082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

314 Chapter 8 Matrix algebra
Table8.1
Iterates produced byJacobi’smethod.
Iterationno. (n) x (n) y (n)
0 0 0
1 2.0000 1.6667
2 1.1667 2.3333
3 0.8333 2.0556
4 0.9722 1.9444
5 1.0278 1.9907
6 1.0047 2.0093
7 0.9954 2.0016
8 0.9992 1.9985
9 1.0008 1.9997
10 1.0002 2.0003
TheresultsofsuccessivelyapplyingtheseformulaeareshowninTable8.1.Thesequence
of values ofx (n) seems toconverge to1while thatofy (n) seems toconverge to2.
Clearly this sort of approach is simple to program and iterative techniques such as
Jacobi’s method are best implemented on a computer. When writing the program a test
should be incorporated so that after each iteration a check for convergence is made by
comparing successive iterates. In many cases, even when convergence does occur, it is
slow and so other techniques are used which converge more rapidly. The Gauss--Seidel
method is attractive for this reason. It uses the most recent approximation to x when
calculatingyleadingtoimprovedratesofconvergenceasthefollowingexampleshows.
Example8.50 Usethe Gauss--Seidel method tosolve the equations ofExample 8.49.
Solution As before wewrite the equations inthe form
x = − 1 2 y +2
y = 1 3 x + 5 3
Withx (0) = 0,y (0) = 0 as our initial guess,wefind
x (1) = − 1 2 (0)+2=2
To findy (1) weuse the mostrecent approximation toxavailable, thatisx (1) :
y (1) = 1 3 (2) + 5 3 = 2.3333
Generally, wefind
x (n+1) = − 1 2 y(n) +2
y (n+1) = 1 3 x(n+1) + 5 3