Page 239 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes The third property and the continuity of the function f(x) imply that f(a) 0 and that f(b) 0.
The crucial observation is the fact that the fourth property implies that a = b. Consequently, f (a)
= f (b) = 0, and we are done.

Example: Let’s compute numerical approximations for the with the help of the
2
bisection method. We set f(x) = x – 2. Let us start with an interval of length one: a = 1 and b =
0 1

2. Note that f(a ) = f(1) = –1 < 0, and f(b ) = f(2) = 2 > 0. Here are the first 20 applications of the
0 0
bisection algorithm:

Bisection is the division of a given curve, figure, or interval into two equal parts
(halves).

A simple bisection procedure for iteratively converging on a solution which is known to lie inside
some interval [a, b] proceeds by evaluating the function in question at the midpoint of the original
interval x = (a + b)/2 and testing to see in which of the subintervals [a, (a + b)/2] or [(a + b)/2,
b] the solution lies. The procedure is then repeated with the new interval as often as needed to
locate the solution to the desired accuracy.
n
n
Let a and b be the endpoints at the th iteration (with a = a and b = b) and let r be the th
n n 1 1 n
approximate solution. Then the number of iterations required obtaining an error smaller than
is found by noting that

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