Q:

Given

Find u ₁₁, u ₁₂, u ₂₂, and l ₂₁

In general,

Forward and Backward Substitution. The solution of a simultaneous set of algebraic linear equations by triangular factorization involves two main steps:

• formation of the table of factors, and

• forward/backward substitutions.

Having found the table of factors as indicated previously, the following relations are used to complete the solution:

Vector b is destroyed in this process and converted in situ to vector w. Since l ₁₁ is unity, w ₁ is simply b ₁ as can be seen from the matrix solution

Subsequent rows of w are found using

As this equation is used, w ₁ replaces b ₁, w ₂ replaces b ₂, and so on.

Having found w ₁ a similar process is used to find x:

By inspection,

or

Again, in situ technique is used and x _n replaces w _n.

In row n ? 1:

or

Element x _{n ? 1} replaces w _{n ? 1}. The process continues for rows n ? 2, n ? 3, , 1. In row r

This completes the solution.

Q:

Solution

Given

Find u ₁₁, u ₁₂, u ₂₂ and l ₂₁.

In general:

Since the first row...