Asymptotically fast polynomial matrix algorithms for multivariable systems

We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer algebra techniques, minimal basis computations and matrix fraction expansion/reconstruction, and to polynomial matrix multiplication. Such reductions eventually imply that all these problems can be solved in about the same amount of time as polynomial matrix multiplication

Algorithms for Simultaneous Pad\'e Approximations

We describe how to solve simultaneous Pad\'e approximations over a power series ring $K[[x]]$ for a field $K$ using $O~(n^{\omega - 1} d)$ operations in $K$, where $d$ is the sought precision and $n$ is the number of power series to approximate. We develop two algorithms using different approaches. Both algorithms return a reduced sub-bases that generates the complete set of solutions to the input approximations problem that satisfy the given degree constraints. Our results are made possible by recent breakthroughs in fast computations of minimal approximant bases and Hermite Pad\'e approximations.Comment: ISSAC 201

Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

Given a nonsingular $n \times n$ matrix of univariate polynomials over a field $\mathbb{K}$, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use $\widetilde{\mathcal{O}}(n^\omega \lceil s \rceil)$ operations in $\mathbb{K}$, where $s$ is bounded from above by both the average of the degrees of the rows and that of the columns of the matrix and $\omega$ is the exponent of matrix multiplication. The soft-$O$ notation indicates that logarithmic factors in the big-$O$ are omitted while the ceiling function indicates that the cost is $\widetilde{\mathcal{O}}(n^\omega)$ when $s = o(1)$. Our algorithms are based on a fast and deterministic triangularization method for computing the diagonal entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm

Fast Computation of Common Left Multiples of Linear Ordinary Differential Operators

We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coefficients. We analyze the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM computation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output.Comment: The final version will appear in Proceedings of ISSAC 201

Algorithms for Linearly Recurrent Sequences of Truncated Polynomials

Linear recurrent sequences are those whose elements are defined as linear combinations of preceding elements, and finding recurrence relations is a fundamental problem in computer algebra. In this paper, we focus on sequences whose elements are vectors over the ring A = K[x]/ of truncated polynomials. Finding the ideal of their recurrence relations has applications such as the computation of minimal polynomials and determinants of sparse matrices over A. We present three methods for finding this ideal: a Berlekamp-Massey-like approach due to Kurakin, one which computes the kernel of some block-Hankel matrix over A via a minimal approximant basis, and one based on bivariate Pade approximation. We propose complexity improvements for the first two methods, respectively by avoiding the computation of redundant relations and by exploiting the Hankel structure to compress the approximation problem. Then we confirm these improvements empirically through a C++ implementation, and we discuss the above-mentioned applications

Automatic Generation of Fast and Certified Code for Polynomial Evaluation

International audienceDesigning an efficient floating-point implementation of a function based on polynomial evaluation requires being able to find an accurate enough evaluation program, exploiting at most the target architecture features. This article introduces CGPE, a tool dealing with the generation of fast and certified codes for the evaluation of bivariate polynomials. First we discuss the issue underlying the evaluation scheme combinatorics before giving an overview of the CGPE tool. The approach we propose consists in two steps: the generation of evaluation schemes by using some heuristics so as to quickly find some of low latency; and the selection that mainly consists in automatically checking their scheduling on the given target and validating their accuracy. Then, we present on-going development and ideas for possible improvements of the whole process. Finally, we illustrate the use of CGPE on some examples, and show how it allows us to generate fast and certified codes in a few seconds and thus to reduce the development time of libms like FLIP