How to find initial guess for Newton-Raphson method

Question

I have a polynomial equation $x^3-2x^2-5=0$ for which I have to find solutions accurate to $10^{-4}$ in the interval $[1,4]$. How can I find initial guess $P_0$?

Answer 1

As far as I know, there is no general criterion to decide the “best” initial point. The algorithm will (eventually) converge to one of the possible roots. As a rule of thumb, you should start from a point inside an interval $I=[a,b]$, such that the function is continuous in $I$ and $f(a)\cdot f(b)<0$. This ensures that at least a solution there exists in $I$. If you guarantee that in $I$ the function is smooth, namely with continuous high-order derivatives, the algorithm will run smoother.

In your case, I suggest you choose the middle point $x_0=2.5$. FYI, cubic polynomial admits closed-form solution, have a look on the “Cardano-Tartaglia’s Formula”, e.g. Click here

Remark:

The algorithm just gives you one root, say $x_1$. If $p(x)=x^3-2x^2-5$ is your polynomial function, then you can find $p_1(x)=p(x)/(x-x_1)$ by using the polynomial division algorithm. Applying the Newton-method to $p_1(x)$ will give you another root, say $x_2$. Then, define $p_2(x)=p_1(x)/(x-x_2)$ gives another root $x_3$, and so on... In your case, upon finding the first root, $p_1(x)$ is just a parabola whose roots are easy to be found.

Answer 2

Provided the continuity of the second derivative, if you want to avoid overshoot of the solution, you must select $x_0$ such that $$f(x_0)\times f''(x_0) >0$$

This is Darboux theorem.

So, in your case, $x_0=???$

For illustration, run the two possible cases