Application of Bisection Method

Presenting:c

PresenterPresenters :s :

Atif AbbasAtif Abbas

Faisal Zubi

Oldest known approximation Problem

Numerical Computation;

Root - finding

Problem: Eq. :- Root Approximate

1700 B.C. traced back

Intermediate Value Theorem----1st technique

BISECTION Methodz

Find_ root ‘x’ ? Eq.—form : f(x) = 0 given ‘f’

Number ‘x’ -> zero of ‘f’

Kind of Binary Search

Efficient than stepwise search

m

. Function----continuous

. F(x) +/- [ _,_ ]

F(x) = 0. Put ‘x’ and

When root lies :-

Bisection Method :

‘ f ’ __ continuous function

l

x3 f(x3) = 0

f(x1) & f(x2) ________ +/-

Procedure __ work only when

Remember

[x1,x2]

[x1,x2]

f(x1) & f(x2) opposite sign

x3 midpoint [x1,x2]

= 1 / 2 (x1 + x2)

f (x3) = 0 ___ x = x3 ROOT

if

else f(x3) ------ + /+ , - / - f(x1) or f(x2)

if f(x3) & f(x1) ___ same sign x (x3,x2)

set x1 = x3

if f(x3) & f(x1) ___ opposite sign x (x1,x3)

set x2 = x3 z

Drawback :

Slow in converging

i.e. # of iterations ___ quite large

Property :

Always converges solution

z

Demonstration

g

x1

f(x1)

x2

f(x2)

_

+

k

x3

f(x3)

_

x1

f(x1)

x2

f(x2)

+

g

x4

f(x4)

x3

f(x3)

_

x1

f(x1)

x2

f(x2)

+

m

x5

f(x5)

x4

f(x4)

x3

f(x3)

_

x1

f(x1)

x2

f(x2)

+

Applicationt Zero degree Celsius

With time , temp.

Temp. 0 CMachine Shut down

_ operation time 9 hrs.

_ Machine switch off

_ Temp. 0 C

‘machine’

if

r x (time,hr)

03579

1113

15

f(x) (temp. Celsius)

- 30

- 25 - 21.5

- 16 - 12.5

- 07- 3.5

+ 0 2

Table of the machine:

s

From divided difference method

Get the function of the plant

P7(x) = -30 + (x-0)(1.66) + (x-0)(x-3)(0.068) +

(x-0)(x-3)(x-5)(0.0170)+ (x-0)(x-3)(x-5)(x-7)(-0.0098) +

(x-0)(x-3)( x-5)(x-7)(x-9)(0.0025) + (x-0)(x-3)(x-5)(x-7)(x-9)(x-11)(-0.0002) +

(x-0)(x-3)(x-5)(x-7)(x-9)(x-11)(x-13)(-0.0000066) . K

Where ‘K’……..any constant, representing the environmental temp.

-3.5

-07

-12.5

-16

-21.5

-25

-30

+2 3 5 7 9 11 13 15

l

hrs.

temp.

[ 0,15 ]

w

Atif Abbas