Lab for the LU Factorization

Module for the LU Factorization

Part I involves the factorization A = LU .
Part II involves solving LU = B.

A = LU Factorization of the matrix A. If row interchanges are not needed to solve the linear system AX = B, then A has the LU factorization (illustrated with 4x4 matrices)
=

Back Substitution. To solve the upper-triangular system:

Proceed with the method only if all the diagonal elements are non-zero.
First compute and then use the rule
for k = n-1, n-2, ... , 1.

Forward Substitution. To solve the lower-triangular system:

Proceed with the method only if all the diagonal elements are non-zero. Use the rule
for k = 1,2,...,n.

Report to be handed in.

Computer Problems.

Exercises 1-2 involve the A = LU Factorization of the matrix A.

Exercise 1. Find the LU factorization of the matrix A.

Exercise 2. Find the LU factorization of the matrix A.

Remark. To save typing, the matrix A can be generated with the command Table.

Exercises 3-6 involve back substitution and forward substitution.

Exercise 3. Solve the lower triangular linear system LY = B.

Exercise 4. Solve the upper triangular linear system UX = Y.

Exercise 5. Solve the linear system LUX = Y, where

Exercise 6. Solve the linear system LUX = Y, where

(c) John H. Mathews, 1998