How can I use Numerical-Methods to solve that equation ...

Question

Use the Newton-Raphson iteration method (NR) to find one positive root ( Xr > 0 ) of the equation $ X^3 +X -1 =0 $ ??

Answer 1

Yes. Set $f(x)=x^3+x-1$ and iterate with the Newton Raphson formula:

$$ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. $$ You need to guess a value for $x_0$ to start the iteration. Also, $f'(x_n)$ is the derivative of your function $f$ evaluated at $x_n$. If your iterations don't converge, try a different starting value $x_0$.

Hope this helps some.

Cheers,

Paul Safier

Answer 2

Approach outline:

A plot (courtesy of Wolfram-Alpha) of the function shows:

Choose a starting point, say:

$$x_0 = 1$$

Write the Newton Iteration formula as:

$$x_{n+1} = x_n - \dfrac{f(x)}{f'(x)} = x_n - \dfrac{x_n^3 + x_n -1}{3 x_n^2 + 1}$$

Write out the iterates $x_0, x_1, x_2, \ldots$ and find:

$$x_5 = 0.682328$$