Introduction to Computational Linear Algebra

Chapter 6

Iterative Methods for Systems of

Linear Equations

In this chapter we discuss iterative methods for solving

Ax = b, (6.1)

where x ∈ R

n

, A ∈ R

n×n

and b ∈ R

n

are given. We assume that A is non-

singular, so that problem (6.1) has a unique solution

x = A

−1

b. (6.2)

Starting from an initial guess x

(0)

, iterative methods generate a sequence

{x

(0)

, x

(1)

, . . . , x

(k)

, . . . , } which converges towards A

−1

b. We define the resid-

uals as

r

(k)

= b − Ax

(k)

, (6.3)

and the errors as

e

(k)

= x − x

(k)

= A

−1

r

(k)

. (6.4)

Thus the sequence of residuals and the sequence of errors converge towards 0

when the iterative method converges. The rate of convergence should be fast ...