# Linear Systems

• Rainer Kress
Part of the Graduate Texts in Mathematics book series (GTM, volume 181)

## Abstract

The solution of systems of linear equations arises in various parts of mathematics and is of central importance in numerical analysis. To illustrate the significance of linear systems, we will start this chapter by providing some examples of their occurrence as part of the numerical solution of differential and integral equations. After seeing the examples, we will proceed with the solution of systems of linear equations. In principle, we have to distinguish between two groups of methods for the solution of linear systems:
1. 1

In the so-calleddirect methodsorelimination methodsthe exact solution, in principle, is determined through a finite number of arithmetic operations (in real arithmetic leaving aside the influence of roundoff errors).

2. 2

In contrast to thisiterative methodsgenerate a sequence of approximations to the solution by repeating the application of the same computational procedure at each step of the iteration. Usually, they are applied for large systems with special structures that ensure convergence of the successive approximations.

### Keywords

Manifold Expense