Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.3 SIMULTANEOUS LINEAR EQUATIONS

the case then an iterative method may produce a satisfactory degree of precision

with less calculation. Such a method, known asGauss–Seidel iteration,isbased

upon the following analysis.

The problem is again that of finding the components of the column matrixx

that satisfies

Ax=b (27.23)

whenAandbare a given matrix and column matrix, respectively.

The steps of the Gauss–Seidel scheme are as follows.

(i) Rearrange the equations (usually by simple division on both sides of each

equation) so that all diagonal elements of the new matrixCare unity, i.e.

(27.23) becomes

Cx=d, (27.24)

whereC=I−F,andFhas zeros as its diagonal elements.

(ii) Step (i) produces

Fx+d=Ix=x, (27.25)

and this forms the basis of an iteration scheme,

xn+1=Fxn+d, (27.26)

wherexnis thenth approximation to the required solution vectorξ.

(iii) To improve the convergence, the matrixF, which has zeros on its leading

diagonal, can be written as the sum of two matricesLandUthat have

non-zero elements only below and above the leading diagonal, respectively:

Lij=

{

Fij ifi>j,

0otherwise,

(27.27)

Uij=

{

Fij ifi<j,

0otherwise.

This allows the latest values of the components ofxto be used at each

stage and an improved form of (27.26) to be obtained:

xn+1=Lxn+1+Uxn+d. (27.28)

To see why this is possible, we note, for example, that when calculating,

say, the fourth component ofxn+1, its first three components are already

known, and, because of the structure ofL, these are the only ones needed

to evaluate the fourth component ofLxn+1.