Algorithms for simultaneous Hermite–Padé approximations. (January 2021)

• Record Type: Journal Article
• Authors: Rosenkilde, Johan; Storjohann, Arne
• Is Part Of: Journal of symbolic computation. Volume 102(2020)
• Page Start: 279
• Page End: 303
• Publication Date: 2021-01
• Subjects: Padé approximation -- Approximant bases -- Structured linear systems; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.07.026
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We describe how to compute simultaneous Hermite–Padé approximations, over a polynomial ring K [ x ] for a field K using O ∼ ( n ω − 1 t d ) operations in K, where d is the sought precision, where n is the number of simultaneous approximations using t < n polynomials, and where O ( n ω ) is the cost of multiplying n × n matrices over K . We develop two algorithms using different approaches. Both algorithms return a reduced sub-basis that generates the complete set of solutions to the input approximation problem that satisfy the given degree constraints. Previously, the cost O ∼ ( n ω − 1 t d ) has only been reached with randomized algorithms finding a single solution for the case t < n . Our results are made possible by recent breakthroughs in fast computations of minimal approximant bases and Hermite–Padé approximations for the case t ≥ n .