Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

Chapter 3
Solving Linear Systems
A number of problems in numerical analysis can be reduced to, or approximated by, a system of
linear equations.
3.1 Gaussian Elimination with Naïve Pivoting
Our goal is the automatic solution of systems of linear equations:
a 11 x 1 + a 12 x 2 + a 13 x 3 + · · · + a 1n x n = b 1
a 21 x 1 + a 22 x 2 + a 23 x 3 + · · · + a 2n x n = b 2
a 31 x 1 + a 32 x 2 + a 33 x 3 + · · · + a 3n x n = b 3
. . .
. .. . .
a n1 x 1 + a n2 x 2 + a n3 x 3 + · · · + a nn x n = b n
In these equations, the a ij and b i are given real numbers. We also write this as
Ax = b,
where A is a matrix, whose element in the i th row and j th column is a ij , and b is a column vector,
whose i th entry is b i .
This gives the easier way of writing this equation:
⎡
⎢
⎣
⎤
a 11 a 12 a 13 · · · a 1n
a 21 a 22 a 23 · · · a 2n
a 31 a 32 a 33 · · · a 3n
.
. . . ..
⎥
. ⎦
a n1 a n2 a n3 · · · a nn
⎡
⎢
⎣
⎤
x 1
x 2
x 3
⎥
. ⎦
x n
⎡
=
⎢
⎣
⎤
b 1
b 2
b 3
⎥
. ⎦
b n
(3.1)
3.1.1 Elementary Row Operations
You may remember that one way to solve linear equations is by applying elementary row operations
to a given equation of the system. For example, if we are trying to solve the given system of
equations, they should have the same solution as the following system:
23