Chapter 1.4, Problem 3CP

Apply Newton’s Method to find the only root to as much accuracy as possible, and find the root’s multiplicity. Then use Modified Newton’s Method to converge to the root quadratically. Report the forward and backward errors of the best approximation obtained from each method.

(a) $f\left(x\right)=27x^3+54x^2+36x+8$

(b) $f\left(x\right)=36x^4-12x^3+37x^2-12x+1$