MDB3053_Root Finding Using Open Methods

7/26/2019 MDB3053_Root Finding Using Open Methods

1/24

Chap6/1

CHAPTER 6 : ROOTS OF EQUATIONS:Open Methods

LESSON OUTCOMES

To calculate roots of equation using

Open Methods:1. Newton-Raphson Method

2. Secant Method

3. Fixed-point Iteration

y

x

- when a function is zero

- when a function crossesx-axis

roots


2/24

Chap6/2

OPEN METHODS : INTRODUCTION

Open methods are based on

formulas that require only onesingle or two starting value(s)

of x that do not need to

bracket the root.

Not always work as

sometimes it can diverge,

depending on the initial guess

When it converges, it

reaches solution much more

faster than Bracketing

method.

Bisection method Open method


3/24

Chap6/3

NEWTON-RAPHSON METHOD

Also called as Newtons method, it is a powerful and

widely used method.

Based on Taylor series expansion; truncate the series after

1st order derivative term:

)(

)(

)(0

Re

0)(whenofvaluetheisrootThe

...)()()(

1

1

11

1

i

iii

iiii

ii

niii

xf

xfxx

xx)(xf)f(x

arranging,

xfx

Rxxfxfxf

Newton-Raphson formula

Solve forxi+1


4/24

Chap6/4


Convenient method for functions

whose derivatives can be evaluatedanalytically.

Not suitable for functions whose

derivatives cannot be evaluatedanalytically

It converges very fast as it is

quadratically convergent (seebook in Box 6.2, p141)

Pitfalls: cant find multiple roots, may not

converge if guess is near zero slope !

)(

)(1

i

iii

xf

xfxx

Fig. 6.5


5/24

Chap6/5


Rootis estimated byextending a tangent atx1

down tox-axis

)(

)(

0)()(

1

1

i

iii

ii

ii

xf

xfxx

xxxfxf

:Tangent


6/24

Chap6/6

NEWTON-RAPHSON METHODALGORITHM

i

iii

xf

xfxx

1

STEP 1: Assume initial value forxi x0

STEP 2: Calculate the value ofxi+1using equation

STEP 3 :Calculate the approx. error |a|

STEP 4: Repeat STEP 2until a< spre-specified value,

then root =xi+1

1001

1

i

iia

x

xx

i= 0,1,2,3,


7/24

Chap6/7

Newton-Raphson method Poor Convergence

Inflection point,f '' = 0 inthe vicinity of a root

Oscillation of slopes

Guess is near zero slope

Zero slope is encountered


8/24

Chap6/8

CLASS ACTIVITY

Determine the highest real root off(x) =x3 6x2 + 11x 6.1

using Newton-Raphson Method. Consider three iterationswith guessxo=3.5. Also calculate the estimated error, aafter

each iteration.

i

iii

xf

xfxx

1

1001

1

i

iia

x

xx


9/24

Chap6/9

ii

ii xf

xfxx

1

i x(i+1) xi f(xi) f'(xi) epsilon(a) %

1 3.1913 3.5000 1.7750 5.7500

2 3.0687 3.1913 0.3994 3.2576 4.00

3 3.0473 3.0687 0.0519 2.4264 0.70

SOLUTION:


10/24

Chap6/10

SECANT METHOD

For functions whose derivatives are difficult to evaluate,

the derivatives f '(x) can be approximated byBackwardFinite-Divided Differencing scheme :

1

1)()(

)(

ii

ii

i xx

xfxf

xf

ii

ii

xf

xfxx

1:RaphsonNewton

)()(

))((:methodSecant

1

1

1

ii

iii

ii

xfxf

xxxfxx


11/24

Chap6/11

SECANT METHOD

Requires 2 initial guesses:xi&

xi-1 that do not necessarilybracket the root. Hence, NOT abracketing method !

Similarity to false-positionmethod:

)()())((

1

11

ii

iiiii

xfxfxxxfxx

Definition: Secantis astraight line that intercepts acurve at 2 or more points

2initial guesses ofxi

andxi-1

uL

uLuur

xfxf

xxxfxx

Fig. 6.7


12/24

Chap6/12

SECANT METHOD

Secant method requires

TWOguesses:x1andx2

Chord / Secant


13/24

STEP 1: Assume 2 initial guesses forxi andxi-1 ati=0

STEP 2: Calculate the value ofxi+1using

STEP 3: Calculate the error |a|

STEP 4: RepeatSTEP 2until a< pre-specified value, then

set root =xi+1

NOTE: Convergenceis not guaranteed for allxoChap6/13

SECANT METHODALGORITHM

)()(

))((

1

11

ii

iiiii

xfxf

xxxfxx

1001

1

i

iia

x

xx

i= 0,1,2,3,


14/24

Chap6/14

CLASS ACTIVITY

Use three iterations to determine the root off(x) =exx by

using Secant Method with initial estimate of x-1=0 andx0=1.0. Also calculate the estimated error, a after eachiteration.

1001

1

i

iia

x

xx

)()(

))((

1

11

ii

iiiii

xfxf

xxxfxx


15/24

Chap6/15

f(x) = exx

x-1=0 andx0=1.0 )()(

))((

1

1

1ii

iii

ii xfxf

xxxf

xx

iter x(i+1) x(i-1) xi x(i-1)-xi f(xi) f(i-1) eps(a) %

1 0.6127 0.0000 1.0000 -1.0000 -0.6321 1.0000

2 0.5638 1.0000 0.6127 0.3873 -0.0708 -0.6321 8.67

3 0.5672 0.6127 0.5638 0.0489 0.0052 -0.0708 0.59


16/24

Chap6/16

BRACKETING VS OPEN METHOD

BRACKETING Method

Bracketing methods

can alwaysconvergeto

a root.

Bracketing method is

slowto converge

OPEN Method

Open method might

divergeor move away

from the true root

depending on the initialguess (see pitfalls)

When it converges, it

reaches results fasterthan the bracketing

methods.


17/24

Rearrange the function so thatxis on the left side of the

equation: x= g(x) For example: x2 x 2 = 0 can be manipulated to yield

x=x2 2

or x= (x+ 2)1/2

or x= (x+ 2)/x

From an initial estimate ofxi, the next estimate,xi+1can be

ITERATIVELYcomputed by the function ofxi+1= g(xi) The approximate percent error:

Chap6/17

FIXED-POINT ITERATION

To obtain x= g(x)

%1001

1

i

iia

x

xx


18/24

Chap6/18

EXAMPLE


19/24

Chap6/19

CLASS ACTIVITY

Use fixed-point iteration to find the root of x2 - 3x+2=0


20/24

Chap6/20

Xi eps(a)

1 0.0000

2 0.6667 100.00

3 0.8148 18.18

4 0.8880 8.24

5 0.9295 4.47

6 0.9547 2.64

7 0.9705 1.63

8 0.9806 1.03

9 0.9872 0.67

SOLUTION


21/24

x=g(x) can be expressed as a

pair of equations:f1(x) =x and

f2(x) = g(x)

Plot these two equationsseparately the intersectionof the two curves will be the

root! The method is slower to

converge because it is

linearly convergent Chap6/21

FIXED-POINT ITERATION CONVERGENCE

Figure 6.2


22/24

Fixed-point methods may sometime diverge, depending

on the starting point (initial guess) and how the functionbehaves (see Sec 6.1.1).

It will converge if the absolute slope

When the method converges, the error is roughly

proportional to the error of the previous step, therefore it

is called linearly convergent.Chap6/22


1)( xg the absolute slope ofy = g(x) is lessthan the slope ofy=x, see Fig 6.3 (p137)


23/24

Chap6/23


|g(x)|1

diverge

|g(x)|>1

|g(x)|


24/24

THANK YOU

MDB3053_Root Finding Using Open Methods

Documents

Transcript of MDB3053_Root Finding Using Open Methods