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Chapter 2Roots of Equations

- Simple Fixed-PointIteration

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Learning Outcome t the end the lecture students should

!e a!le to use the Simple Fixed-Point

Iteration to estimate the root of theequation"

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Open #ethodsOpen methods in$ol$e one or more

initial guesses% !ut there is no need for

them to !rac&et the root" 'he openmethods do not al(a)s (or& *the) candi$erge+ !ut (hen the) do the) usuall)con$erge quic&l)"

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Open #ethods'he three main open methods are

a+ Simple Fixed-Point Iteration #ethod

!+ ,e(ton-Raphson #ethod

c+ Secant #ethod

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Simple Fixed-PointIteration formula can !e de$eloped for simple

xed-point iteration !) rearranging the

function f(x)=0 so that xis on one side of theequation.

'his transformation can !e accomplished

either !) alge!raic manipulation or !)simpl) adding xto !oth sides of the originalequation"

)(xgx =

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Simple Fixed-PointIteration Example. Con$ert in the

form of

3or

32or

32or

2

3

032

2

2

2

+=

+=

=

+=

=+

xxx

xx

xx

xx

xx

0322 =+ xx

)(xgx =

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Simple Fixed-PointIteration'he utilit) of is that it pro$ides a

formula to predict a ne( $alue of xas a

function of an old $alue of x" 'hus% gi$enan initial guesses the equation

can !e used to compute a ne(estimate as expressed !) the

iterati$e formula

)(xgx =

ix )(xgx =

1+ix

)(1 ii xgx =+

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Simple Fixed-PointIteration'he algorithm is as follo(s.

Initial guess /

First iteration.

Second iteration.

'hird iteration.

Fourth iteration.

0x

)( 01 xgx =

)( 12 xgx =

)( 23 xgx =

)( 34 xgx =

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Simple Fixed-Point

Iteration'his method is depicted in the diagram

!elo(.

'his graph is called oscillating or spiralpattern"

'he root estimate (e o!tained for

is also the root estimate for f(x)=0"

root0x

)(xg

xy =

)( 0xg

10 )( xxg =

)( 1xg

21)( xxg =

)( 2xg

)(xgx =

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Simple Fixed-PointIteration'he Simple xed-point iteration can also !e

depicted as follo(s.

'his graph is called monotone pattern

root ix

xy =

)(xg

)( ixg

1)( += ii xxg

)( 1+ixg21)( ++ = ii xxg

)( 2+ixg32 )( ++ = ii xxg

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Simple Fixed-PointIteration'he graphical is either a spiral or

monotone depends on the function g(x)

that (e emplo)"'he approximate error for this equation

can !e determined using the errorestimator.

%1001

1=

+

+

i

iia

xxx

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0i$ergence of Fixed-PointIteration'his method is not al(a)s con$erged to

the root% the) sometimes di$erge or

mo$e a(a) from the true root as thecomputation progresses"

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Con$ergence of Fixed-PointIteration

*a+ and *!+con$erged to

true root" *c+ and *d+

di$erged a(a)from true root"

Con$ergenceoccurs if in theregion of interest

1)('

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Example 1 se simple xed-point iteration to locate

the root of (here

xis in radians" se an initial guess ofand iterate until "

xxxf = )sin(2)(

5.00 =x

%001.0a

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Solution Rearranging the equation

as

'he formula for simple xed-point iterationis

sing an initial guess of % the rstiteration is

0)sin(2)( == xxxf

)sin(2 xx=

)sin(21 ii xx =+5.00 =x

299274.1)5.0sin(2

)sin(2 01

==

= xx

%517.61%100299274.1

5.0299274.1%100

1

01=

=

=

x

xxa

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Solution'he remaining iterations are displa)ed in the

follo(ing ta!le.

fter 3 iterations and the approximate

$alue of the root is 1"4526378"

7 7"8

1 1"244259 :1"815

2 1"315193 23"944

6 1"487859 :"397

9 1"4:4596 7"456

8 1"4527:4 7"113

: 1"452699 7"719

5 1"452655 7"772

3 1"4526378 7"77713

(%)aixi

%001.0a

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Example 2 se simple xed-point iteration to locate

the root of

emplo)ing an initial guess of "Iterate until " se the true$alue 7"8:519624 to compute "

xexf

x=

)(00 =x

%5.1a

t

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Solution Rearranging the equation

as

'he formula for simple xed-point iterationis

sing an initial guess of %

0)( == xexf x

xex =

ix

i ex

+ =1

00 =x

%100%100

56714329.0

056714329.0=

=t

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Solution First iteration.

Second iteration.

%3.76%10056714329.0

156714329.0

%100%1001

01

1010

=

=

=

=

===

t

a

xeex

%1.35%10056714329.0

367879.056714329.0

%8.171%100367879.0

1367879.0

367879.0121

=

=

=

=

===

t

a

xeex

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Solution ll the remaining iterations are displa)ed in

the follo(ing ta!le.

7 7 177"7

1 1 177"7 5:"6

2 7"6:5354 151"3 68"1

6 7":42271 9:"4 22"1

9 7"877956 63"6 11"3

8 7":7:299 15"9 :"34: 7"89864: 11"2 6"36

5 7"854:12 8"4 2"27

3 7"8:7118 6"93 1"29

4 7"851196 1"46 7"51

17 7"8:9354 1"11 7"97

(%)(%) taixi

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Solution fter 17 iterations% and the

computation is terminated" 'he

approximate $alue of the root is7"8:9354" 'hus% each iteration !ringsthe estimate closer to the true $alue ofthe root 7"8:519624"

%5.1a