Can Initial Guesses Improve Newton-Raphson's Root Predictions?

[danong]

Uhm i mean, actually how do we predict the estimated roots before we implement this?

Because i wonder if the estimated root is too big enough to predict and sometimes time-consuming, is there any way to predict the roots better and accurate before we implement the Newton Raphson's iterative methods? I am using plain subsitution into the function which sometimes have to check through all the numbers in order to get one estimated root beforehand.


Thanks in advance.

Regards,
Daniel.

 

[HallsofIvy]

We can't. One of the most famous "fractals" is the graph formed by looking at the three roots of z³= 1. These are, of course, 1, [itex]-(1/2)+ i\sqrt{3}/2[/itex], and [itex]-(1/2)-i\sqrt{3}/2[/itex]. Treat the x,y-plane as the complex plane, (x,y) corresponding to x+ iy. Taking each x+ iy as the "initial estimate" for Newton-Raphson, color the point "red", "blue", "green", or "black" according to whether the sequence converges to 1, [itex]-(1/2)+ i\sqrt{3}/2[/itex], and [itex]-(1/2)-i\sqrt{3}/2[/itex], or does not converge respectively. You will see large patches of "red", "blue", and "green" close to those respective roots but the boundary is extremely complex ("fractal"). In fact, every boundary point is a boundary point of all four sets simultaneously. It is possible that very tiny variations in choice of initial point will cause the iteration to converge to a different answer, or not converge at all.

(By the way, when I first programmed a computer (with a "graphics" terminal) to do that, it took almost an hour. Now the same program runs in less that 10 seconds!)

 

[tacman]

There are a few ways to predict estimated roots before implementing the Newton Raphson's iterative method. One approach is to use a graphing calculator or software to plot the function and visually estimate the roots. This can give you a general idea of where the roots may be located and help guide your implementation of the method.

Another approach is to use interval bisection or other root-finding methods, such as the bisection method or the secant method, to find initial guesses for the roots. These methods can provide a good starting point for the Newton Raphson's method and potentially save time in the iterative process.

Additionally, it may be helpful to consider the properties of the function, such as its behavior near the roots, to make an educated guess on the estimated roots. For example, if the function is continuous and changes signs at the roots, you can use the intermediate value theorem to narrow down the possible range of the roots.

Ultimately, predicting estimated roots before implementing the Newton Raphson's method may require some trial and error, but using these strategies can help improve the accuracy and efficiency of the method.