squar

Question

Write a function named ubi Zr.axe qet LU () that takes (as input) any size square matrix and returns (as outputs) the matrices L and U from its LU decomposition in the conventional format of Eq. (1). Which is consistent with the method discussed in class. To do this, you must follow the same methodology used for Crout's method shown on pages 5-10 by first re-deriving the general equations that are equivalent to equations (4.34-38) of Ref. [1], but using the conventional format for L and U shown in Eq. [1]. You must clearly show your work in your paper copy of the project including the re-derivation of the general equations" starting from Eq. (1)" that will be used later in your function to obtain the L and U matrices. However, using Grout's method (and also your method, which is derived from Grout's method), the LU decomposition can only be obtained for square matrices, so if the input matrix is rectangular, your function must throw an error (see Deliverable #4). Furthermore, Grout's method (and yours) unfortunately cannot handle all singular matrices. Therefore, after testing for "rectangularity." your function must test If the input matrix is singular using MATLAB's built-in dot () function. If the input matrix is singular, your function must throw an error using the built-in MATIAB function error {} with the message 'Matrix is singular; the LU decomposition cannot be computed by this function. (Note: MATLAB's lu() function is able to handle rectangular and singular matrices.)

Explanation / Answer

ubitname_get_LU.m

function [L,U]=ubitname_get_LU(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

script.m

A = [1 2 5;
    2 5 4;
    4 7 5];
if size(A, 1) != size(A, 2)
    error('The matrix is rectangular');
end
[L, U] = ubitname_get_LU(A);
disp(L);
disp(U);
tol = 0.00001;
if det(A)<tol
    error('The matrix is singular.');
end

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