Question

Working on the Newton approximation for the root of a polynomial. (PS2, Question 3) Anyone have any hints regarding what the answer space I should search is? I can't think of a good justification for choosing the upper bound and lower bound for the root.

Thanks for any help that might be offered!

Answer 1

You need the upper and lower bound since Newton's method is only conditionally stable. This means that for some functions the method will not converge (not single in on the root) and might even go away to infinity. So either you could put the upper and lower bound as parameters to the function and let the user provide them, or you could choose some upper and lower bound based on the initial guess somehow.

Answer 2

I myself would just put the upper and lower bounds as parameters.

Answer 3

An example of one of these functions is that Newton's method runs away on is \(f(x) = x^{1/3}\)

In fact, it won't even approach \(+\infty\) or \(-\infty\), rather it oscilate back and forth (within the restraints of computers, though, if you are using an IEEE 754 double-precision floating point number, the highest you can go is \(\pm\left(2^{2048} - 1\right)\left(1 - 2^{-51}\right)\), which I think is about \(\pm10^{615}\), about 32000 x "google to the sixth".

Answer 4

Thanks for the feedback- are there any thoughts on what might be a reasonable initial guess? Should I just start with some default each time or should I adjust my guess based on some other factor, e.g. the coefficient values... ?

Thanks again!