Algorithms for simultaneous Hermite-Padé approximations

Abstract

Abstract We describe how to compute simultaneous Hermite–Pade 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–Pade approximations for the case t ≥ n .