iterative methods to solve equation system

8/9/2019 Iterative Methods to solve equation system

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System of linear equations

Iterative Methods


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TOPICS

Jacobi Method

Gauss-Seidel Method Gauss-Seidel Method amended

Nonlinear equation systems


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Description All the presentation is going to manage

the follow notation to write the matrixes

-

!

-

y

-

m

i

n

i

mnmm

inii

n

n

b

b

b

b

x

x

x

x

aaa

aaa

aaa

aaa

2

1

2

1

21

21

22221

11211

.

/

.

//

.

.

A x b

Ax=bAx=b

Given a square

system ofn

linear equations

with unknown x:


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Jacobi method Jacobi method is an algorithm for determining the solutions of a

system of linear equations with largest absolute values in each

row and column dominated by the diagonal element.

Each diagonal element is solved for, and an approximate value

plugged in. The process is then iterated until it converges

This algorithm is a stripped-down version of the Jacobi

transformation method of matrix diagonalization.


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Given a square system of n linear equations

-

!

mnmm

inii

n

n

aaa

aaa

aaa

aaa

A

.

/

.

//

.

.

21

21

22221

11211

Ax=bAx=b

-

!

n

i

x

x

x

x

x

2

1

-

!

m

i

b

b

b

b

b

2

1

, ,


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ThenA can be decomposed into a diagonal component D,

and the remainder R:

-

mna

aa

D

.

/

.

//

.

.

00

000

0000

22

11

-

0

0

0

0

21

21

221

112

.

/

//

.

.

mm

inii

n

n

aa

aaa

aa

aa

R,

A = D + RA = D + R


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The system of linear equations may be rewritten as:

(D + R) x = b(D + R) x = b DxDx + Rx = b+ Rx = b

And finally

DxDx = b= b -- RxRx


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The Jacobi method is an iterative technique that solves

the left hand side of this expression for x, using previous

value for x on the right hand side. Analytically, this may

be written as

)( )(1)1( kk

i RxbDx !

The element-based formula is thus

)(1 )(

,

,

)1( k

j

ij

jii

ii

k

i xaba

x {

!

ii = 1,2,= 1,2,


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the computation of xi(k+1) requires each element in x(k)

except itself. Unlike the GaussSeidel method, we can't

overwrite xi(k) with xi

(k+1), as that value will be needed

by the rest of the computation.

This is the most meaningful difference between theJacobi and GaussSeidel methods, and is the reason

why the former can be implemented as a parallel

algorithm, unlike the latter. The minimum amount of

storage is two vectors of size n.


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Gauss Seidel Method Also known as the Liebmann method or the method of

successive displacement, is an iterative method used to solve a

linear system of equations.

Though it can be applied to any matrix with non-zero elements on

the diagonals, convergence is only guaranteed if the matrix is

either diagonally dominant, or symmetric and positive definite.


11/19

Given a square system of n linear equations

-

!

mnmm

inii

n

n

aaa

aaa

aaa

aaa

A

.

/

.

//

.

.

21

21

22221

11211

Ax=bAx=b

-

!

n

i

x

x

x

x

x

2

1

-

!

m

i

b

b

b

b

b

2

1

, ,


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A can be decomposed into a lower triangular component L

and a strictly upper triangular component U

-

!

000

000

000

2

112

.

/

//

.

.

in

n

n

a

aaa

U

-

!

mnmm

ii

aaa

aa

aa

a

L

.

/

//

.

.

21

21

2221

11

0...

0

00

,

A = L + UA = L + U Lx = bLx = b -- UxUx


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The GaussSeidel method is an iterative technique that

solves the left hand side of this expression for x, using

previous value for x on the right hand side. Analytically,

this may be written as

)( )(1)1( kk

i UxbLx !

However, by taking advantage of the triangular form ofL*, the

elements ofx(k+1) can be computed sequentially using forward

substitution

)(1 )1(

,

)(

,

,

)1(

"

!k

j

ij

ji

k

j

ij

jii

ii

k

i xaxaba

x

Similar to

Jacobi

Method

ii = 1,2,= 1,2,


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The computation of xi(k+1) requires each element in x(k)

except itself. Unlike the GaussSeidel method, we can't

overwrite xi(k) with xi

(k+1), as that value will be needed

by the rest of the computation.


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Convergence criterion to Gauss- Seidel method

To ensure the convergence for the method, the

diagonal coefficient of each equation must behigher than the sum of the absolute value from the

others coefficients of the equation

{!

"

n

ijj

jiij aa1

, ||||

The last criterion it is enough but not necessary to theconvergence.

The convergence is ensure when the restriction is

satisfied. Systems that meet the restriction are known

as diagonally dominant


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Gauss-Seidel Method amended

(relaxation)

Relaxation is an improvement made tothe Gauss-Seidel Method to achieve

faster the convergence.


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Improvement of convergence with relaxation

The relaxation represents a soft modification to Gauss-

Seidel method to improve the convergence. After all theprocess and the calculation of each x, that value is modify

through an average of the results of each iteration made

before an the actual one.

last

i

new

i

new

i xxx )1( PP !

Where is a weighted factor that has a value between 0 and

2. If has a value between 0 and 1, the result is a weightedaverage. These type of modifications are known as sub-

relaxation.

To values of between 1 and 2, is given an extra value to the

actual one. This modification are called over-relaxation.


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NON-LINEAR EQUATIONS

Generally, nonlinear algebraic problems are often exactly solvable, and if notthey usually can be thoroughly understood through qualitative and numerical

analysis. As an example, the equation

012 ! xx

Could be written as

cxf !)( Where xxxf ! 2)( And C = 1C = 1

and is nonlinear because f(x) satisfies neither additively nor

homogeneity (the nonlinearity is due to the x2). Though nonlinear,

this simple example may be solved exactly (via the quadratic

formula) and is very well understood


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Bibliography

CHAPRA, Steven C. y CANALE, Raymond P.: Mtodos

Numricos para Ingenieros. McGraw Hill 2002.

Black, Noel and Moore, Shirley, "Gauss-Seidel Method"

from MathWorld

http://www.slideshare.net/nestorbalcazar/mtodos-

numricos-06

Khalil, Hassan K. (2001). Nonlinear Systems. Prentice

Hall. ISBN 0-13-067389-7.

iterative methods to solve equation system

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