Question

6) (a) Denote the successive intervals that arise in the bisection method by [a₀,b₀, [a₁,b₁] , [a₂,b₂], and so on

   Show that   b_n – a_n = 2^-n(b₀-a₀)

(b) The bisection method is said to have a linear convergence. Explain as clearly as possible what that means

c) For the bisection method, prove that |c_n – c_n+1| = 2^-n-2 (b₀ – a₀)

   where cn is the midpoint of each interval (ie) c_n = (a_n + b_n) /2

Answer 1

The first and third parts require proofs- the first, or (a) has been proved using mathematical induction, and the third, (c) uses the results of the proof (a).