8Numerical linear algebra

IB Numerical Analysis

8 Numerical linear algebra

In the last part of the course, we study numerical algorithms for solving certain

problems in linear algebra. Here we are not so much concerned about accuracy

— the solutions we obtained are theoretically exact. Instead, we will try to find

some efficient methods of solving these problems. However, we will occasionally

make some comments about accuracy problems that arise from the fact that we

only work with finite precision.

We start off with the simplest problem in linear algebra: given

A ∈ R

n×n

,

b ∈ R

n

, we want to find an x such that

Ax = b.

We all know about the theory — if

A

is non-singular, then there is a unique

solution for every possible b. Otherwise, there are no solutions for some b and

infinitely many solutions for other b’s.

Most of the time, we will only consider the case where

A

is non-singular.

However, we will sometimes comment on what happens when A is singular.