Fixed$-$Point Method USE PYTHON FOR CODE

$($a$)$ For each of the following functions, design two $(2)$ different functions $g(x)$ such that solving the

corresponding fixed$-$point problem $x=g(x)$ is equivalent to solving the root finding problem $f(x)=0.$

Show that the root finding problem and the corresponding fixed$-$point problem are equivalent. Note:

All $g(x)$ must be distinct and provably convergent.

$f(x)=x^5+5x^3-x^2+1$ on the interval $-1,2.$

$f(x)=x^4-8x^3+24x^2-32x+16$ on the interval $0,3.$

$($b$)$ Implement in code the fixed$-$point iterations for the problems you designed and solve the root finding

problem up to accuracy of $|f(x)|$ftol$={10}^{-12}.$ For each of these problems compute the actual

rate of convergence of the iteration.

$($c$)$ Compare and describe any differences you observe between the two formulations of $g(x)$ you chose.