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

1

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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Simple Fixed-point Iteration

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

1

okk xxgxxxgxf

Bracketing methods are convergent.

Fixed-point methods may sometime

diverge, depending on the stating point

(initial guess) and how the function behaves.

Rearrange the function so that x is on the

left side of the equation:


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xxg

or

xxgor

xxg

xxxxf

21)(

2)(

2)(

02)(

2

2

Example:


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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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6

Conclusion

Fixed-point iteration converges if

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

When 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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7

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


8/19Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

8

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



9

Fig. 6.6



10

The Secant Method

A slight variation of Newtons method forfunctions whose derivatives are difficult toevaluate. For these cases the derivative can beapproximated by a backward finite divided

difference.

,3,2,1)()(

)(

)()(

)(

1

11

1

1

ixfxf

xxxfxx

xfxf

xxxf

ii

iiiii

ii

iii



11

Requires two initial

estimates of x , e.g, xo,

x1. However, becausef(x) is not required to

change signs between

estimates, it is not

classified as a

bracketing method.

The scant method has the

same properties as

Newtons method.

Convergence is not

guaranteed for all xo,

f(x).

Fig. 6.7



12

Fig. 6.8



13

Multiple Roots

None of the methods deal with multiple roots

efficiently, however, one way to deal with problems

is as follows:

)(

)(1xfindThen

)()()(Set

i

i

i

i

ii

xu

xu

xfxfxu

This function has

roots at all the same

locations as the

original function



14

Fig. 6.13



15

Multiple root corresponds to a point where a

function is tangent to the x axis. Difficulties

Function does not change sign at the multiple root,

therefore, cannot use bracketing methods. Both f(x) and f(x)=0, division by zero with

Newtons and Secant methods.



16

Systems of Linear Equations

0),,,,(

0),,,,(

0),,,,(

321

3212

3211

nn

n

n

xxxxf

xxxxf

xxxxf



17

Taylor series expansion of a function of more than

one variable

)()(

)()(

11111

11111

iii

ii

iii

iii

ii

iii

yyyvxx

xvvv

yyy

uxx

x

uuu

The root of the equation occurs at the value of xand y where ui+1and vi+1equal to zero.



18

y

vy

x

vxvy

y

vx

x

vy

uy

x

uxuy

y

ux

x

u

ii

iiii

ii

i

ii

iiii

ii

i

11

11

A 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

1

1

Determinant of

the Jacobianof

the system.

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