If a function f(x) is continuous from a to b
and f(a) and f(b) are of opposite signs, then there exists at
least one x such that f(x) = 0. This fact is used in finding the
root of the given equation by bisection method. Here, let us assume that f(a)
is positive and f(b) is negative. Then the root lies between a
and b and let us approximate it as
If f(x0)
= 0, we conclude that x0 is the root. Otherwise root lies
between x0 and b (if f(x0) is
positive) or between a and x0 (if f(x0)
is negative) and accordingly we have to choose the interval. We have to repeat
the process till the required accuracy is reached. In each iteration the width
of the interval becomes half of the previous iteration. So width of the
interval after nth iteration is given by
If
e is the desired accuracy then we
have,
On
simplification,
Using
this inequality, one can estimate the number of iterations required to achieve
accuracy e.
Graphical representation of bisection
method
References:
1.
Introductory Methods of Numerical
Analysis, S.S. Sastry, Prentice Hall of India (1983).
2.
Numerical Methods, E Balagurusamy, Tata
McGraw-Hill Publishing Company Ltd (1999).
An excellent beginning!!! All the best
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