10/17/2015 1 Gauss-Siedel Method Industrial Engineering Majors Authors: Autar Kaw

04/20/23http://

numericalmethods.eng.usf.edu 1

Gauss-Siedel Method

Industrial Engineering Majors

Authors: Autar Kaw

http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM

Undergraduates

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Gauss-Seidel Method

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Gauss-Seidel MethodAn iterative method.

Basic Procedure:

-Algebraically solve each linear equation for xi

-Assume an initial guess solution array

-Solve for each xi and repeat

-Use absolute relative approximate error after each iteration to check if error is within a pre-specified tolerance.


Gauss-Seidel MethodWhy?

The Gauss-Seidel Method allows the user to control round-off error.

Elimination methods such as Gaussian Elimination and LU Decomposition are prone to prone to round-off error.

Also: If the physics of the problem are understood, a close initial guess can be made, decreasing the number of iterations needed.


Gauss-Seidel MethodAlgorithm

A set of n equations and n unknowns:

11313212111 ... bxaxaxaxa nn

2323222121 ... bxaxaxaxa n2n

nnnnnnn bxaxaxaxa ...332211

. . . . . .

If: the diagonal elements are non-zero

Rewrite each equation solving for the corresponding unknown

ex:

First equation, solve for x1

Second equation, solve for x2



Rewriting each equation

11

131321211 a

xaxaxacx nn

nn

nnnnnnn

nn

nnnnnnnnnn

nn

a

xaxaxacx

a

xaxaxaxacx

a

xaxaxacx

11,2211

1,1

,122,122,111,111

22

232312122

From Equation 1

From equation 2

From equation n-1

From equation n



General Form of each equation

11

11

11

1 a

xac

x

n

jj

jj

22

21

22

2 a

xac

x

j

n

jj

j

1,1

11

,11

1

nn

n

njj

jjnn

n a

xac

x

nn

n

njj

jnjn

n a

xac

x

1



General Form for any row ‘i’

.,,2,1,1

nia

xac

xii

n

ijj

jiji

i

How or where can this equation be used?


Gauss-Seidel MethodSolve for the unknowns

Assume an initial guess for [X]

n

-n

2

x

x

x

x

1

1

Use rewritten equations to solve for each value of xi.

Important: Remember to use the most recent value of xi. Which means to apply values calculated to the calculations remaining in the current iteration.


Gauss-Seidel MethodCalculate the Absolute Relative Approximate Error

100

newi

oldi

newi

ia x

xx

So when has the answer been found?

The iterations are stopped when the absolute relative approximate error is less than a prespecified tolerance for all unknowns.

Example: Production Optimization

To find the number of toys a company should manufacture per day to optimally use their injection-molding machine and the assembly line, one needs to solve the following set of equations. The unknowns are the number of toys for boys, x1, number of toys for girls, x2, and the number of unisexual toys, x3.

0

1260

756

00.000.105.1

3333.06667.01667.0

6667.01667.03333.0

3

2

1

x

x

x

Find the values of x1, x2,and x3 using the Gauss-Seidel Method


The system of equations is:

Initial Guess: Assume an initial guess of

100

1000

1000

3

2

1

x

x

x

0

1260

756

00.000.105.1

3333.06667.01667.0

6667.01667.03333.0

3

2

1

x

x

x



0

1260

756

00.000.105.1

3333.06667.01667.0

6667.01667.03333.0

3

2

1

x

x

x

3333.0

6667.01667.0756 321

xxx

6667.0

3333.01667.01260 312

xxx

0

)00.1(05.10 213

xxx


0

)00.1(05.10 213

xxx

The Equation for x3 is divided by 0 which is undefined. Therefore the order of the equations will need to be

reordered. Equation 3 and equation 1 will be switched. By switching equations 3 and 1, the matrix will also become

diagonally dominant.

The system of equations becomes:

756

1260

0

666701667033330

333306667016670

000001051

3

2

1

x

x

x

...

...

...



756

1260

0

666701667033330

333306667016670

000001051

3

2

1

x

x

x

...

...

...

051

00010 321 .

xx.x

6667.0

1667.03333.0756 213

xxx

6667.0

3333.01667.01260 312

xxx


Applying the initial guess and solving for xi

Initial Guess

When solving for x2, how many of the initial guess values were used?

100

1000

1000

3

2

1

x

x

x

38.95205.1

10001000)00.1(0x1

8.16016667.0

1003333.038.9521667.01260x2

32.2576667.0

8.16011667.038.9523333.0756x3


Finding the absolute relative approximate error

100

newi

oldi

newi

ia x

xx

At the end of the first iteration

The maximum absolute relative approximate error

is 61.138%

%..

.a 00005100

38952

1000389521

%..

.a 57037100

81601

1000816012

%138.6110032.257

10032.2573

a

32257

81601

38952

3

2

1

.

.

.

x

x

x


Iteration 2

Using from Iteration 1,

the values of xi are found:

32257

81601

38952

3

2

1

.

.

.

x

x

x

51525

051

3225708160100101 .

.

...x

8137966670

3225733330515251667012602 .

.

....x

2952666670

813791667051525333307563 .

.

....x


Finding the absolute relative approximate error

At the end of the second iteration

The maximum absolute relative approximate error

is 878.59%

%570.371005.1525

38.9525.15251

a

%085.161008.1379

8.16018.13792

a

%59.878100295.26

32.257295.263

a

29526

81379

51525

3

2

1

.

.

.

x

x

x

Iteration x1 x2 x3

123456

952.381525.51314.11474.51406.01451.9

537.57016.08510.8744.86863.1618

1601.81379.81548.21476.31524.51501.9

37.57016.08510.8744.86863.16181.5021

257.3226.29589.87627.69449.86332.554

61.138878.5970.743224.5344.45953.170


Repeating more iterations, the following values are obtained%1a %

2a %3a

! Notice – After six iterations, the absolute relative approximate errors are decreasing, but are still

high.

Iteration x1 x2 x3

2021

1439.81439.8

0.000642760.00034987

1511.81511.8

0.000349870.00019257

36.11536.114

0.00914950.0049578


Repeating more iterations, the following values are obtained%1a %

2a %3a

The value of closely approaches the true value of

11336

81511

81439

3

2

1

.

.

.

x

x

x

11436

8.1511

81439

3

2

1

.

.

x

x

x


Gauss-Seidel Method: Pitfall

Even though done correctly, the answer may not converge to the correct answer

This is a pitfall of the Gauss-Siedel method: not all systems of equations will converge.

Is there a fix?

One class of system of equations always converges: One with a diagonally dominant coefficient matrix.

Diagonally dominant: [A] in [A] [X] = [C] is diagonally dominant if:

n

jj

ijaa

i1

ii

n

ijj

ijii aa1

for all ‘i’ and for at least one ‘i’


Gauss-Seidel Method: Pitfall

116123

14345

3481.52

A

Diagonally dominant: The coefficient on the diagonal must be at least equal to the sum of the other coefficients in that row and at least one row with a diagonal coefficient greater than the sum of the other coefficients in that row.

1293496

55323

5634124

]B[

Which coefficient matrix is diagonally dominant?

Most physical systems do result in simultaneous linear equations that have diagonally dominant coefficient matrices.


Gauss-Seidel Method: Example 2

Given the system of equations

15312 321 x- x x

2835 321 x x x 761373 321 x x x

1

0

1

3

2

1

x

x

x

With an initial guess of

The coefficient matrix is:

1373

351

5312

A

Will the solution converge using the Gauss-Siedel method?



1373

351

5312

A

Checking if the coefficient matrix is diagonally dominant

43155 232122 aaa

10731313 323133 aaa

8531212 131211 aaa

The inequalities are all true and at least one row is strictly greater than:

Therefore: The solution should converge using the Gauss-Siedel Method



76

28

1

1373

351

5312

3

2

1

a

a

a


12

531 321

xxx

5

328 312

xxx

13

7376 213

xxx


1

0

1

3

2

1

x

x

x

50000.0

12

150311

x

9000.4

5

135.0282

x

0923.3

13

9000.4750000.03763

x



The absolute relative approximate error

%00.10010050000.0

0000.150000.01

a

%00.1001009000.4

09000.42a

%662.671000923.3

0000.10923.33a

The maximum absolute relative error after the first iteration is 100%



8118.3

7153.3

14679.0

3

2

1

x

x

x

After Iteration #1

14679.0

12

0923.359000.4311

x

7153.3

5

0923.3314679.0282

x

8118.3

13

900.4714679.03763

x

Substituting the x values into the equations

After Iteration #2

0923.3

9000.4

5000.0

3

2

1

x

x

x



Iteration #2 absolute relative approximate error

%61.24010014679.0

50000.014679.01a

%889.311007153.3

9000.47153.32a

%874.181008118.3

0923.38118.33a

The maximum absolute relative error after the first iteration is 240.61%

This is much larger than the maximum absolute relative error obtained in iteration #1. Is this a problem?

Iteration a1 a2 a3

123456

0.500000.146790.742750.946750.991770.99919

100.00240.6180.23621.5464.53910.74307

4.90003.71533.16443.02813.00343.0001

100.0031.88917.4084.49960.824990.10856

3.09233.81183.97083.99714.00014.0001

67.66218.8764.00420.657720.0743830.00101



Repeating more iterations, the following values are obtained

%1a %

2a %3a

4

3

1

3

2

1

x

x

x

0001.4

0001.3

99919.0

3

2

1

x

x

xThe solution obtained is close to the exact solution of .



Given the system of equations

761373 321 xxx

2835 321 xxx

15312 321 xxx


1

0

1

3

2

1

x

x

x

Rewriting the equations

3

13776 321

xxx

5

328 312

xxx

5

3121 213

xxx

Iteration a1 A2 a3

123456

21.000−196.15−1995.0−201492.0364×105

−2.0579×105

95.238110.71109.83109.90109.89109.89

0.8000014.421−116.021204.6−121401.2272×105

100.0094.453112.43109.63109.92109.89

50.680−462.304718.1−476364.8144×105

−4.8653×106

98.027110.96109.80109.90109.89109.89



Conducting six iterations, the following values are obtained

%1a %

2a %3a

The values are not converging.

Does this mean that the Gauss-Seidel method cannot be used?


Gauss-Seidel MethodThe Gauss-Seidel Method can still be used

The coefficient matrix is not diagonally dominant

5312

351

1373

A

But this is the same set of equations used in example #2, which did converge.

1373

351

5312

A

If a system of linear equations is not diagonally dominant, check to see if rearranging the equations can form a diagonally dominant matrix.


Gauss-Seidel MethodNot every system of equations can be rearranged to have a diagonally dominant coefficient matrix.

Observe the set of equations

3321 xxx

9432 321 xxx

97 321 xxx

Which equation(s) prevents this set of equation from having a diagonally dominant coefficient matrix?


Gauss-Seidel MethodSummary

-Advantages of the Gauss-Seidel Method

-Algorithm for the Gauss-Seidel Method

-Pitfalls of the Gauss-Seidel Method


Gauss-Seidel Method

Questions?

Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

http://numericalmethods.eng.usf.edu/topics/gauss_seidel.html

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10/17/2015 1 Gauss-Siedel Method Industrial Engineering Majors Authors: Autar Kaw