The Islamic University of Gaza

Faculty of Engineering

Civil Engineering Department

Numerical Analysis

ECIV 3306

Chapter 5 Bracketing Methods

Associate Prof. Mazen Abualtayef Civil Engineering Department, The Islamic University of Gaza

PART II: 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

Study Objectives for Part Two

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

• Non-computer methods:

- Closed form solution (not always available)

- Graphical solution (inaccurate)

• Numerical systematic methods suitable for computers

ROOTS OF EQUATIONS

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

c

f(c)

c=14.75 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) <0.

(The function changes sign on opposite sides of the root)

Bracketing Methods 1. Bisection Method

• Generally, if f(x) is real and continuous in the interval xl to xu and f (xl).f(xu)<0, then there is at least one real root between xl and xu to this function.

• The bisection method, which is alternatively called binary chopping, interval halving, or Bolzano’s method, is one type of incremental search method in which the interval is always divided in half.

• The interval at which the function changes sign is located. Then the interval is divided in half with the root lies in the midpoint of the subinterval. This process is repeated to obtained refined estimates.

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)<0): xu = xr

xr = ( xl + xu )/2

Step 1: Choose lower xl and upper xu

guesses for the root such that:

f(xl).f(xu)<0

Step 2: The root estimate is:

xr = ( xl + xu )/2

Step 3: Subdivide the interval according to:

– If (f(xl).f(xr)<0) the root lies in the

lower subinterval; set xu = xr and go to

step 2.

– If (f(xl).f(xr)>0) the root lies in the

upper subinterval; set xl = xr and go to

step 2.

– If (f(xl).f(xr)=0) the root is xr and stop

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) = 6.067•1.569=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 <0;

the root lies bet. 14 and 15.

6. Set xl = 14 and xu=15, thus the new root

xr=(14+ 15)/2= 14.5

and so on…...

15

40)1(38.667

)( 146843.0 cec

cf

True Value at Xr = 14.7802007701481

• In the previous example, if the stopping criterion is et =

0.5%; what is the root?

Bisection method: Example

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

%100

:

true

eapproximattrue

tX

XX

ErrorrelaiveTrue

e

Bisection method

Flow Chart –Bisection Start

Input: xl , xu , es, maxi

f(xl). f(xu)<0

i=0 ea=1.1es

False

while ea> es & i <maxi

2

1

u r

r

x xx

i i

False

Stop

Print: xr , f(xr ) ,ea , i

xu+xl =0

100%u l

a

u l

x x

x xe

True

Test=f(xl). f(xr)

ea=0.0 Test=0

xu=xr Test<0

xl=xr

True

True

False

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).

• The fact that the replacement of the curve by a straight line gives a “false position” of the root is the origin of the name, method of false position. It is also called the linear interpolation method.

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(xr)

2. False-position Method -Procedure

f(xr)

2. False-position Method -Procedure

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

root such that: f(xl).f(xu)<0

Step 2: The root estimate is:

Step 3: Subdivide the interval according to:

– If (f(xl).f(xr)<0) the root lies in the lower subinterval; xu = xr and go to step 2.

– If (f(xl).f(xr)>0) the root lies in the upper subinterval; xl = xr and go to step 2.

– If (f(xl).f(xr)=0) the root is xr and stop

False-position Method -Procedure

• 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

False position method: Example

f(x)

x 16 12

-2.269

6.067

14.91

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 between 12 and 14.9113.

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

and f(xu)=-0.2543 f(xl).f(xu)<0

5. The new root xr= 14.7942

6. This has an approximate error of 0.79%

False position method: Example 40)1(

38.667)( 146843.0 ce

ccf

False position method: Example

Flow Chart –False Position Start

Input: xl , x0 , es, maxi

f(xl). f(xu)<0

i=0 ea=1.1es

False

while ea> es & i <maxi

( )( )

( ) ( )

1

u l u

r u

l u

f x x xx x

f x f x

i i

False

Stop

Print: xr , f(xr ) ,ea , i

i=1 or

xr=0

0 100%r r

a

r

x x

xe

True

Test=f(xl). f(xr)

ea=0.0 Test=0

xu=xr

xr0=xr

Test<0

xl=xr

xr0=xr

True

True

False

False Position Method-Example 2

False Position Method-Example 2

Excel – Goal Seek Tool

Roots of Polynomials: Using Software Packages

MS Excel:Goal seek

f(x)=x-cos x