che 555 open methods

by Lale Yurttas, Texas A&M University

Chapter 2 1

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

Chapter 2ROOTS OF EQUATION:

OPEN METHODS


Chapter 2 2

Open Methods

• Open methods are based on formulas that require only a single starting value of x or two starting values that do not necessarily bracket the root.

Figure 6.1

SIMPLE FIXED-POINT ITERATION

• Sometime “diverge”• Depending on the stating point (initial

guess) and how the function behaves.


Chapter 2 3

Procedures


Chapter 2 4

xxgxf )(0)(1. Rearrange the function so that x is on

the left side of the equation:2. Guess the initial at the root xi to

compute xi+1 (new). Thus xi+1 = g (xi).3. Compare es with ea . If ea< es, stop.

Otherwise repeat the process.%100

1

1

i

iia x

xx


Chapter 2 5

xxg

orxxg

orxxg

xxxxf

21)(

2)(

2)(

02)(2

2

Example:

Convergence• Fixed-point iteration converges if

• When the method converges, the error is roughly proportional to or less than the error of the previous step, therefore it is called “linearly convergent.”


Chapter 2 6

x)f(x) line theof (slope 1)( xg


Chapter 2 7

Procedures• x = g(x) can be expressed

as a pair of equations:y1 = xy2 = g(x) (component equations)

• Plot them separately.

Figure 6.2

Example 1a) Use simple fixed-point iteration to

find the root of f(x) = e-x – x.b) Separate the equation e-x – x = 0

into two parts and determine its root graphically.


Chapter 2 8


Chapter 2 9

NEWTON-RAPHSON METHOD

• Most widely used method.• Based on Taylor series expansion:

)()(

)(0g,Rearrangin

0)f(x when xof value theisroot The!2

)()()()(

1

1

1i1i

32

1

i

iii

iiii

iiii

xfxfxx

xx)(xf)f(x

xOxxfxxfxfxf

Newton-Raphson formula

Solve for


Chapter 2 10

• A convenient method for functions whose derivatives can be evaluated analytically.

• It may not be convenient for functions whose derivatives cannot be evaluated analytically.

Fig. 6.5

Pitfalls• Slow convergence → due to nature of

function• No general convergence criterion →

depends on nature of function & accuracy of initial guess.


Chapter 2 11


Chapter 2 12

Fig. 6.6

Solution shoots off horizontally & never hits the x-axis

x0 progressively diverge from the root

Near-zero slope is reached, solution sent far from the area of interestInitial guess close to one root can jump to a location several roots away → tendency to move away from area of interest

Example 2a) Use the Newton-Raphson method to

estimate the root of f(x) = e-x – x, employing an initial guess of x0 = 0.

b) Determine the positive root of f(x) = x10 – 1 using the Newton Raphson method and an initial guess of x0 = 0.5


Chapter 2 13


Chapter 2 14

THE SECANT METHOD• A slight variation of Newton’s method for

functions whose derivatives are difficult to evaluate. For these cases the derivative can be approximated by a backward finite divided difference.

,3,2,1)()(

)(1

11

i

xfxfxxxfxx

ii

iiiii

ii

iii xx

xfxfxf

1

1 )()()('

Formula for the Secant method:


Chapter 2 15

• 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 Newton’s method. Convergence is not guaranteed for all xo, f(x).

Fig. 6.7

Difference between Secant & False-position method


Chapter 2 16

1st iteration – both use the same points

2nd iteration – points used

differ. Secant methods diverge.

• Alternative approach → involves a fractional perturbation (small change) of the independent variable to estimate f’(x)


Chapter 2 17

THE MODIFIED SECANT METHOD

,3,2,1)()(

)(1

i

xfxxfxfxxx

iii

iiii

i

iiii x

xfxxfxf )()()('

Formula for the Modified Secant method:

δ = small perturbation fraction• If δ too small – method swamped by round

off error.• If too big – technique become inefficient

and divergent.• If chosen correctly – nice alternative for

cases where evaluating the derivative is difficult & developing 2 initial guesses is inconvenient.


Chapter 2 18

Example 3a) Use the Secant method to estimate the root

of f(x) = e-x – x. Start with initial estimates of x-1 = 0 and x0 = 1.0.

b) Use the False-position and Secant methods to estimate the root of f(x) = ln x. Start the computation with values of xl = xi-1 = 0.5 and xu = xi = 5.0.

c) Use the modified secant method to estimate the root of f(x) = e-x – x. Use a value of 0.01 for δxi and start with x0 = 1.0.


Chapter 2 19


Chapter 2 20

MULTIPLE ROOTS• Corresponds to a point where a function is tangent to

x-axis.

)()())((x

:methodsecant of version Modified

)(")()]('[)(')(x

:methodRaphson Newton of Modified

)()()(Set

1

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21i

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i

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xuxuxxxux

xfxfxfxfxfx

xfxfxu

This function has roots at all the same locations as the original function


Chapter 2 21

Fig. 6.10

Difficulties


Chapter 2 22

– 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 Newton’s and Secant methods.


Chapter 2 23

SYSTEMS OF NON-LINEAR EQUATIONS

0),,,,(

0),,,,(0),,,,(

321

3212

3211

nn

n

n

xxxxf

xxxxfxxxxf


Chapter 2 24

• Taylor series expansion of a function of more than one variable

)()(

)()(

11111

11111

iii

iii

ii

iii

iii

ii

yyyvxx

xvvv

yyyuxx

xuuu


Chapter 2 25

yvy

xvxvy

yvx

xv

yuy

xuxuy

yux

xu

ii

iiii

ii

i

ii

iiii

ii

i

11

11

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


Chapter 2 26

xv

yu

yv

xu

xuv

xvu

yy

xv

yu

yv

xu

yuv

yvu

xx

iiii

ii

ii

ii

iiii

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1

1

Determinant of the Jacobian of the system.

•A set of two linear equations with two unknowns that can be solved for.

che 555 open methods

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