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

8.13 Iterative techniques for the solution of simultaneous equations 317
Table8.3
Comparisonofthe Jacobi andGauss--Seidel methods.
Iteration
no. (n)
Jacobi’smethod Gauss--Seidel
x (n) y (n) z (n) x (n) y (n) z (n)
0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1 −0.1250 −3.2000 −1.7500 −0.1250 −3.2250 −2.5875
2 −0.7438 −3.5750 −2.5813 −0.8516 −3.8878 −2.9348
3 −0.8945 −3.8650 −2.8297 −0.9778 −3.9825 −2.9901
4 −0.9618 −3.9448 −2.9399 −0.9966 −3.9973 −2.9985
5 −0.9856 −3.9803 −2.9767 −0.9995 −3.9996 −2.9998
6 −0.9946 −3.9924 −2.9915 −0.9999 −3.9999 −3.0000
7 −0.9980 −3.9972 −2.9968
8 −0.9992 −3.9990 −2.9988
9 −0.9997 −3.9996 −2.9996
Then,
x (2) = 1 8 (−3.2250) + 1 8 (−2.5875) − 1 8 = −0.8516
y (2) = 1 5 (−0.8516) + 1 5 (−2.5875) − 16 5 = −3.8878
Finally,
z (2) = 1 4 (−0.8516) + 1 4 (−3.8878) − 7 4 = −2.9348
x (3) = 1 8 (−3.8878) + 1 8 (−2.9348) − 1 8 = −0.9778
y (3) = 1 5 (−0.9778) + 1 5 (−2.9348) − 16 5 = −3.9825
z (3) = 1 4 (−0.9778) + 1 4 (−3.9825) − 7 4 = −2.9901
For completeness, further iterations areshown inTable 8.3.
AsexpectedtheGauss--SeidelmethodconvergesmorerapidlythanJacobi’s.Thisis
generally the case because ituses the mostrecently calculated values ateach stage.
Unfortunately, as with all iterative methods, convergence is not guaranteed. However,
it can be shown that if the matrix of coefficients is diagonally dominant, that is each
diagonal element is larger in modulus than the sum of the moduli of the other elements
inits row, thenthe Gauss--Seidel methodwill converge.
Engineeringapplication8.5
Findingapproximatesolutionsforthenodevoltagesofan
electricalnetwork
When trying to analyse complicated electrical networks it is frequently necessary
to resort to computer-based methods in order to find the voltages and currents. For ➔