Problem 3: Root Finding
Use only Python standard library modules (https://docs.python.org/3/library/) and matplotlib for this problem, i.e. do not import numpy, scikit, or any other non-standard package.
The secant method is an iterative root-finding algorithm. It uses a sequence of secant line roots to approximate \(c\) such that \(f(c) = 0\) for a continuous function \(f\). Unlike Newton’s method it does not require knowledge or evaluation of the derivative \(f'\). The secant method is defined by the recurrence:
\[x_n = x_{n-1} - f(x_{n-1}) \frac{x_{n-1} - x_{n-2}}{f(x_{n-1}) - f(x_{n-2})}.\]
Write a python script that uses the secant method to approximate the root of a continuous function \(f\) in the interval \([a, b]\). You may assume that \(f\) has at most one root in \([a,b]\). Use \(|x_{k+1} - x_{k}| < 10^{-10}\) as the convergence criterion. Let \(N\) be the number of iterations to reach convergence. Output \(N\) followed by the three root approximations \(x_{N-2}\), \(x_{N-1}\), \(x_N\). Output each number to its own line and use precision sufficient to show convergence.
Import the function \(f\) from a file named func.py in the same directory as your script – i.e. from func import f. You may assume that \(f\) is continuous on \([a,b]\) and that func.f(x) returns a scalar float for all \(x \in [a,b]\).
Your script should accept a and b as two numeric command line arguments, i.e. python hw2p3.py "1.1" "1.4". Your script must validate that a and b are numeric, verify that a < b, and check that f(a)f(b) < 0 – see Bolzano’s Theorem. Write “Range error” to STDERR (standard error) if any of these three conditions fail and immediately terminate.
Your script should not produce any output except as described above.