NM2012S-Lecture12-Iterative Methods Gauss Siedel With Relaxation

8/10/2019 NM2012S-Lecture12-Iterative Methods Gauss Siedel With Relaxation

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Iterative Methods

Berlin ChenDepartment of Computer Science & Information Engineering

National Taiwan Normal University

Reference:1. Applied Numerical Methods with MATLAB for Engineers , Chapter 12 & Teaching material


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Chapter Objectives

Understanding the difference between the Gauss-Seideland Jacobi methods

Knowing how to assess diagonal dominance andknowing what it means

Recognizing how relaxation can be used to improve

convergence of iterative methods Understanding how to solve systems of nonlinear

equations with successive substitution and Newton-Raphson

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[ A] { x} = { b}


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Gauss-Seidel Method The Gauss-Seidel method is the most commonly used

iterative method for solving linear algebraic equations[ A]{ x }={b}

The method solves each equation in a system for aparticular variable, and then uses that value in laterequations to solve later variables

For a 3x3 system with nonzero elements along thediagonal , for example, the j th iteration values are foundfrom the j -1 th iteration using:

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x1 j b1 a12 x2 j 1 a13 x3 j 1

a11

x2 j b2 a 21 x1 j a23 x3 j 1

a 22

x3 j b3 a 31 x1 j

a 32 x2 j

a 33

3333232131

2323222121

1313212111

b xa xa xa

b xa xa xab xa xa xa


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Gauss-Seidel Method: Convergence

The convergence of an iterative method can becalculated by determining the relative percent change of

each element in { x }. For example, for the i th element inthe j th iteration,

The method is ended when all elements have convergedto a set tolerance

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a ,i xi

j xi j 1 xi

j 100%


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Gauss-Seidel Method: An Example (1/2)

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Example 12.1


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Gauss-Seidel Method: An Example (2/2)

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Example 12.1


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Tactic of Gauss-Seidel Method

As each new x value is computed for the Gauss-Seidelmethod, it is immediately used in the next equation to

determine another x value Thus, if the solution is converging, the best available

estimates will be employed for the Gauss-Seidel method

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Jacobi Iteration

The Jacobi iteration is similar to the Gauss-Seidelmethod, except the j -1th information is used to update all

variables in the j th iteration: Gauss-Seidel Jacobi

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Diagonal Dominance

The Gauss-Seidel method may diverge, but if thesystem is diagonally dominant, it will definitely

converge

Diagonal dominance means:

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a ii aij j 1 j i

n


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MATLAB Program

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Relaxation

To enhance convergence, an iterative programcan introduce relaxation where the value at aparticular iteration is made up of a combinationof the old value and the newly calculated value:

where is a weighting factor that is assigned a value between 0and 2

0<


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Gauss-Seidel with Relaxation: An Example (1/3)

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Example 12.2


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Example 12.2


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Example 12.2


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Nonlinear Systems

Nonlinear systems can also be solved using the samestrategy as the Gauss-Seidel method

Solve each system for one of the unknowns and update eachunknown using information from the previous iteration

This is called successive substitution

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573

102212

2121

x x x

x x x


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Successive Substitution: An Example (1/2)

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Example 12.3

573

102212

2121

x x x

x x x


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Successive Substitution: An Example (2/2)

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Example 12.3

1. For successive substitution, convergence

often depends on the manner in whichthe equations are formulated

2. Divergence also can occur if the initial guesses are insufficiently close to the truesolution


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Newton-Raphson (1/3)

Nonlinear systems may also be solved using theNewton-Raphson method for multiple variables

For a two-variable system , the Taylor seriesapproximation and resulting Newton-Raphson equationsare:

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2

,2,21,2

1

,2,11,1,21,2

2

,1,21,2

1

,1,11,1,11,1

x f x x

x f x x f f

x

f x x

x

f x x f f

iii

iiiii

iii

iiiii

0

0

1,2

1,1

i

i

f

f

ii

ii

iii

ii

ii

ii

iii

ii

x x

f x x

f f x x

f x x

f

x x

f x

x

f f x

x

f x

x

f

,22

,2,1

1

,1,11,2

2

,11,1

1

,1

,22

,1,1

1

,1,11,2

2

,11,1

1

,1

ii

ii

iiiii

x f

x f x x

f

x f x x x f x f

1

11

)of expansionseriesTaylororder-(first

0

:equationsingleaof rootthe

forRaphon- Newton6,ChapterinRecall


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After algebraic manipulation

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1

,2

2

,1

2

,2

1

,1

1

,2,1

1

,1,2

,21,2

1

,2

2

,1

2

,2

1

,1

2

,1

,22

,2

,1

,11,1

x

f

x

f

x

f

x

f x

f f

x

f f

x x

x

f

x

f

x

f

x

f x

f

f x

f

f x x

iiii

ii

ii

ii

iiii

i

i

i

i

ii

Referred to as the determinant

of the Jacobian of the system

2

,2

1

,2

2

,1

1

,1

x

f

x

f x

f

x

f

J ii

ii

Referred to as the Jacobian of the system


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Matrix notion of Newton-Raphson Also can be generalized to n simultaneous equations

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f J x x

x J f x J

ii

ii

11

1


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Newton-Raphson: An Example

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0573,

010,

573

10

21

2212212

2121211

21

2212

2121

x x x x x x f

x x x x x f

x x x x

x x x

Example 12.4


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MATLAB Program

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NM2012S-Lecture12-Iterative Methods Gauss Siedel With Relaxation

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