BISECTION METHOD

BISECTION METHOD
Sunitha A C
Assistant Professor
Department of Computer Science and Applications
St. Mary’s College Thrissur

Bisection Method, Sunitha A C, St.Mary’s College
INTRODUCTION
 One of the simplest and most reliable iterative method
for the solution of non linear equations.
 Also known as binary chopping or half-interval method.
 Based on the repeated application of the intermediate
value theorem.

 The Bisection Method is given an initial interval [a,b]
that contains a root (We can choose a and b such that
f(a) and f(b) are of opposite sign)
 The Bisection Method will cut the interval into 2 halves
and check which half interval contains a root of the
function
 The Bisection Method will keep cut the interval in
halves until the resulting interval is extremely small.
The root is then approximately equal to any value
in the final interval.

 Graphical Representation
• Suppose the interval [a,b] is as follows:

We cut the interval [a,b] in the middle: m = (a+b)/2

Because sign of f(m) ≠ sign of f(a) , we proceed with the
search in the new interval [a,b]:

 Example execution:
• We will use a simple function to illustrate the execution of
the Bisection Method
• Function used:
f(x) = x3 – 5x+1

We will use the starting interval [0,1] since:
The interval [0,1] contains a root because: sign of f(0) ≠ sign
of f(1)
• f(0) = 03−5×0+1 = 1
• f(1) = 13−5×1+1 = -3

 Iteration 1
Let 𝑥1 = 0 𝑎𝑛𝑑 𝑥2 = 1
then 𝑥0=
𝑥1+ 𝑥2
2
=
0+1
2
= 0.5
𝑓 0.5 = −1.375 𝑎𝑛𝑑 𝑓(𝑥1)𝑓(𝑥0)˂0
Thus, the root lies in the interval (0,0.5)

 Iteration 2
Let 𝑥1 = 0 𝑎𝑛𝑑 𝑥2 = 0.5
then 𝑥0 =
𝑥1+ 𝑥2
2
=
0+0.5
2
= 0.25
𝑓 0.25 = −0.234375 𝑎𝑛𝑑 𝑓(𝑥1))𝑓 𝑥0 ˂0
Thus, the root lies in the interval (0,0.25)

Iteration 𝒙 𝟏 𝒙 𝟐 𝒙 𝟎 𝒇(𝒙 𝟎)
1 0 1 0.5 ˂
2 0 0.5 0.25 ˂
3 0 0.25 0.125 ˃
4 0.125 0.25 0.1875 ˃
5 0.1875 0.25 0.21875 ˂
Hence,the root lies in (0.1875,0.21875).The approximate
root is taken as the midpoint of this interval,that is
0.203125.
Sequence of intervals is as follows:

Simple and easy to implement
One function evaluation per iteration
The size of the interval containing the zero is
reduced after each iteration
No knowledge of the derivative is needed
Slow to converge
Good intermediate approximations may be
discarded
DISADVANTAGES
ADVANTAGES

REFERENCE
1. Numerical Methods for Scientific and Engineering
Computation,M.K.Jain,S.R.K.Iyengar,R.K.Jain,New Age
International Publishers
2. Numerical Methods,E.Balagurusamy,Tata McGraw-Hill

BISECTION METHOD

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BISECTION METHOD