Computer Science

Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the determinant of a matrix with polynomial entries using hybrid symbolic and numerical computation. The algorithm relies on the Newton's interpolation method with error control for solving Vandermonde systems. It is also based on a novel approach for estimating the degree of variables, and the degree homomorphism method for dimension reduction. Furthermore, the parallelization of the method arises naturally.

Let S⊂ R n be a compact basic semi-algebraic set defined as the real solution set of multivariate polynomial inequalities with rational coefficients. We design an algorithm which takes as input a polynomial system defining S and an integer p≥0 and returns the n -dimensional volume of S at absolute precision 2 −p .Our algorithm relies on the relationship between volumes of semi-algebraic sets and periods of rational integrals. It makes use of algorithms computing the Picard-Fuchs differential equation of appropriate periods, properties of critical points, and high-precision numerical integration of differential equations.The algorithm runs in essentially linear time with respect to~ p . This improves upon the previous exponential bounds obtained by Monte-Carlo or moment-based methods. Assuming a conjecture of Dimca, the arithmetic cost of the algebraic subroutines for computing Picard-Fuchs equations and critical points is singly exponential in n and polynomial in the maximum degree of the input.

The class of quasiseparable matrices is defined by a pair of bounds, called the quasiseparable orders, on the ranks of the maximal 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 and quadratic with the quasiseparable order. We show, in this paper, the connection between the notion of quasisepa-rability and the rank profile matrix invariant, presented in [Dumas \& al. ISSAC'15]. This allows us to propose an algorithm computing the quasiseparable orders (rL, rU) in time O(n^2 s^( ω --2)) where s = max(rL, rU) and ω the exponent of matrix multiplication. We then present two new structured representations, a binary tree of PLUQ decompositions, and the Bruhat generator, using respectively O(ns log n/s) and O(ns) field elements instead of O(ns^2) for the previously known generators. We present algorithms computing these representations in time O(n^2 s^( ω --2)). These representations allow a matrix-vector product in time linear in the size of their representation. Lastly we show how to multiply two such structured matrices in time O(n^2 s^( ω --2)).

We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed by a known existence criterion of telescopers. Our approach has the important feature that it avoids the potentially costly computation of certificates. Computational experiments are also provided so as to illustrate the efficiency of our approach.

Ore operators form a common algebraic abstraction of linear ordinary differential and recurrence equations. Given an Ore operator L with polynomial coefficients in x , it generates a left ideal I in the Ore algebra over the field k(x) of rational functions. We present an algorithm for computing a basis of the contraction ideal of I in the Ore algebra over the ring R[x] of polynomials, where R may be either k or a domain with k as its fraction field. This algorithm is based on recent work on desingularization for Ore operators by Chen, Jaroschek, Kauers and Singer. Using a basis of the contraction ideal, we compute a completely desingularized operator for L whose leading coefficient not only has minimal degree in x but also has minimal content. Completely desingularized operators have interesting applications such as certifying integer sequences and checking special cases of a conjecture of Krattenthaler.

Differential-algebraic equation systems (DAEs) are generated routinely by simulation and modeling environments. Before a simulation starts and a numerical method is applied, some kind of structural analysis (SA) is used to determine which equations to be differentiated, and how many times. Both Pantelides's algorithm and Pryce's Σ -method are equivalent: if one of them finds correct structural information, the other does also. Nonsingularity of the Jacobian produced by SA indicates a success, which occurs on many problems of interest. However, these methods can fail on simple, solvable DAEs and give incorrect structural information including the index. This article investigates Σ -method's failures and presents two conversion methods for fixing them. Both methods convert a DAE on which the Σ -method fails to an equivalent problem on which this SA is more likely to succeed.

In a previous article, the authors developed two conversion methods to improve the Σ -method for structural analysis (SA) of differential-algebraic equations (DAEs). These methods reformulate a DAE on which the Σ -method fails into an equivalent problem on which this SA is more likely to succeed with a generically nonsingular Jacobian. The basic version of these methods processes the DAE as a whole. This article presents the block version that exploits block triangularization of a DAE. Using a block triangular form of a Jacobian sparsity pattern, we identify which diagonal blocks of the Jacobian are identically singular and then perform a conversion on each such block. This approach improves the efficiency of finding a suitable conversion for fixing SA's failures. All of our conversion methods can be implemented in a computer algebra system so that every conversion can be automated.

The connection method has earned good reputation in the area of automated theorem proving, due to its simplicity, efficiency and rational use of memory. This method has been applied recently in automatic provers that reason over ontologies written in the description logic ALC. However, proofs generated by connection calculi are difficult to understand. Proof readability is largely lost by the transformations to disjunctive normal form applied over the formulae to be proven. Such a proof model, albeit efficient, prevents inference systems based on it from effectively providing justifications and/or descriptions of the steps used in inferences. To address this problem, in this paper we propose a method for converting matricial proofs generated by the ALC connection method to ALC sequent proofs, which are much easier to understand, and whose translation to natural language is more straightforward. We also describe a calculus that accepts the input formula in a non-clausal ALC format, what simplifies the translation.

While Liouvillian sequences are closed under many operations, simple examples show that they are not closed under convolution, and the same goes for d'Alembertian sequences. Nevertheless, we show that d'Alembertian sequences are closed under convolution with rationally d'Alembertian sequences, and that Liouvillian sequences are closed under convolution with rationally Liouvillian sequences.

We propose a symbolic-numeric algorithm to count the number of solutions of a polynomial system within a local region. More specifically, given a zero-dimensional system f 1 =⋯= f n =0 , with f i ∈C[ x 1 ,…, x n ] , and a polydisc Δ⊂ C n , our method aims to certify the existence of k solutions (counted with multiplicity) within the polydisc. In case of success, it yields the correct result under guarantee. Otherwise, no information is given. However, we show that our algorithm always succeeds if Δ is sufficiently small and well-isolating for a k -fold solution z of the system. Our analysis of the algorithm further yields a bound on the size of the polydisc for which our algorithm succeeds under guarantee. This bound depends on local parameters such as the size and multiplicity of z as well as the distances between z and all other solutions. Efficiency of our method stems from the fact that we reduce the problem of counting the roots in Δ of the original system to the problem of solving a truncated system of degree k . In particular, if the multiplicity k of z is small compared to the total degrees of the polynomials f i , our method considerably improves upon known complete and certified methods. For the special case of a bivariate system, we report on an implementation of our algorithm, and show experimentally that our algorithm leads to a significant improvement, when integrated as inclusion predicate into an elimination method.