Systems of Linear Algebraic Equations - Numerical Methods in Engineering with MATLAB - page 92

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. . .

2 417 04 2 720 76 3 07753

x 2 = x 1 + α 1 s 1 = − 0 . 201 42 0 . 24276 4 . 10380 0 . 79482

1 . 00710 − 1 . 182 68 0 . 71999

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12
−

4 − 11 3 . 07753 − 0 . 23529

r 2 = b − Ax 2 =

1
5

− 1 4 − 2 0 . 79482 0 . 33823

1 − 2 4 0 . 71999 0 . 63215

r 2 As 1
s 1 As 1

β 1 =−

( − 0 . 23529)(5 . 59656) + 0 . 33823(16 . 05980) + 0 . 63215( − 10 . 21760)

=−

2 . 72076(5 . 59656) + 4 . 10380(16 . 05980) + ( − 1 . 18268)( − 10 . 21760)

= 0 . 0251452

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− 0 . 23529 2 . 72076 − 0 . 166876

s 2 = r 2 + β 1 s 1 = 0 . 33823 0 . 0251452 4 . 10380 0 . 441421

0 . 63215 − 1 . 18268 0 . 602411

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4 − 11 − 0 . 166876 − 0 . 506514

As 2 = − 1 4 − 2 0 . 441421 0 . 727738

1 − 2 4 0 . 602411 1 . 359930

r 2 s 2
s 2 As 2

α 2 =

( − 0 . 23529)( − 0 . 166876) + 0 . 33823(0 . 441421) + 0 . 63215(0 . 602411)

=

( − 0 . 166876)( − 0 . 506514) + 0 . 441421(0 . 727738) + 0 . 602411(1 . 359930)

= 0 . 46480

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3 . 07753 − 0 . 166876 2 . 99997

x 3 = x 2 + α 2 s 2 = . 0 . 46480 . .

0 79482 0 441421 0 99999

0 . 71999 0 . 602411 0 . 99999

The solution x 3 iscorrect to almost five decimal places. The small discrepancyis
causedbyroundoff errors in the computations.

EXAMPLE 2.17
Write a computerprogram to solve the following n simultaneousequations 3 by
the Gauss-Seidel methodwith relaxation (the program shouldwork with any

3 Equationsofthisform arecalled cyclic tridiagonal. They occur in the finite difference formulation
of second-orderdifferentialequations with periodic boundary conditions.