Chap 06 of Numerical Methods for Engineers by Chapra and Canale2

8/9/2019 Chap 06 of Numerical Methods for Engineers by Chapra and Canale2

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Copyright 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

by Lale Yurttas, TexasA&M University

Chapter 6 1

Chapter 6

Open Methods


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Chapter 6 2

Open methods are based on formulas that require onlya single starting value of x or two starting values that

do not necessarily bracket the root.

Figure 6.1


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Chapter 6 3

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 starting 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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Chapter 6 4

/

xxg

or

xxg

or

xxg

xxxxf

21)(

2)(

2)(

02)(

2

2

!

!

!

"!

Example:


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Chapter 6 5

Convergence

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

y1=xy2=g(x) (componentequations)

Plot them separately.

The root is the x-valueoccurring at theintersection

Figure 6.2


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Chapter 6 6

Conclusion

Fixed-point iteration converges if

x)f(x)linetheof(slope1)( !dxg

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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by Lale Yurttas, TexasA&M University Chapter 6 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

i

ii

iiii

iiii

xf

xfxx

xx)(xf)f(x

xOxxfxxfxfxf

d!

d!

!

((dd(d!

Newton-Raphson formula

Solve for


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A convenient method forfunctions 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

Deficiencies

(a) Points of inflection

local max or mins

f(x)=0(b) Zero or near zero slope

(c) Multiple roots and

near zero slope

(d) Zero slope


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Chapter 6 10

Summary of the Newton-Raphson

Requires: 1 initial guess sufficiently close

to the root and explicit representation of

the functions derivative.

Advantage: Quadratic rate of

convergence as the root is approached.

Problems: multiple roots, points of

inflection, zero or near zero slope, not

guaranteed convergence.


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The Secant Method

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

-,3,2,1)()(

)(

)()(

)(

1

11

1

1

!

!

$d

ixfxf

xxxfxx

xfxf

xxxf

ii

ii

iii

ii

ii

i


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Requires two initial

estimates of x , e.g, xo,

x1. However, becausefix) 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,

fix).

Fig. 6.7


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Fig. 6.8


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Chapter 6 14

Modified Secant Method

Employs a fractional factor,H to estimate the

derivative.

IfHis too small, swamped with round-off error.

IfHis too large, inefficient or divergent

)()(

)(1

iii

iiiixfxxf

xfxxx

!

H

H


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Chapter 6 15

Summary of the Secant Method

Requires: 2 initial guesses

Advantage: Similar to Newton-Raphson

except the derivative is replaces with afinite divided difference.

Problems: not guaranteed convergence.


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Chapter 6 16

)1)(1)(3(375)( 23 !! xxxxxxxf

Fig. 6.10

Multiple Roots

Graphically the curve touchesthe x-axis tangentially at the

double root.(fig.a)

(fig.c))1)(1)(1)(1)(3()(

(fig.b))1)(1)(1)(3()(

!

!

xxxxxxf

xxxxxf

Difficulties: (1) no sign change, must use

open methods that may diverge.

(2) f(x) & f(x) goes to 0 at the root(3) both Secant and Newton-Raphson

methods are linearly convergent for

multiple roots.


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Multiple Roots

None of the methods deal with multiple roots

efficiently, however, one way to deal with

problems is as follows:

)(

)(indThen

)(

)()(Set

1

i

i

ii

i

i

i

xu

xuxx

xf

xfxu

d

!

d!

This function has

roots at all the same

locations as the

original function


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Modified Newton-Raphson

Differentiate u(x) using the chain rule

Modified

NewtonRaphson

Preferable for multiple roots but less efficient.

)('')())('(

)(')(

simpli yandsubstitute

))('(

)('')()(')(')('

21

2

iii

ii

iixfxfxf

xfxfxx

xf

xfxfxfxfxu

!

!


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Modified Secant Method

)()())((

yieldoequation tSecantintosub)('

)(

similarly,

1

11

ii

iii

iixuxuxxxuxx

xf

xfu(x)

!

!

Modified Secant


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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.


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Systems ofNon-LinearEquations

0),,,,(

0),,,,(

0),,,,(

321

3212

3211

!

!

!

nn

n

n

xxxxf

xxxxf

xxxxf

-

/

-

-


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Newton Raphson for Nonlinear

Equations

Taylor series expansion of a function of more

than one variable

)()(

)()(

11111

11111

ii

i

ii

i

ii

iii

ii

iii

yy

y

vxx

x

vvv

yyyuxx

xuuu

x

x

x

x!

xx

xx!

The root of the equation occurs at the value of x

and y where ui+1 and vi+1 equal to zero.


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y

vy

x

vxvy

y

vx

x

v

y

uy

x

uxuy

y

ux

x

u

i

i

i

iii

i

i

i

i

i

i

iii

i

i

i

x

x

x

x!

x

x

x

x

x

x

x

x!

x

x

x

x

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

uxuv

xvu

yy

x

v

y

u

y

v

x

uy

uv

y

vu

xx

iiii

ii

ii

ii

iiii

i

i

i

i

ii

x

x

x

x

x

x

x

x

x

x

x

x

!

x

x

x

x

x

x

x

x

x

x

x

x

!

1

1

Employing Cramers Rule

Problem:

Diverges if initial guesses are not sufficiently

close to the true root.

Documents

Chap 06 of Numerical Methods for Engineers by Chapra and Canale2