Designing an algorithm, numerical analysis

Question

Suppose you need to numerically ?nd a value of x on an interval [0, 5] for which function f(x) = x³ is equal to 2.197.

(a) Design an algorithm for this problem which is (i) guaranteed to converge, (ii) converges linearly and (iii) allows you to theoretically estimate the number of required iterations to achieve accuracy of 10− 5 a priori, i.e. without actually carrying out iterations, estimate this number of iterations.

(b) Design another algorithm for this problem that guarantees quadratic convergence and explain why it converges quadratically.

(c) Implement both of the algorithms in part (a) and (b), solve the problem and demonstrate numerically that the order of convergence is as expected.

Answer 1

You have bisection method a.k.a binary search method

and you have newton's method

The exact answer is x=1.3

There are ton's of calculators for Newton's method and bisection method online.

You can find the algorithms too. I may have some old code lying around.