My question follows from

Fast root finding for strictly decreasing function

I am a bit surprised from the above page that there is even no efficient root finding algorithm (RFA) for a strictly monotonic function. Consider $f: \mathbb R\to\mathbb R$ defined by $f(z)=\sum_{k=1}^n p_k y_k e^{z y_k} -x$, where $p_k>0$ for all $k=1,\ldots, n\ge 2$. Assume further $y_1<y_2<\cdots<y_n$, and there exists a root $z_*$ for $f$, i.e. $f(z_*)=0$. Then it follows from the assumptions that $f$ is strictly increasing and thus $z^*$ is unique. **My concern is to find a numerical approximation of $z^*$.**

A straightforward computation yields 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 any competitive computational method designed for $f$? Comments or remarks are highly appreciated!

PS: My first attempt is to apply Newton's method. Actually, it is easy to remark that

- if $y_1\ge 0$, then $f$ is convex;
- if $y_n\le 0$, then $f$ is concave;

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