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
2. Bisection Method
ď It is a Derivative Based Method for Optimization
ď Requirements for Bisection Method
ďĄ f -> câ i.e. f is continuous for the first derivative.
ďĄ There exists a minima in the level of uncertainty [a
b]
ďĄ Function must be unimodal.
PowerPoint Presentation by Sumita Das, GHRCE
3. Algorithm
ď Initialize Level of uncertainty [a b]
k=1
ak =a
bk =b
Ďľ > 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 â 2x
fâ(x)=6x-2
Put midpoint value in
derivative.
fâ(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 â(x)=2x-4
k ak bk ck fâ(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