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The Islamic University of Gaza

Faculty of EngineeringCivil Engineering Department

Numerical AnalysisECIV 3306

Chapter 11Special Matrices and Gauss-Siedel

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Introduction

• Certain matrices have particular structures that canbe exploited to develop efficient solution schemes.

• A banded matrix is a square matrix that has allelements equal to zero, with the exception of a bandcentered on the main diagonal. These matricestypically occur in solution of differential equations.

• The dimensions of a banded system can bequantified by two parameters: the band width BWand half-bandwidth HBW. These two values arerelated by BW=2HBW+1.

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Banded matrix

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4

Tridiagonal Systems•

A tridiagonal system has a bandwidth of 3:

4

3

2

1

4

3

2

1

44

333

222

11

r r

r

r

x x

x

x

f e g f e

g f e

g f

• An efficient LU decomposition method, called Thomas

algorithm , can be used to solve such an equation. Thealgorithm consists of three steps: decomposition, forwardand back substitution, and has all the advantages of LUdecomposition.

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Fig 11.2

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Cholesky Decomposition

6

This method is suitable for only symmetric systemswhere:

T ij jia a and A A

11 12 13 11 11 21 31

21 22 23 21 22 22 32

31 32 33 31 32 33 33

0 0

0 * 0

0 0

a a a l l l l

a a a l l l l

a a a l l l l

11

21 22

31 32 33

0 0

[ ] 0

l

L l l

l l l

* T A L L

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Cholesky Decomposition

1

1 1,2, , 1

i

ki ij kj j

ki

ii

a l l l for i k

l

12

1

k

kk kk kj j

l a l

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Pseudocode for Cholesky ’ s LUDecomposition algorithm (cont ’ d)

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Gauss-Siedel

• Iterative or approximate methods provide analternative to the elimination methods. The Gauss-Seidel method is the most commonly used iterative

method.• The system [A]{X}={B}is reshaped by solving the

first equation for x 1, the second equation for x 2,and the third for x

3, …and n th equation for x

n. We

will limit ourselves to a 3x3 set of equations.

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Gauss-Siedel

33

23213131

22

32312122

11

31321211

a xa xab x

a

xa xab x

a

xa xab x

3333232131

2323222121

1313212111

b xa xa xa

b xa xa xa

b xa xa xa

Now we can start the solution process by choosing

guesses for the x’

s. A simple way to obtain initialguesses is to assume that they are zero. Thesezeros can be substituted into x 1 equation tocalculate a new x 1 =b 1 /a 11 .

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Gauss-Siedel

• New x 1 is substituted to calculate x 2 and x 3. Theprocedure is repeated until the convergencecriterion is satisfied:

snew

i

old

i

new

iia

x

x x %100,

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Jacobi iteration Method

An alternative approach, called Jacobi iteration,

utilizes a somewhat different technique. This

technique includes computing a set of new x ’s on the

basis of a set of old x ’s. Thus, as the new values are

generated, they are not immediately used but are

retained for the next iteration.

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Gauss-Siedel

The Gauss-Seidel method The Jacobi iteration method

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Convergence Criterion for Gauss-Seidel Method

• The gauss-siedel method is similar to the technique offixed-point iteration.

• The Gauss-Seidel method has two fundamentalproblems as any iterative method:

1. It is sometimes non-convergent, and

2. If it converges, converges very slowly.• Sufficient conditions for convergence of two linear

equations, u(x,y) and v(x,y) are:

1

1

y

v

x

v

yu

xu

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Convergence Criterion for Gauss-SeidelMethod (cont ’ d)

• Similarly, in case of two simultaneous equations, theGauss-Seidel algorithm can be expressed as:

1 121 2 2

11 11

2 211 2 1

22 22

12

1 2 11

21

1 22 2

( , )

( , )

0

0

b au x x x

a a

b av x x x

a a

au u

x x a

av v

x a x

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Convergence Criterion for Gauss-SeidelMethod (cont ’ d)

Substitution into convergence criterion of two linearequations yield:

In other words, the absolute values of the slopes must beless than unity for convergence:

That is, the diagonal element must be greater than the off-diagonal element for each row.

1,122

21

11

12

a

a

a

a

2122

1211

aa

aan

i j j

jiii aa1

,

equationsnFor

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Gauss-Siedel Method- Example 1

10

x20 x304717

x30 x103193

x20 x10857

x

x

x

471

319

857

x

x

x

102030

307 10

20103

21

31

32

3

2

1

3

2

1

...

...

...

..

.

....

..

• Guess x1, x2, x3= zero for the first guessIter. x 1 x2 x3 | a,1 |(%) | a,2 | (%) | a,3 | (%)

0 0 0 0 - - -

1 2.6167 -2.7945 7.005610 100 100 1002 2.990557 -2.499625 7.000291 12.5 11.8 0.076

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Improvement of Convergence UsingRelaxation

• Where is a weighting factor that is assigned a

value between [0, 2]• If = 1 the method is unmodified.• If is between 0 and 1 (under relaxation) this is

employed to make a non convergent system toconverge.

• If is between 1 and 2 (over relaxation) this isemployed to accelerate the convergence.

1new new old i i i x x x

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Gauss-Siedel Method- Example 2

3862

3473

2028

321

321

321

x x x

x x x

x x x

1 2 3

1 2 3

1 2 3

8 2 20

2 6 38

3 7 34

x x x

x x x

x x x

Rearrange so thatthe equations are

diagonally dominant

8

22032

1

x x x

6

23831

2

x x x

7

33421

3

x x x

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Gauss-Siedel Method- Example 2

iteration unknown value a maximum a 0 x 1 0

x 2 0

x 3 0

1 x 1 2.5 100.00%x

2 7.166667 100.00%x

3 -2.7619 100.00% 100.00%

2 x 1 4.08631 38.82%x

2 8.155754 12.13%x

3 -1.94076 42.31% 42.31%

3 x 1 4.004659 2.04%x

2 7.99168 2.05%

x 3 -1.99919 2.92% 2.92%

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Gauss-Siedel Method- Example 2

The same computation can be developed with relaxation where = 1.2

First iteration:

Relaxation yields:

Relaxation yields:

Relaxation yields :

5.28

)0(20208

220 321

x x

x

3)0(2.0)5.2(2.11 x

1 32

38 2 38 2(3) 07.333333

6 6

x x x

8.8)0(2.0)333333.7(2.12

x

3142857.27

8.8)3(3347334 21

3

x x x

7771429.2)0(2.0)3142857.2(2.13 x

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Gauss-Siedel Method- Example 2

Iter. unknown value relaxation a maximum a 1 x 1 2.5 3 100.00%

x 2 7.3333333 8.8 100.00%

x 3 -2.314286 -2.777143 100.00% 100.000%

2 x 1 4.2942857 4.5531429 34.11%x

2 8.3139048 8.2166857 7.10%x

3 -1.731984 -1.522952 82.35% 82.353%

3 x 1 3.9078237 3.7787598 20.49%x

2 7.8467453 7.7727572 5.71%x

3 -2.12728 -2.248146 32.26% 32.257%

4 x 1 4.0336312 4.0846055 7.49%x

2 8.0695595 8.12892 4.38%x

3 -1.945323 -1.884759 19.28% 19.280%