7eCh06 Open Methods

5/19/2018 7eCh06 Open Methods

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Copyright The McGraw-Hill Companies, Inc. Permission required or reproduction or display.

!

Chapter 6


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"

Open MethodsChapter 6

Open methods

are based on

formulas that

require only asingle starting

value of x or two

starting values

that do notnecessarily

bracket the root.

Figure 6.1


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#

imple !ixed"point #teration

...$%&%k%given'('()'(

& ==

==

okk xxgxxxgxf

*racketing methods are +convergent,.

!ixed"point methods may sometime

+diverge,% depending on the stating point

(initial guess' and how the function behaves.

-earrange the function so that x is on the

left side of the equation


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$

xxg

or

xxgor

xxg

xxxxf

$&'(

$'(

$'(

)$'(

$

$

+=

+=

=

=

/xample


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%

Convergence

x0g(x' can be expressed

as a pair of equations

y&0x

y$0g(x' (component

equations'

1lot them separately.

Figure 6.2


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&

Conclusion

!ixed"point iteration converges if

x'f(x'linetheof(slope&'( = xg

2hen the method converges% the error is

roughly proportional to or less than the error of

the previous step% therefore it is called +linearlyconvergent.,


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'

3ewton"-aphson Method

Most widely used method.

*ased on 4aylor series expansion

'(

'(

'()

g%-earrangin

)'f(xwhenxofvaluetheisroot4he

5$'('('('(

&

&

&i&i

$

&

i

iii

iiii

iiii

xf

xfxx

xx)(xf)f(x

xOx

xfxxfxfxf

=

+=

=

+

++=

+

+

++

+

3ewton"-aphson formula

olve for


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(

7 convenient method for

functions whose

derivatives can be

evaluated analytically. #t

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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!*

4he ecant Method

7 slight variation of 3ewton8s method forfunctions whose derivatives are difficult toevaluate. !or these cases the derivative can beapproximated by a backward finite divided

difference.

%%$%&'('(

'(

'('(

'(

&

&&

&

&

=

=

+

ixfxf

xxxfxx

xfxf

xxxf

ii

iiiii

ii

iii


11/19Copyright The McGraw-Hill Companies, Inc. Permission required or reproduction or display.

!!

-equires two initial

estimates of x % e.g% xo%

x&. 9owever% becausef(x' is not required to

change signs between

estimates% it is not

classified as a

+bracketing, method.

4he scant method has the

same properties as

3ewton8s method.

Convergence is not

guaranteed for all xo%

f(x'.

Fig. 6.7



!"

Fig. 6.8



!#

Multiple -oots

3one of the methods deal with multiple roots

efficiently% however% one way to deal with problems

is as follows

'(

'(&xfind4hen

'('('(et

i

i

i

i

ii

xu

xu

xfxfxu

+

= 4his function has

roots at all the same

locations as the

original function



!$

Fig. 6.13



!%

+Multiple root, corresponds to a point where a

function is tangent to the x axis. :ifficulties

;!unction does not change sign at the multiple root%

therefore% cannot use bracketing methods.;*oth f(x' and f



!&

ystems of >inear /quations

)'%%%%(

)'%%%%(

)'%%%%(

$&

$&$

$&&

=

=

=

nn

n

n

xxxxf

xxxxf

xxxxf



!'

4aylor series expansion of a function of more than

one variable

'('(

'('(

&&&&&

&&&&&

iii

ii

iii

iii

ii

iii

yyyvxx

xvvv

yyy

uxx

x

uuu

+

+=

+

+=

+++

+++

4he root of the equation occurs at the value of xand y where ui?&and vi?&equal to =ero.



!(

y

vy

x

vxvy

y

vx

x

vy

uy

x

uxuy

y

ux

x

u

ii

iiii

ii

i

ii

iiii

ii

i

+

+=

+

+

+=

+

++

++

&&

&&

7 set of two linear equations with two

unknowns that can be solved for.


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!)

x

v

y

u

y

v

x

ux

uvx

vu

yy

x

v

y

u

y

v

x

uy

uv

y

vu

xx

iiii

iiii

ii

iiii

ii

ii

ii

=

=

+

+

&

&

:eterminant of

the Jacobianof

the system.

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7eCh06 Open Methods