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PPS-UB-PAT-TM-2010 Chapter 6 1

Simple Fixed-point Iteration

...2,1,k,given)()(0)(

1

okk xxgxxxgxf

Bracketing methods are convergent.

Fixed-point methods may sometimediverge, depending on the stating point(initial guess) and how the function

behaves.

Rearrange the function so that x is on theleft side of the equation:

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PPS-UB-PAT-TM-2010 Chapter 6 2

xxg

or

xxgor

xxg

xxxxf

21)(

2)(

2)(

02)(

2

2

Example:

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PPS-UB-PAT-TM-2010 Chapter 6 3

Convergence

x=g(x) can be expressed

as a pair of equations:

y1=x

y2=g(x) (component

equations)

Plot them separately.

Figure 6.2

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PPS-UB-PAT-TM-2010 Chapter 6 4

Conclusion Fixed-point iteration converges if

x)f(x)linetheof(slope1)( xg

When the method converges, the error isroughly proportional to or less than the error

of the previous step, therefore it is calledlinearly convergent.

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PPS-UB-PAT-TM-2010 Chapter 6 5

Newton-Raphson Method Most widely used method.

Based on Taylor series expansion:

)(

)(

)(0

g,Rearrangin

0)f(xwhenxofvaluetheisrootThe

!2)()()()(

1

1

1i1i

3

2

1

i

iii

iiii

iiii

xf

xfxx

xx)(xf)f(x

xOx

xfxxfxfxf

Newton-Raphson formula

Solve for

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PPS-UB-PAT-TM-2010 Chapter 6 6

A convenient method for

functions whose

derivatives can be

evaluated analytically. It

may not be convenientfor functions whose

derivatives cannot be

evaluated analytically.

Fig. 6.5

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Fig. 6.6

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The Secant Method

A slight variation of Newtons method for functionswhose derivatives are difficult to evaluate. For thesecases the derivative can be approximated by a

backward finite divided difference.

,3,2,1)()(

)(

)()()(

1

1

1

1

1

1

ixfxf

xxxfxx

xfxf

xx

xf

ii

ii

iii

ii

ii

i

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Requires two initial

estimates of x , e.g, xo,

x1. However, because f(x)is not required to change

signs between estimates,

it is not classified as a

bracketing method.

The secant method has

the same properties as

Newtons method.

Convergence is not

guaranteed for all xo, f(x).

Fig. 6.7

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Fig. 6.8