Algorithms for simultaneous Hermite–Padé approximations

We describe how to compute simultaneous Hermite–Padé approximations, over a polynomial ring K[x] for a field K using O∼(nω−1td) operations in K, where d is the sought precision, where n is the number of simultaneous approximations using t

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∼(nω−1td) has only been reached with randomized algorithms finding a single solution for the case t

Our results are made possible by recent breakthroughs in fast computations of minimal approximant bases and Hermite–Padé approximations for the case t≥n.