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# Lecture03 - Iterative Methods

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• 8/9/2019 Lecture03 - Iterative Methods

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Solution of Linear System of Equations

Lecture 3:

Iterative Methods

MTH2212 Computational Methods and Statistics

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Dr. M. HrairiDr. M. Hrairi MTH2212MTH2212 -- Computational Methods and StatisticsComputational Methods and Statistics 22

Objectives

Introduction

Jacobi Method

Gauss-Seidel Method

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Introduction

To solve the linear system Ax = b we may use either:

Direct Methods

- Gaussianelimination

- PLU decomposition Iterative Methods

- Jacobi Method

- Gauss-Seidel Method

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Iterative Methods

Suppose we solve Ax = b for a given matrix A by finding the

PLU decomposition

Ifwe change the vectorb,we may continue to use the PLU

Ifwe change A,we now have to re-compute the PLUdecomposition: expensive

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Iterative Methods

Instead, suppose we have solved the system

Ax = b

for a given matrix A

Suppose we change A slightly, e.g., modify a single resistor

in a circuit

Ifwe call that new matrix Amod

, is it possible to use the

solution to Ax = b to solve Amod

x = b?

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Iterative Methods

They provide an alternative to the elimination method.

Let Ax = b be the set ofequations to be solved.

The system Ax = b is reshaped by solving the first equation

for x1, the second equation for x2, and the third for x3, and

nth equation for xn.

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Iterative Methods

For ease ofcomputation, lets assume we have a 3x3

system ofequations to solve.

If the diagonal elements are all non-zero then:

!!

!

3333232131

2323222121

1313212111

bxaxaxa

bxaxaxa

bxaxaxa

33

23213133

22

32312122

11

31321211

a

xaxabx

a

xaxabx

a

xaxabx

!

!

!

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Jacobi Iteration Method

1. Assume all the xs are zero

2. Substitute the zeros into the three equations to get:

3. Repeat the procedure until the error criterion is satisfied:

11

1

1 a

bx !

22

2

2 a

bx !

33

3

3 a

bx !

si

j

i

j

i

jja

xxx Iev

!I

%00,

33

23233

3

22

32322

2

3322

a

xaxabx

a

xaxabx

a

xaxabx

ii

i

iii

ii

i

!

!

!

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Gauss-Seidel Method

It is the most commonly used iterative method.

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Gauss-Seidel Procedure

1. Assume all the xs are zero

2. Substitute the zeros into the first equation i.e. equation (1)to give:

3. Substitute the new value ofx1 and x3 = 0 into equation (2)to compute x2

4. Substitute the value ofx1 and the new value ofx2 inequation (3) to estimate x3

11

11

a

bx !

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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1111

Gauss-Seidel Procedure

5. Return to equation (1) and repeat the entire procedure untilthe error criterion is satisfied:

33

1

232

1

13131

3

22

323

1

12121

2

11

31321211

1

a

aab

a

aab

a

aab

ii

i

ii

i

ii

i

si

j

i

j

i

jja

xxx Iev

I

%00,

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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1212

Example 1

Use Gauss-Seidel method to solve the following set of

linear equations:

3x1 0.1x2 0.2x3 = 7.85 (1)

0.1x1 + 7x2 0.3x3 = -19.3 (2)

0.3x1 0.2x2 + 10x3 = 71.4 (3)

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Example 1 - Solution

First we have:

10

2.03.04.71

7

3.01.03.19

3

2.01.085.7

213

31

2

321

xx

x

xx

x

xx

x

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Example 1 - Solution

1st iteration

Assumethat x2 = 0 and x3 = 0, weobtain

Substitute x1 = 2.616667 and x3 = 0 intoequation (2)

Substitute x1

= 2.616667 and x2

= -2.794524 intoequation (3)

This completes the first iteration

6 6667.23

85.7!!x

794524.27

0)616667.2(1.03.192

x

0056 0.70

)794524.2(2.0)6 6667.2(3.04.73 !

!x

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Example 1 - Solution

2nd iteration

990557.)005610.7(.0)79454.(1.085.7

1 !

!x

499625.27

)005610.7(3.0)990557.2(1.03.192 !

!x

000291.710

)499625.2(2.0)990557.2(3.04.713 !

!x

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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1616

Example 1 - Solution

Error estimate

Forx1

Forx2

Forx3

%12%12

1221 !

! aI

%8.11%100499625.2

)794524.2(499625.22, !

! aI

%% !

! aI

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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1717

Convergence

Gauss-Seidel is similar in spirit to the simple fixed-pointiteration.

Gauss-Seidel will converge iffor every equation of the

system,w

e have:

Such system is said to be diagonally dominant.

This criterion is sufficient but not necessary forconvergence.

{!

"n

ijj

ijii aa1

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Relaxation

Designed to Enhance convergence.

After each new value ofx is computed, that value ismodified using:

Where is a weighting factor.

The choice of is problem-specific and is often determinedempirically.

oldi

new

i

new

ixxx PP! 1

20 ePe

P

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Gauss-Seidel/Jacobi Iteration Methods

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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 2020

Gauss-Seidel iteration converges more rapidly than the

Jacobi iteration does; since, it uses the latest updates.

But there are some cases that Jacobi iteration does

converge but Gauss-Seidel does not.

Gauss-Seidel/Jacobi Iteration Methods

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Assignment #1

Computational Methods

12.11, 12.30, 12.33

Statistics

2.2, 2.14, 2.22, 2.26, 2.28, 2.37, 2.45, 2.52, 2.65, 2.74

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