Numerical on bisection method

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Bisection Method is a Derivative Based Method for Optimization. It is one of the classical optimization techniques. Numerical on Bisection method is discussed in this Presentation

NUMERICAL ON
BISECTION METHOD
BY Sumita Das

Bisection Method
 It is a Derivative Based Method for Optimization
 Requirements for Bisection Method
 f -> c’ i.e. f is continuous for the first derivative.
 There exists a minima in the level of uncertainty [a
b]
 Function must be unimodal.
PowerPoint Presentation by Sumita Das, GHRCE

Algorithm
 Initialize Level of uncertainty [a b]
k=1
ak =a
bk =b
ϵ > 0
l : Allowable level of uncertainty such that
(1/2) n <= (1/(b`-a`))
PowerPoint Presentation by Sumita Das, GHRCE

While k<= n
ck= (ak +bk )/2
if f(ck)=0
Stop with ck as the
solution
if f(ck)>0
ak+1 = ak
bk+1 = ck
else
ak+1 = ck
bk+1 = bk
end if
k=k+1
end while
Find midpoint
c is now b.
a remains same
c is now a.
b remains same
Midpoint is the minima
PowerPoint Presentation by Sumita Das, GHRCE

In Simple words
Midpoint
m
a b
Example: f(x)=3x2 – 2x
f’(x)=6x-2
Put midpoint value in
derivative.
f’(x)=6*50-2=298
1. if f(m)=0, Midpoint is
minima
2. if f(m)>0, Level of
uncertainty will be [a, m]
3. if f(m)<0, Level of
uncertainty will be [m, b]
50 9010
So, 298>0, level of uncertainty will be
[10, 50] ,Follow the procedure.
PowerPoint Presentation by Sumita Das, GHRCE

Example
Que: Find Minima f(x)=(x-2)2
[0 6]
Solution: f ‘(x)=2x-4
k ak bk ck f’(ck )
1 0 6 3 2
2 0 3 1.5 -1
3 1.5 3 2.25 0.5
4 1.5 2.25 1.875 -0.25
5 1.875 2.25 2.062 0.123
6 1.875 2.062 1.9685 -0.063
7 1.9685 2.062 2.015 0.0305
8 1.9685 2.015 1.99175 -0.016
9 1.99175 2.015 2.003 0.006
PowerPoint Presentation by Sumita Das, GHRCE

References
[1] Singiresu S. Rao, “Engineering Optimization, Chapter 5:
Nonlinear Programming I: One-Dimensional Minimization
Methods”, 4th Edition
PowerPoint Presentation by Sumita Das, GHRCE

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Numerical on bisection method

1. NUMERICAL ON BISECTION METHOD BY Sumita Das

2. Bisection Method  It is a Derivative Based Method for Optimization  Requirements for Bisection Method  f -> c’ i.e. f is continuous for the first derivative.  There exists a minima in the level of uncertainty [a b]  Function must be unimodal. PowerPoint Presentation by Sumita Das, GHRCE

3. Algorithm  Initialize Level of uncertainty [a b] k=1 ak =a bk =b ϵ > 0 l : Allowable level of uncertainty such that (1/2) n <= (1/(b`-a`)) PowerPoint Presentation by Sumita Das, GHRCE

4. While k<= n ck= (ak +bk )/2 if f(ck)=0 Stop with ck as the solution if f(ck)>0 ak+1 = ak bk+1 = ck else ak+1 = ck bk+1 = bk end if k=k+1 end while Find midpoint c is now b. a remains same c is now a. b remains same Midpoint is the minima PowerPoint Presentation by Sumita Das, GHRCE

5. In Simple words Midpoint m a b Example: f(x)=3x2 – 2x f’(x)=6x-2 Put midpoint value in derivative. f’(x)=6*50-2=298 1. if f(m)=0, Midpoint is minima 2. if f(m)>0, Level of uncertainty will be [a, m] 3. if f(m)<0, Level of uncertainty will be [m, b] 50 9010 So, 298>0, level of uncertainty will be [10, 50] ,Follow the procedure. PowerPoint Presentation by Sumita Das, GHRCE

6. Example Que: Find Minima f(x)=(x-2)2 [0 6] Solution: f ‘(x)=2x-4 k ak bk ck f’(ck ) 1 0 6 3 2 2 0 3 1.5 -1 3 1.5 3 2.25 0.5 4 1.5 2.25 1.875 -0.25 5 1.875 2.25 2.062 0.123 6 1.875 2.062 1.9685 -0.063 7 1.9685 2.062 2.015 0.0305 8 1.9685 2.015 1.99175 -0.016 9 1.99175 2.015 2.003 0.006 PowerPoint Presentation by Sumita Das, GHRCE

7. References [1] Singiresu S. Rao, “Engineering Optimization, Chapter 5: Nonlinear Programming I: One-Dimensional Minimization Methods”, 4th Edition PowerPoint Presentation by Sumita Das, GHRCE

Numerical on bisection method

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