LUP Decomp with Partial Pivoting

(1) Find L, U, and P using Gaussian Elimination with partial pivoting.

(2) Find the d, using forward substitution, by solving L d = P b

(3) Find the solution x using backward substitution by solving U x = d

Function: lup_decomp.m Write an m-file function called lup_decomp.m that decomposes a matrix A into L, U, and P. U is found using Gaussian Elimination with partial pivoting. To find P and L:

(1) Start with P = I, and L = 0

(2) We set the elements of L as we do in L U decomposition (using the factors calculated from Gaussian Elimination).

(3) Whenever we swap rows during the course of partial pivoting, we also swap the same rows in L and P.

(4) When we are all finished, we set the diagonal elements of L to 1. Inputs:

A The matrix to decompose

Outputs:

L The lower triangular matrix

U The upper triangular matrix

P The permutation matrix

Driver: ch2_1.m Solve the following system of equations using P

T L U decomposition:

−5 −3 −5 1 −2 1

5 1 −1 −4 3 −1

−4 −3 3 4 −4 3

0 −2 −3 2 −3 2

2 1 −2 −5 0 −1

1 3 1 2 4 0

x =

1 1 1 1

Print L, U, P, and x to the Command Window, and confirm you have the correct results using the lu built-in function in Ma t lab:

1 [ L U P ] = lu(A)

. . . . The driver isnt so important but this is what I have so far:

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

%outputs of function L,U,P; function of matrix A

n= length(A);

L= eye(n);

U= zeros(n);

P= eye(n);

for k=1:n-1

    [~,r]= max(abs(A(k:end,k)));

    r= n-(n-k+1)+r;

A([k r], :)= A([r k],:);

P([k r],:)= P([r k],:);

L([k r],:)= L([r k],:);

L(k+1:n,k)= A(k+1:n,k)/A(k,k);

U(k,1:n)= A(k,1:n);

A(k+1:n,1:n)= A(k+1:n,1:n)- L(k+1:n,k)*A(k,1:n);

end

U(:,end)= A(:,end);

end . . . ... .. I just have no idea where to go from here