Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix

We reduce the problem of computing the rank and a nullspace basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n x n matrix of degree d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension n and degree d. If the latter multiplication is done in MM(n,d)=softO(n^omega d) operations, with omega the exponent of matrix multiplication over K, then the algorithm uses softO(MM(n,d)) operations in K. The softO notation indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of matrix Hensel high-order lifting and matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-moduleComment: Research Report LIP RR2005-03, January 200

Sparse approaches for the exact distribution of patterns in long state sequences generated by a Markov source

We present two novel approaches for the computation of the exact distribution of a pattern in a long sequence. Both approaches take into account the sparse structure of the problem and are two-part algorithms. The first approach relies on a partial recursion after a fast computation of the second largest eigenvalue of the transition matrix of a Markov chain embedding. The second approach uses fast Taylor expansions of an exact bivariate rational reconstruction of the distribution. We illustrate the interest of both approaches on a simple toy-example and two biological applications: the transcription factors of the Human Chromosome 5 and the PROSITE signatures of functional motifs in proteins. On these example our methods demonstrate their complementarity and their hability to extend the domain of feasibility for exact computations in pattern problems to a new level

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

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

Deterministic Unimodularity Certification and Applications for Integer Matrices

The asymptotically fastest algorithms for many linear algebra problems on integer matrices, including solving a system of linear equations and computing the determinant, use high-order lifting. For a square nonsingular integer matrix A, high-order lifting computes B congruent to A^{-1} mod X^k and matrix R with AB = I + RX^k for non-negative integers X and k. Here, we present a deterministic method -- "double-plus-one" lifting -- to compute the high-order residue R as well as a succinct representation of B. As an application, we give a fully deterministic algorithm to certify the unimodularity of A. The cost of the algorithm is O((log n) n^{omega} M(log n + log ||A||)) bit operations, where ||A|| denotes the largest entry in absolute value, M(t) the cost of multiplying two integers bounded in bit length by t, and omega the exponent of matrix multiplication. Unimodularity certification is then applied to give a heuristic, but certified, algorithm for computing the determinant and Hermite normal form of a square, nonsingular integer matrix. Though most effective on random matrices, a highly optimized implementation of the latter algorithm demonstrates the techniques' effectiveness across a variety of inputs: empirical running times grow as O(n^3log n). A comparison against the fastest known Hermite normal algorithms -- those available in Sage and Magma -- show our implementation is, in all cases, highly competitive, and often surpasses existing, state-of-the-art implementations

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

Fast computation of power series solutions of systems of differential equations

We propose new algorithms for the computation of the first N terms of a vector (resp. a basis) of power series solutions of a linear system of differential equations at an ordinary point, using a number of arithmetic operations which is quasi-linear with respect to N. Similar results are also given in the non-linear case. This extends previous results obtained by Brent and Kung for scalar differential equations of order one and two