Problem in code for Newton Raphson Method

if true
  % code
function x = newton(x0)
tolerance = 10^(-8);
error = 999;
x = x0;
while error>tolerance
  f = x-x^(1/3)-2; % Function value
  dfdx = 1-(1/3)*x^(-2/3); % Derivative value
  x = x-f/dfdx; % Newton-Raph Equation
  error = abs(((x-x0)/x)*100);
  x0 = x;
end
end
end

While executing this code I'm getting x = 0, which is not the intended answer. Please help.

Answer 1

[John D'Errico]

It would seem that knowing what your starting value was would be of importance.

When I run that code, I get

x =
     3.5214

But then, maybe my starting guess was more intelligently chosen than yours. So, plot your function. ALWAYS plot EVERYTHING. Think about what will happen if you give it some arbitrary starting point in x0. In fact, I wonder if you tried starting at x0=0. In hindsight, think about how Newton's method works. Then look at the plot. Do you think that 0 is a good choice here?

fun = @(x) x - nthroot(x,3) - 2;
ezplot(un,[-5,5])
grid on

Failing to carefully think about what you are doing tends to yield randomly poor results.

Finally, note that I used nthroot. It is a bit safer then x^(-1/3). In fact, I'd suggest that a better computational choice for x^(-2/3) may also be nthroot(x,3)^2. Why is that? What happens when x happens to be a negative number?

(-1)^(-1/3)
ans =
          0.5 -    0.86603i

(-1)^(-2/3)
ans =
         -0.5 -    0.86603i

[Joseph Achou]

Is there a way to generalize this code andd make it into a function?