Numerical Analysis/LU decomposition exercises

From Wikiversity
Jump to navigation Jump to search

LU Decomposition exercises and quiz[edit]

Exercise 1[edit]


Problem
Given the 3x3 matrix:

Find LU decomposition of A


Solution
The procedure of LU decomposition runs similar to the process of Gaussian Elimination. Firstly A is reduced to upper triangular form, which is U, using just the third elementary row operation, namely: add to one row of matrix a scalar time another row of that same matrix. Those scalar used during this process are co-efficient in the L matrix.

The final result will look like this: .

Here are the solution for this problem:

According to Gaussian Elimination, the first number in row 2 must be zero-out by adding the first row of matrix a scalar times second row. This scalar, fortunately, is .

Therefore:

.
And .
In the similar manner we have:

.
And

.
And .
Eventually, A has been factorized to LU:
.


Exercise 2[edit]

Use exercise 1's result to solve the system:

Solution:

The idea of using LU decomposition to solve systems of simultaneous linear equations Ax=b is rewriting the systems as L(Ux)=b. To solve x, we first solve the systems Ly=b for y, and then, once y is determined, we solve the systems: Ux=y for x. Both systems are easy to solve, the first by forward substitution and the second by backward substitution.

Here is the solution for this exercise:

This system has the matrix form:

=

Since , by Exercise 1 we have:

Lets , we have:

.

Use forward substitution we have:

Next, since , we have
.
Use backward substitution we have: