Numerical Analysis By Shanker Rao

22 NUMERICAL ANALYSIS
2.3 METHOD OF BISECTION
Consider the equation
f(x) = 0, refer (1)
where f(x) is continuous on (a, b) and f(a) f(b) < 0. In order to find a root of (1) lying in the interval
(a, b). We shall determine a very small interval (a 0
, b 0
) (by graphical method) in which f(a 0
) f(b 0
)
< 0 and f ′ ( x) maintains the same sign in (a 0
, b 0
), so that there is only one real root of the equation
f(x) = 0.
Divide the interval in half and let
x
1
a
=
+ b
2
0 0
If f(x) = 0 then x 1
is a root of the equation. If f(x 1
) ≠ 0 then either f(a 0
) f(x 1
) < 0 of f(b 0
)
f(x 1
) < 0. If f(a 0
) f(x 1
) < 0 then the root of the equation lies in (a 0
, x 1
) otherwise the root of the
equation lies in (x 1
, b 0
). We rename the interval in which the root lies as (a 1
, b 1
) so that
1
b1
– a1 = bb0 − a0g,
2
now we take
x
2
a
=
+ b
2
1 1
If f(x 2
) = 0 then x 2
is the root of f(x) = 0. If f(x 2
) ≠ 0 and f(x 2
) f(a 1
) < 0, then the root lies
in (a 1
, x 2
). In which case we rename the interval as (a 2
, b 2
), otherwise (x 2
, b 1
) is renamed as (a 2
,
b 2
) where
Proceeding in this manner, we find
a2 − b2 = 1 b a
2
b 0 − 0g.
2
x
n+ 1 =
a
n
+ b
2
n
which gives us the (n + 1)th approximation of the root of f(x) = 0, and the root lies (a n
, b n
) where
1
bn − an = b − a
n
b 0 0g,
2
since the left end points a 1
, a 2
, …, a n
, … form a monotonic non-decreasing bounded sequence, and
the right end points b 1
, b 2
, b 2
, …, b n
, … form a monotonic non-increasing bounded sequence, then
there is a common limit
c=
lim
n →∞
a
n
=
n →∞
such that f(c) = 0 which means that c is a root of equation (1).
The bisection method is well suited to electronic computers. The method may be conveniently
used in rough approximations of the root of the given equation. The bisection method is a simple
but slowly convergent method.
lim
b
n