function x = NRCubeRoot(R, x0, nmax, delta, eps)

% Applies Newton's Method to find cube root of R.

%

% Input:

% R = number whose cube root we want.

% x0 = initial iterate.

% nmax = maximum number of iterates to use.

% delta = tolerance for small derivatives (break if df(x) is too small).

% eps = stopping tolerance. Stop when relative magnitude of f is small

% enough.

% Output:

% x = approximate cube root of f.

%% Initialization.

% Initalize x. Stores sequence of iterates.

x = zeros(nmax + 1,1);

x(1) = x0;

%% Run Newton's Method.

for n = 2 : nmax + 1

% Check if df is small.

df = ?; % CHANGE THIS. % df is derivative evaluated at last iterate x(n-1).

if (% INSERT STOPPING CONDITION)

x = x(n-1); % extract root from sequence of approximate solutions.

break % leave the loop.

end

% Compute step length to check for convegence.

d = ?; % CHANGE THIS.

% Stop if absolute step length is sufficiently small.

if (% INSERT STOPPING CONDITION.)

x = x(n-1); % extract root from sequence of approximate solutions.

break % leave the loop.

end

% Update x using the formula from Part(a).

x(n) = ?; % INSERT UPDATE FORMULA.

end

We can use Newton's Method to evaluate the cube root of a positive number as follows. The number a is the cube root of the positive number R if and only if That is, the cube root(s) of R are the roots of the polynomial equation we can use the Newton- Raphson Method to mumerically approximate the This implies that of any positive number by finding the roots of (6.1.1)