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- earrange t e unct on so t at x s on t e

left side of the equation:

)(0)( xxgxf

...2,1,k,given)( 1 okk xxgx

rac e ng me o s are convergen .

diverge, depending on the stating point

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Example:

xxxxf 02)(

2

or

xxg 2)(

x 21

x

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gure .

x=g(x) can be expressedas a pair of equations:

=x

y2=g(x) (component

Plot them separately.

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Fixed-point iteration converges if

xxneeos ope xg

When the method converges, the error is

the previous step, therefore it is called linearlyconvergent.

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- Most widely used method.

Based on Taylor series expansion:

)()()()( 32

1 iiii xOx

xfxxfxfxf

0)f(xwhenxofvaluetheisrootThe 1i1i

0

g,Rearrangin

xx)(xf)f(x Solve for

)(1

iii

xfxx

Newton-Raphson formula

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i

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A convenient method for

derivatives can be .

may not be convenient

derivatives cannot be

.

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Fig. 6.6

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A sli ht variation of Newtons method for

functions whose derivatives are difficult to

evaluate. For these cases the derivative can beapproximated by a backward finite divideddifference.

1 xx

)()()(1

xfxfxfiii

,3,2,1)( 11

i

xx

xxxfxx iiiii

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Re uires two initial

Fig. 6.7

estimates of x , e.g, xo,

x1. However, because

f(x) is not required to

change signs between,

classified as a

bracketin method.

The secant method has

Newtons method.

Convergence is notguaranteed for all xo,

f(x).

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Fig. 6.8

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Chapter 6 11