Question

Let xbe the exact zero of a given function f(x). We are looking at the bisection algorithm with the starting interval [a,b]. Let cₙ=(a+b)/2 be the middle of the interval in iteration n of the algorithm.

Derive a formula for the size of the interval S(n) after step n of the bisection algorithm (at step n=0 it is S(0)=b−a because we start with [a,b];S(1) is 1/2 of that.

Answer 1

The size of the interval in the bisection algorithm can be calculated using the formula: S(n) = (b - a) / (2^n).

Step-by-step explanation:

The bisection algorithm is a method used to find the zero of a function. In each iteration, the algorithm divides the interval [a, b] into two halves by finding the middle point, cₙ = (a + b) / 2. The size of the interval S(n) after step n can be calculated using the formula: S(n) = (b - a) / (2^n).

For example, at step n=0, we start with [a, b], so S(0) = b - a. At step n=1, the interval is halved, so S(1) = (b - a) / 2.