# Na.Numeric Analysis – Fast Root Search Algorithm for a Special Function

My question stems from

Fast root search for a strictly decreasing function

I am a little surprised at the page above that there is not even an effective root search algorithm (RFA) for a strictly monotonous function. Consider $$f: mathbb R to mathbb R$$ Defined by $$f (z) = sum_ {k = 1} ^ n p_k y_k e ^ {z y_k} -x$$, or $$p_k> 0$$ for everyone $$k = 1, ldots, n 2$$. Assume further $$y_1 and there is a root $$z _ *$$ for $$f$$, that is to say. $$f (z _ *) = 0$$. Then, it follows from the assumptions that $$f$$ is strictly growing and so $$z ^ *$$ is unique. My concern is to find a numerical approximation of $$z ^ *$$.

A simple calculation gives that

$$f (z) = sum_ {k = 1} ^ n p_k y_k ^ 2 e ^ {z y_k} ~> ~ 0, quad mbox {for all} z in mathbb R.$$

Is there a competitive calculation method designed for $$f$$? Comments or remarks are very appreciated!

PS: My first attempt is to apply Newton's method. In fact, it's easy to notice that

1. if $$y_1 ge 0$$then $$f$$ is convex;
2. if $$y_n the 0$$then $$f$$ is concave;

For these two cases, I think that Newton's method can work. As for $$y_1 <0 , I think it holds $$inf_ {z in mathbb R} f (z)> 0$$.