25284_mws_gen_sle_ppt_seidel2

8/8/2019 25284_mws_gen_sle_ppt_seidel2

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Gauss-Siedel Method


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http://numericalmethods.eng.usf.edu

Gauss-Seidel MethodAn iterative method.

Basic Procedure:

-Algebraically solveeach linearequation for xi

-Assumean initial guesssolution array

-Solve foreach xi and repeat

-Useabsolute relativeapproximateerroraftereachiterationtocheckiferroris within apre-specifiedtolerance.


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Gauss-Seidel MethodWhy?

The Gauss-Seidel Methodallowstheusertocontrol round-offerror.

Elimination methodssuchas Gaussian Elimination and LU

Decomposition arepronetoproneto round-offerror.

Also: Ifthephysicsoftheproblemareunderstood,acloseinitialguesscan bemade,decreasing the numberofiterations needed.


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Gauss-Seidel MethodAlgorithm

A setofn equationsandn unknowns:

11313212111... bxaxaxaxa nn !

2323222121... bxaxaxaxa n2n !

nnnnnnn bxaxaxaxa ! ...332211

. .

. .

. .

If: thediagonal elementsare

non-zero

Rewrite eachequation solvingforthecorresponding unknown

ex:

Firstequation,solve for x1

Secondequation,solve for x2


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Rewriting eachequation

11

13132121

1

a

xaxaxacx nn

!

--

nn

nnnnnn

n

nn

nnnnnnnnnn

nn

a

xaxaxacx

a

xaxaxaxacx

a

xaxaxacx

11,2211

1,1

,122,122,111,111

22

23231212

2

!

!

!

--

--

///

--

FromEquation 1

Fromequation 2

Fromequation n-1

Fromequation n


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General Formofeachequation

11

11

11

1a

xac

x

n

jj

jj

{!

!

22

21

22

2a

xac

x

j

n

jj

j

{!

!

1,1

11

,11

1

{!

!

nn

n

njj

jjnn

na

xac

x

nn

n

njj jnjn

na

xac

x{!

!

1


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General Form forany row i

.,,2,1,

1

nia

xac

xii

n

ijj

jiji

i -!

!

{!

How or wherecan thisequation beused?


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Gauss-Seidel MethodSolve fortheunknowns

Assumean initial guess for [X]

-

n

-n

2

x

x

x

x

1

1

/

Use rewritten equationstosolve for

each valueof xi.

Important:Remembertousethe

most recent valueof xi.Which

meanstoapply valuescalculatedto

thecalculations remaining in the

current iteration.


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Gauss-Seidel MethodCalculatethe AbsoluteRelative ApproximateError

100x

xxnew

i

old

i

new

i

ia v

!I

So when hastheanswerbeen found?

Theiterationsarestopped when theabsolute relative

approximateerroris lessthan aprespecifiedtolerance forall

unknowns.


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Gauss-Seidel Method: Example 2Given thesystemofequations

15x-3x12x321!

283x5xx321

!

7613x7x3x321 !

-

!

-

1

0

1

3

2

1

x

x

x

Withan initial guessof

Thecoefficientmatrix is:

? A

-

!

1373

351

5312

A

Will thesolution convergeusing the

Gauss-Siedel method?


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Gauss-Seidel Method: Example 2

? A

-

!

1373

351

5312

A

Checking ifthecoefficientmatrix isdiagonally dominant

43155232122

!!u!! aaa

10731313 323133 !!u!! aaa

8531212 131211 !!u!! aaa

Theinequalitiesareall trueandat leastone row isstrictlygreaterthan:

Therefore: Thesolution shouldconvergeusing the Gauss-Siedel Method


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-

!

-

-

7628

1

1373351

5312

3

2

1

aa

a

Rewriting eachequation

12

531 321

xxx

!

5

328 312

xxx

!

13

7376 213

xxx

!

Withan initial guessof

-

!

-

1

0

1

3

2

1

xx

x

50000.0

12

150311

!

!x

9000.

45

135.028

2!

!

x

0923.3

13

9000.4750000.03763 !

!x


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Gauss-Seidel Method: Example 2Theabsolute relativeapproximateerror

%662.6710050000.0

0000.150000.01a

!v

!

%00.1001009000.4

09000.42a

!v

!

%662.671000923.3

0000.10923.33a

!v!

Themaximumabsolute relativeerrorafterthe firstiteration is 100%


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-

!

-

8118.3

7153.3

14679.0

3

2

1

x

x

x

After Iteration #1

14679.0

12

0923.359000.4311

!

!x

7153.35

0923.3314679.0282 !

!x

8118.3

13

900.4714679.03763

!

!x

Substituting the x valuesintotheequations After Iteration #2

-

!

-

0923.3

9000.4

5000.0

3

2

1

x

x

x


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Gauss-Seidel Method: Example 2Iteration #2 absolute relativeapproximateerror

%62.24010014679.0

50000.014679.01

!v

!a

%887.311007153.3

9000.47153.32

!v

!a

%876.18100

8118.3

0923.38118.33

!v

!a

Themaximumabsolute relativeerrorafterthe firstiteration is 240.62%

Thisismuch largerthan themaximumabsolute relativeerrorobtainedin

iteration #1. Isthisaproblem?


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Gauss-Seidel Method: Example 2Repeating moreiterations,the following valuesareobtained

1aI

2aI

3aI

Iteration a1 a2 a3

1

2

3

4

5

6

0.50000

0.14679

0.74275

0.94675

0.99177

0.99919

67.662

240.62

80.23

21.547

4.5394

0.74260

4.900

3.7153

3.1644

3.0281

3.0034

3.0001

100.00

31.887

17.409

4.5012

0.82240

0.11000

3.0923

3.8118

3.9708

3.9971

4.0001

4.0001

67.662

18.876

4.0042

0.65798

0.07499

0.00000

-

!

-

4

3

1

3

2

1

x

x

x

-

!

-

0001.4

0001.3

99919.0

3

2

1

x

x

xThesolution obtained

isclosetotheexactsolution of

25284_mws_gen_sle_ppt_seidel2

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