Section: New Results
Algebraic computing and high-performance kernels
Algebraic Diagonals and Walks: Algorithms, Bounds, Complexity
The diagonal of a multivariate power series
Multiple Binomial Sums
Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with algebraic generating function. We study the representation of the generating functions of binomial sums by integrals of rational functions. The outcome is twofold. Firstly, we show that a univariate sequence is a multiple binomial sum if and only if its generating function is the diagonal of a rational function. Secondly, we propose algorithms that decide the equality of multiple binomial sums and that compute recurrence relations for them. In conjunction with geometric simplifications of the integral representations, this approach behaves well in practice. The process avoids the computation of certificates and the problem of the appearance of spurious singularities that afflicts discrete creative telescoping, both in theory and in practice [7].
Fast and Accurate Computation of Orbital Collision Probability for Short-Term Encounters
We provide a new method for computing the probability of collision between two spherical space objects involved in a short-term encounter under Gaussian-distributed uncertainty. In this model of conjunction, classical assumptions reduce the probability of collision to the integral of a two-dimensional Gaussian probability density function over a disk. The computational method is based on an analytic expression for the integral, derived by use of Laplace transform and D-finite functions properties. The formula has the form of a product between an exponential term and a convergent power series with positive coefficients. Analytic bounds on the truncation error are also derived and are used to obtain a very accurate algorithm. Another contribution is the derivation of analytic bounds on the probability of collision itself, allowing for a very fast and — in most cases — very precise evaluation of the risk. The only other analytical method of the literature — based on an approximation — is shown to be a special case of the new formula. A numerical study illustrates the efficiency of the proposed algorithms on a broad variety of examples and favorably compares the approach to the other methods of the literature [20].
Efficient Algorihtms for Mixed Creative Telescoping
Creative telescoping is a powerful computer algebra paradigm — initiated by Doron Zeilberger in the 90's — for dealing with definite integrals and sums with parameters. We address the mixed continuous-discrete case, and focus on the integration of bivariate hypergeometric-hyperexponential terms. We design a new creative telescoping algorithm operating on this class of inputs, based on a Hermite-like reduction procedure. The new algorithm has two nice features: it is efficient and it delivers, for a suitable representation of the input, a minimal-order telescoper. Its analysis reveals tight bounds on the sizes of the telescoper it produces [26].
Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables
Analytic combinatorics studies the asymptotic behaviour of sequences through the analytic properties of their generating functions. This article provides effective algorithms required for the study of analytic combinatorics in several variables, together with their complexity analyses. Given a multivariate rational function we show how to compute its smooth isolated critical points, with respect to a polynomial map encoding asymptotic behaviour, in complexity singly exponential in the degree of its denominator. We introduce a numerical Kronecker representation for solutions of polynomial systems with rational coefficients and show that it can be used to decide several properties (0 coordinate, equal coordinates, sign conditions for real solutions, and vanishing of a polynomial) in good bit complexity. Among the critical points, those that are minimal—a property governed by inequalities on the moduli of the coordinates—typically determine the dominant asymptotics of the diagonal coefficient sequence. When the Taylor expansion at the origin has all non-negative coefficients (known as the `combinatorial case') and under regularity conditions, we utilize this Kronecker representation to determine probabilistically the minimal critical points in complexity singly exponential in the degree of the denominator, with good control over the exponent in the bit complexity estimate. Generically in the combinatorial case, this allows one to automatically and rigorously determine asymptotics for the diagonal coefficient sequence. Examples obtained with a preliminary implementation show the wide applicability of this approach [43].
Tableau sequences, open diagrams, and Baxter families
Walks on Young’s lattice of integer partitions encode many objects of algebraic and combinatorial interest. Chen et al. established connections between such walks and arc diagrams. We show that walks that start at
On 3-dimensional lattice walks confined to the positive octant
Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in
Asymptotic Lattice Path Enumeration Using Diagonals
We consider
Asymptotics of lattice walks via analytic combinatorics in several variables
We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set
Linear Time Interactive Certificates
With J.G. Dumas (LJK, Grenoble), E. Kaltofen (NCSU, USA), and E. Thomé (Inria Nancy) we work on interactive certificates. Computational problem certificates are additional data structures for each output, which can be used by a (possibly randomized) verification algorithm that proves the correctness of each output. In [32] we give a new certificate for the minimal polynomial of sparse or structured matrices whose Monte Carlo verification complexity requires a single matrix-vector multiplication and a linear number of extra field operations (sufficiently large cardinality field). We also propose a novel preconditioner that ensures irreducibility of the characteristic polynomial of the generically preconditioned matrix. This preconditioner takes linear time to be applied and uses only two random entries. We combine these two techniques to give algorithms that compute certificates for the determinant, and thus for the characteristic polynomial, whose Monte Carlo verification complexity is therefore also linear.
Computing minimal interpolation bases
With É. Schost (U. Waterloo, Canada), we consider the problem of computing
minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Padé approximation and constrained multivariate interpolation, and has applications in coding theory and security.
The problem is classically solved using iterative algorithms based on recurrence relations.
First, we discuss in [62] a fast, divide-and-conquer version of this recurrence, taking advantage of fast matrix computations over the scalars and over the polynomials. This new algorithm is deterministic, and for computing shifted minimal bases of relations between
Fast computation of shifted Popov forms of polynomial matrices via systems of modular polynomial equations
In [46] we give a Las Vegas algorithm which computes the shifted Popov form
of an
Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix
With G. Labahn and W. Zhou (U. Waterloo, Canada)
we give in [64] fast and deterministic algorithms to compute the determinant
and Hermite normal form of a nonsingular
Fast Computation of the Rank Profile Matrix and the Generalized Bruhat Decomposition
The row (resp. column) rank profile of a matrix describes the stair-case shape of its row (resp. column) echelon form.
With J. G. Dumas and Z. Sultan (LJK, Grenoble), we propose in [11] a new matrix invariant, the rank profile matrix, summarizing all information on the row and column rank profiles of all the leading sub-matrices. We show that this normal form exists and is unique over any ring, provided that the notion of McCoy's rank is used, in the presence of zero divisors. We then explore the conditions for a Gaussian elimination algorithm to compute all or part of this invariant, through the corresponding PLUQ decomposition. This enlarges the set of known Elimination variants that compute row or column rank profiles. As a consequence a new Crout base case variant significantly improves the practical efficiency of previously known implementations over a finite field. With matrices of very small rank, we also generalize the techniques of Storjohann and Yang to the computation of the rank profile matrix, achieving an
Computing with quasiseparable matrices
The class of quasiseparable matrices is defined by a pair of bounds, called the quasiseparable orders,
on the ranks of the sub-matrices entirely located in their strictly lower and upper triangular parts.
These arise naturally in applications, as e.g. the inverse of band matrices, and are widely used for they admit structured representations allowing to compute with them in time linear in the dimension.
In [47] we show the connection between the notion of quasiseparability
and the rank profile matrix invariant of Dumas et al.
This allows us to propose an algorithm computing the quasiseparable orders
A Real QZ Algorithm for Structured Companion Pencils
With Y. Eidelman (U. Tel Aviv) and L. Gemignani (U. Pisa), we design in [54]
a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfind-ing problems.
The modified QZ algorithm computes the generalized eigenvalues of an
Efficient Solution of Parameter Dependent Quasiseparable Systems and Computation of Meromorphic Matrix Functions
In [55], with Y. Eidelman (U. Tel Aviv) and L. Gemignani (U. Pisa), we focus on the solution of shifted quasiseparable systems and of more general parameter dependent matrix equations with quasiseparable representations. We propose an efficient algorithm exploiting the invariance of the quasiseparable structure under diagonal shifting and inversion. This algorithm is applied to compute various functions of matrices. Numerical experiments show the effectiveness of the approach.