Bracketing Methods

ROOTS OF EQUATIONS

Roots of Equations

Bracketing Methods

Bisection method

False Position Method

Open Methods

Simple fixed point iteration

Newton Raphson

Secant

Modified Newton Raphson

System of Nonlinear Equations

Roots of polynomials

Muller Method

ROOTS OF EQUATIONS

Root of an equation: is the value of the equation variable which

make the equations = 0.0

But

a

acbbxcbxax

2

40

22

?0sin

?02345

xxx

xfexdxcxbxax

ROOTS OF EQUATIONS

Non-computer methods:

- Closed form solution (not always available)

- Graphical solution (inaccurate)

Numerical systematic methods suitable for

computers

Graphical Solution

roots

The roots exist where f(x) crosses the x-axis.

f(x)

x f(x)=0 f(x)=0

Plot the function f(x)

Graphical Solution: Example

The parachutist velocity is

What is the drag coefficient c needed to reach a velocity of

40 m/s if m=68.1 kg, t =10 s, g= 9.8 m/s2

)(t

m

c

e1c

mgv

40)1(38.667

)(

)1()(

146843.0

c

tm

c

ec

cf

vec

mgcf

Check: F (14.75) = 0.059 ~ 0.0

v (c=14.75) = 40.06 ~ 40 m/s

Numerical Systematic Methods

I. Bracketing Methods

f(x)

x

roots

f(xl)=+ve

f(xu)=+ve

xl xu

No roots or even

number of roots

f(x)

x

roots

f(xl)=+ve

f(xu)=-ve xl xu

Odd number of roots

Bracketing Methods (cont.)

Two initial guesses (xl and xu) are required for the

root which bracket the root (s).

If one root of a real and continuous function, f(x)=0,

is bounded by values xl , xu then f(xl).f(xu)

Special Cases

Effect of computer scale resolution

Bracketing Methods 1. Bisection Method

Generally, if f(x) is real and continuous in the interval xl to xu

and f (xl).f(xu)

f(x)

x xu xl

f(xu)

f(xu)

xr1

f(x)

x xu xl

f(xu)

xr2

xr = ( xl + xu )/2

f(xu)

f(xr1)

f(xr2)

(f(xl).f(xr)

Bisection Method - Termination Criteria

For the Bisection Method ea > et

The computation is terminated when ea becomes less than a certain criterion (ea < es)

%100

:

true

eapproximattrue

tX

XX

ErrorrelaiveTrue

e

1

:

100%

100% (Bisection)

n n

r ra n

r

u la

u l

Approximate relative Error

X X

X

X X

X X

e

e

Bisection method: Example

The parachutist velocity is

What is the drag coefficient c needed to reach a velocity of 40

m/s if m = 68.1 kg, t = 10 s, g= 9.8 m/s2

f(c)

c

)(t

m

c

e1c

mgv

40e1c

38667cf

ve1c

mgcf

c1468430

tm

c

)(.

)(

)()(

.

f(x)

x 16 12

-2.269

6.067

14

f(x)

x 14 16 -0.425 -2.269

1.569

1.569

(f(12).f(14)>0): xl = 14

1. Assume xl =12 and xu=16

f(xl)=6.067 and f(xu)=-2.269

2. The root: xr=(xl+xu)/2= 14

3. Check f(12).f(14) = 6.0671.569=9.517 >0;

the root lies between 14 and 16.

4. Set xl = 14 and xu=16, thus the new root

xr=(14+ 16)/2= 15

5. Check f(14).f(15) = 1.569-0.425= -0.666

Iter. Xl Xu Xr ea% et% 1 12 16 14 5.279 --

2 14 16 15 6.667 1.487

3 14 15 14.5 3.448 1.896

4 14.5 15 14.75 1.695 1.204

5 14.75 15 14.875 0.84 0.641

6 14.74 14.875 14.813 0.422 0.291

In the previous example, if the stopping criterion is et = 0.5%; what is the root?

Bisection method: Example

Bisection method

Flow Chart Bisection Start

Input: xl , xu , es, maxi

f(xl). f(xu) es & i

xu+xl =0

100%u lau l

x x

x xe

True

Test=f(xl). f(xr)

ea=0.0 Test=0

xu=xr Test

Bracketing Methods 2. False-position Method

The bisection method divides the interval xl to xu in

half not accounting for the magnitudes of f(xl) and

f(xu). For example if f(xl) is closer to zero than f(xu),

then it is more likely that the root will be closer to

f(xl).

False position method is an alternative approach

where f(xl) and f(xu) are joined by a straight line; the

intersection of which with the x-axis represents and

improved estimate of the root.

2. False-position Method

False position method is an

alternative approach where

f(xl) and f(xu) are joined by

a straight line; the

intersection of which with

the x-axis represents and

improved estimate of the

root.

f(x)

x xu xl

f(xu)

f(xl)

xr

f(xr)

)()(

))((

)()(

ul

uluur

ur

u

lr

l

xfxf

xxxfxx

xx

xf

xx

xf

False-position Method -Procedure

Step 1: Choose lower xl and upper xu guesses for the

root such that: f(xl).f(xu)

False position method: Example

The parachutist velocity is

What is the drag coefficient c needed to reach a

velocity of 40 m/s if m =68.1 kg, t =10 s, g= 9.8 m/s2

f(c)

c

)(t

m

c

e1c

mgv

40)1(38.667

)(

)1()(

146843.0

c

tm

c

ec

cf

vec

mgcf

f(x)

x 16 12

-2.269

6.067

14.91

False position method: Example

1. Assume xl = 12 and xu=16

f(xl)= 6.067 and f(xu)= -2.269

2. The root: xr=14.9113

f(12) . f(14.9113) = -1.5426 < 0;

3. The root lies bet. 12 and 14.9113.

4. Assume xl = 12 and xu=14.9113, f(xl)=6.067 and

f(xu)=-0.2543

5. The new root xr= 14.7942

6. This has an approximate error of 0.79%

False position method: Example

Flow Chart False Position Start

Input: xl , x0 , es, maxi

f(xl). f(xu) es & i

i=1 or

xr=0

0 100%r rar

x x

xe

True

Test=f(xl). f(xr)

ea=0.0 Test=0

xu=xr xr0=xr

Test

False Position Method-Example 2

False Position Method - Example 2

Pitfalls of the False Position Method

Although a method such as false position is often superior to bisection, there are some cases (when function has significant curvature that violate this general conclusion.

In such cases, the approximate error might be misleading and the results should always be checked by substituting the root estimate into the original equation and determining whether the result is close to zero.

major weakness of the false-position method: its one sidedness That is, as iterations are proceeding, one of the bracketing points will tend stay fixed which lead to poor convergence.

Modified Fixed Position

One way to mitigate the "one-sided" nature of false position is to make the algorithm detect when one of the bounds is stuck. If this occur, the function value at the stagnant bound is divided in half. This is thought to fasten the convergence.