Chapter 6, Problem 32P

One numerical method for calculating the cubic root of a number, $\sqrt[3]{P}$ is Halley’s method. The solution process starts by choosing a value x₁as a first estimate of the solution. Using this value, a second, more accurate value x₂is calculated with $x_2=x_1(x_1^3+2P)/(2x_1^3+P)$ , which is then used for calculating a third, still more accurate value x_3,and so on. The general equation for calculating the value of x_i+1 is from the value of x₁is $x_{i+1}=x_i(x_i^3+2P)/(2x_i^3+P)$. Write a MATLAB program that calculates the cubic root of a number. In the program use x₁= P for the first estimate of the solution. Then, by using the general equation in a loop, calculate new, more accurate values. Stop the looping when the estimated relative error defined by $E=\left|\frac{x_{i+1}-x_i}{x_i}\right|$ is smaller than 0.00001. Use the program to calculate:

(a) $\sqrt[3]{800}$(b) $\sqrt[3]{59071}$(c) $\sqrt[3]{-31.55}$