Computing functions of matrices via composite rational functions

Most algorithms for computing a matrix function $f(A)$ are based on finding a rational (or polynomial) approximant $r(A)≈f(A)$ to the scalar function on the spectrum of $A$. These functions are often in a composite form, that is, $f(z)≈r(z)=r_k(⋯r_2(r_1(z)))$ (where $k$ is the number of compositions, which is often the iteration count, and proportional to the computational cost); this way $r$ is a rational function whose degree grows exponentially in $k$. I will review algorithms that fall into this category and highlight the remarkable power of composite (rational) functions.