Download - 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.
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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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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.