Shared-Pole Carathéodory–Fejér Approximations for Linear Combinations of φ-Functions

We develop a shared denominator Carathéodory–Fejér (CF) method for efficiently evaluating linear combinations of φ-functions for matrices whose spectrum lies in the negative real axis, as required in exponential integrators for large stiff ODE systems. This entire family is approximated with a single set of poles (a common denominator). The shared pole set is obtained by assembling a stacked Hankel matrix from Chebyshev boundary data for all target functions and computing a single SVD; the zeros of the associated singular-vector polynomial, mapped via the standard CF slit transform, yield the poles. With the poles fixed, per-function residues and constants are recovered by a robust least squares fit on a suitable grid of the negative real axis. For any linear combination of resolvent operators applied to right-hand sides, the evaluation reduces to one shifted linear solve per pole with a single combined right-hand side, so the dominant cost matches that of computing a single φ-function action. Numerical experiments indicate geometric convergence at a rate consistent withHalphen’s constant, and for highly stiff problems our algorithm outperforms existing Taylor and Krylov polynomial-based algorithms.