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(Solved): Root Finding Methods and Bisection Method (a) Deduce x^(*), convergence order, and asymptotic error ...
Root Finding Methods and Bisection Method
(a) Deduce x^(*), convergence order, and asymptotic error constant for the following:
i. x_(k)=2^(-2^(k)).
ii. x_(k)=2^(-2^((k)/(2))). Hint: Consider p=\sqrt(2).
(b) Consider the function f(x)=e^(-x)-sinx.
i. Using the Intermediate Value Theorem, deduce the number of roots for f on 0,2\pi .
ii. Devise a Python program to compute all roots of f on 0,2\pi using the Bisection
method with a tolerance of 10^(-14) for the interval length. Give brief explanations on
how you chose the starting intervals.
iii. What is the actual number of iterations needed to find all the roots? How do they
compare with the bound derived in class (Lecture 14 - Slide 12)?