Question

How do you find the number of iterations in the bisection method?

Answer 1

Answer:

Solution: The bisection method generates a sequence {pn} approximating a root p of f(x) = 0 with

Step-by-step explanation:

|pn − p| ≤ b − a . 2n

To converge to within an absolute error tolerance of ε means we need to have |pn−p| ≤ ε, or

b − a ≤ ε. (1) 2n

Solving (1), we obtain: 2n ≥ b−a,

εb−a nlog2 ≥ log ε ,

logb−a n≥ε.

log 2

Togetsomeintuition,plugina=0,b=1,andε=0.1. Then,wewouldget n >= 3.3219. Thus, n = 4 iterations would be enough to obtain a solution pn that is at most 0.1 away from the correct solution. Note that dividing the interval [0, 1] three consecutive times would give us a subinterval of 0.0625 in length, which is smaller than 0.1.