1. (Numerical Method: Iteration Methods)
As you may already know that 3 numerical methods, namely Fixed-point iteration
(FPI), Newton-Raphson method (NRM), and Secant method (SM), are used for solving
nonlinear equation, $f(x) = 0$. By the comparison, the FPI requires a value of $p$ for which $g(p) =$
p that requires the appropriate $g(x) = x$ and initial value of $x$. Meanwhile, the NRM is a famous
method, however, it requires the derivative of the function to be solved, whereas the SM uses
an approximation for that derivative, but unfortunately, it needs two initial values.
Consider solving
$f(x) = x^2 + \sin(x) - 5\cos(x)$,
using FPI, NRM, and SM by taking the same initial value in the given range of [-2, 2]. Find
the root of $f(x) = 0$ and the maximum number of iteration ($N$), if the solution is converged.
Which one yields the fastest convergence?
