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.