Interval Halving (Bisection)

An algorithm for halving the interval (Bisection):

To determine a root of $f(x)=0$ that is accurate within a ... $x_l=x_3$ End If\\
Until $(\vert x_1 - x_2\vert) < 2*tolerance value$\\

>> format long e
>> fa=1e-120;fb=-2e-300;
>> fa*fb
ans =     0
>> sign(fa)~=sign(fb)
ans =     1

Table 3.1: The bisection method for $ f(x)=3x + sin(x) - e^x$, starting from $ x_1=0,x_2=1$, using a tolerance value of 1E-4.


The root is (almost) never known exactly, since it is extremely unlikely that a numerical procedure will find the precise value of $ x$ that makes $ f(x)$ exactly zero in floating-point arithmetic.

Cem Ozdogan 2011-12-27