Question

Similar to√b, on most computers, the computation of3√b is also based on
Newton’s method. Set up the Newton’s iteration for solving x^3 − b = 0,and write down the formula for xn+1 in terms of xn.

Answer 1

According to the Newton's method,

x(n+1) = x(n) - f(x(n))/f'(x(n)).

So, in the case of f(x) = x³-b we have the following formula:

x(n+1) = x(n) - ((x(n))³-b)/(3(x(n))²).