I need to solve numerically the secular equation
$$f(t) = 1 + \rho \sum_{i=1}^n \frac{z_i^2}{d_i-t}$$
where $\{z_i\}_{i=1}^n$ and $\{d_i\}_{i=1}^n$ are known. I implemented an algorithm that alternates between the bisection method and Newton method (see here, bottom of page 460). The algorithm converges, but it takes many iterations to do so.
How is the secular equation solved numerically nowadays? I know this is done in variations of divide and conquer algorithms for eigenvalues.
Written as $$\sum_{i=1}^n \frac{\alpha_i\, z_i}{\alpha_i- \theta}=1-q$$ this is an equation we worked extensively in my research group because of its importance in chemical process simulation (in a single simulation, we have to solve it zillions of times). Our most recnt work was puplished in 2014 (see here) where we propose extremely good estimates of the solution; typically, we need only between $3$ and $5$ iterations because of the convex transformations we proposed.
On this site, have a look at this question.
