Chapter 5.7, Problem 6E

(Numerical analysis) Here’s a challenging problem for those who know a little calculus. The Newton-Raphson method can be used to find the roots of any equation $y\left(x\right)=0.$ In this method, the $\left(i+1\right)stapproximation,xi+1,toarootofy\left(x\right)=0$ is given in terms of the ith approximation, xi, by the following formula, where y’ denotes the derivative of y(x) with respect to x:

$x_{i+1}=x_i-y(x_i)/y\text{'}(x_i)$

For example, if $y\left(x\right)=3x2+2x-2,theny’\left(x\right)=6x+2$ , and the roots are found by making a reasonable guess for a first approximation x1 and iterating by using this equation:

$x_{i+1}=x_i-(3x_i^2+2x_i-2)/(6x_i+2)$

a. Using the Newton-Raphson method, find the two roots of the equation $3x^2+2x-2=0.$ (Hint: There’s one positive root and one negative root.)

b. Extend the program written for Exercise 6a so that it finds the roots of any function $y\left(x\right)=0,$ when the function for y(x) and the derivative of y(x) are placed in the code.