2025/09/30 by Awad H. Al-Mohy, Al-Mohy, Awad H.
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #FOS: Mathematics #Numerical Analysis (math.NA) #Polynomial and algebraic computation #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2509.26475
openalex publication_date 2025/09/30 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
We propose a matrix-free algorithm for evaluating linear combinations of φ-function actions, wi := ∑j=0p αi j φj(ti A)vj for i=1\colon r, arising in exponential integrators. The method combines the scaling and recovering method with a truncated Taylor series, choosing a spectral shift and a scaling parameter by minimizing a power-based objective of the shifted operator. Accuracy is user-controlled and ultimately limited by the working precision. The algorithm decouples the stage abscissae ti from the polynomial weights αij, and a block variant enables simultaneous evaluation of \wi\i=1r. Across standard benchmarks, including stiff and highly nonnormal matrices, the algorithm attains near-machine accuracy (IEEE double precision in our tests) for small step sizes and maintains reliable accuracy for larger steps where several existing Krylov-based algorithms deteriorate, providing a favorable balance of reliability and computational cost.