This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast high-precision numerical evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our descriptions contain improvements over existing algorithms. We also provide references to relevant ideas not currently used in NumGfun.

The complexity of matrix multiplication has attracted a lot of attention in the last forty years. In this paper, instead of considering asymptotic aspects of this problem, we are interested in reducing the cost of multiplication for matrices of small size, say up to 30. Following previous work in a similar vein by Probert & Fischer, Smith, and Mezzarobba, we base our approach on previous algorithms for small matrices, due to Strassen, Winograd, Pan, Laderman,... and show how to exploit these standard algorithms in an improved way. We illustrate the use of our results by generating multiplication code over various rings, such as integers, polynomials, differential operators or linear recurrence operators. Keywords: matrix multiplication, small matrix, complexity.