Products of Ordinary Differential Operators by Evaluation and Interpolation

TL;DR

This paper introduces a faster evaluation-interpolation algorithm for multiplying differential operators, proving their computational equivalence to matrix multiplication in characteristic zero and nearly optimal multiplication in positive characteristic.

Contribution

It presents a new, more efficient algorithm for multiplying differential operators and establishes their computational equivalence to matrix multiplication over fields of characteristic zero.

Findings

New evaluation-interpolation algorithm is faster than previous methods

Multiplication of differential operators is computationally equivalent to matrix multiplication in characteristic zero

Differential operators can be multiplied in nearly optimal time in positive characteristic

Abstract

It is known that multiplication of linear differential operators over ground fields of characteristic zero can be reduced to a constant number of matrix products. We give a new algorithm by evaluation and interpolation which is faster than the previously-known one by a constant factor, and prove that in characteristic zero, multiplication of differential operators and of matrices are computationally equivalent problems. In positive characteristic, we show that differential operators can be multiplied in nearly optimal time. Theoretical results are validated by intensive experiments.