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

8.14 Computer solutions of matrix problems 319
Table8.5
Node voltages derived from the Gauss--Seidel method.
Iterationno.(n)
V (n)
a
V (n)
b
V (n)
c
V (n)
d
V (n)
e
0 0.0000 0.0000 0.0000 0.0000 0.0000
1 2.0000 −0.8710 0.9326 0.9048 0.5018
.
20 2.9665 0.5226 2.1985 1.8569 1.1212
21 2.9674 0.5233 2.1989 1.8573 1.1215
EXERCISES8.13
1 Performthree iterationsofthe methods ofJacobiand
Gauss--Seidel to obtain approximate solutionsofthe
following.Ineach case, use an initialguess of
x (0) =y (0) =z (0) =0
(a) 4x+y+z=−1
x+6y+2z=0
x+2y+4z=1
(b) 5x+y−z=4
x−4y+z=−4
2x+2y−4z=−6
(c) 4x+y+z=17
x+3y−z=9
2x−y+5z=1
Solutions
1 (a) Jacobi
x 1 = −0.2500,y 1 = 0,z 1 = 0.2500
x 2 = −0.3125,y 2 = −0.0417,z 2 = 0.3125
x 3 = −0.3177,y 3 = −0.0521,z 3 = 0.3490
Gauss--Seidel
x 1 = −0.2500,y 1 = 0.0417,z 1 = 0.2917
x 2 = −0.3333,y 2 = −0.0417,z 2 = 0.3542
x 3 = −0.3281,y 3 = −0.0634,z 3 = 0.3637
(b) Jacobi
x 1 = 0.8000,y 1 = 1,z 1 = 1.5000
x 2 = 0.9000,y 2 = 1.5750,z 2 = 2.4000
x 3 = 0.9650,y 3 = 1.8250,z 3 = 2.7375
Gauss--Seidel
x 1 = 0.8000,y 1 = 1.2000,z 1 = 2.5000
x 2 = 1.0600,y 2 = 1.8900,z 2 = 2.9750
x 3 = 1.0170,y 3 = 1.9980,z 3 = 3.0075
(c) Jacobi
x 1 = 4.2500,y 1 = 3,z 1 = 0.2000
x 2 = 3.4500,y 2 = 1.6500,z 2 = −0.9000
x 3 = 4.0625,y 3 = 1.5500,z 3 = −0.8500
Gauss--Seidel
x 1 = 4.2500,y 1 = 1.5833,z 1 = −1.1833
x 2 = 4.1500,y 2 = 1.2222,z 2 = −1.2156
x 3 = 4.2484,y 3 = 1.1787,z 3 = −1.2636
8.14 COMPUTERSOLUTIONSOFMATRIXPROBLEMS
Theuseoftechnicalcomputinglanguageshasvastlyimprovedtheabilityofengineersto
calculate the solutions to complex engineering problems. As has been demonstrated by
theexamplesgiveninpreviouschapters,highlevelmathematicalfunctionsareprovided
such as the ability to calculate the roots of an equation and the eigenvalues of a matrix.
In electrical engineering problems it is frequently necessary to solve matrix equations.
Consider again Example 8.32. The equations areof the form
AX=B