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 <y_2 < cdots <y_n $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

- if $ y_1 ge 0 $then $ f $ is convex;
- if $ y_n the $ 0then $ f $ is concave;

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