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

8.13 Iterative techniques for the solution of simultaneous equations 315
Table8.2
Iterates produced bythe Gauss--Seidel
method.
Iterationno. (n) x (n) y (n)
0 0 0
1 2.0000 2.3333
2 0.8334 1.9445
3 1.0278 2.0093
4 0.9954 1.9985
5 1.0008 2.0003
6 0.9999 2.0000
andtheresultsofsuccessivelyapplyingtheseformulaeareshowninTable8.2.Asbefore,
we see that the sequence x (n) seems to converge to 1 and y (n) seems to converge to 2,
although more rapidly than before.
Both ofthese techniques generalize tolarger systems of equations.
Example8.51 Perform three iterations of Jacobi’s method and three iterations of the Gauss--Seidel
method tofind anapproximate solution of
−8x+y+z=1
x−5y+ z =16
x+ y−4z =7
with aninitial guess ofx =y=z=0.
Solution Werewritethesystemtomakex,yandzthesubjectofthefirst,secondandthirdequation,
respectively:
x = 1 8 y + 1 8 z − 1 8
y = 1 5 x + 1 5 z − 16
5
(8.11)
z = 1 4 x + 1 4 y − 7 4
To apply Jacobi’s method we substitute the initial guessx (0) = y (0) = z (0) = 0 into
the r.h.s. of Equation (8.11) to obtainx (1) ,y (1) andz (1) , and then repeat the process. In
general,
x (n+1) = 1 8 y(n) + 1 8 z(n) − 1 8
y (n+1) = 1 5 x(n) + 1 5 z(n) − 16 5
z (n+1) = 1 4 x(n) + 1 4 y(n) − 7 4