Computer Science

This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain a necessary and sufficient condition for the equivalence of a square polynomial matrix and a diagonal matrix. Based on the constructive proof of the new criteria, we give a factorization algorithm and prove the uniqueness of the factorization. We implement the algorithm on Maple, and two illustrative examples are given to show the effectiveness of the algorithm.

The sparse difference resultant introduced in \citep{gao-2015} is a basic concept in difference elimination theory. In this paper, we show that the sparse difference resultant of a generic Laurent transformally essential system can be computed via the sparse resultant of a simple algebraic system arising from the difference system. Moreover, new order bounds of sparse difference resultant are found. Then we propose an efficient algorithm to compute sparse difference resultant which is the quotient of two determinants whose elements are the coefficients of the polynomials in the algebraic system. The complexity of the algorithm is analyzed and experimental results show the efficiency of the algorithm.

We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial p of degree d with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for multiple roots of a polynomial given by a black box for the approximation of its coefficients, and their complexity decreases at least proportionally to the number of roots in a region of interest (ROI) on the complex plane, such as a disc or a square, but we greatly strengthen the main ingredient of the previous algorithms. Namely our new counting test essentially amounts to the evaluation of a polynomial p and its derivative p ′ , which is a major benefit, e.g., for sparse polynomials p . Moreover with evaluation at about log(d) points (versus the previous record of order d ) we output correct number of roots in a disc whose contour has no roots of p nearby. Moreover we greatly soften the latter requirement versus the known subdivision algorithms. Our second and less significant contribution concerns subdivision algorithms for polynomials with real coefficients. Our tests demonstrate the power of the proposed algorithms.

It is known since the 1970s that no more than 23 multiplications are required for computing the product of two 3 x 3-matrices. It is not known whether this can also be done with fewer multiplications. However, there are several mutually inequivalent ways of doing the job with 23 multiplications. In this article, we extend this list considerably by providing more than 13 000 new and mutually inequivalent schemes for multiplying 3 x 3-matrices using 23 multiplications. Moreover, we show that the set of all these schemes is a manifold of dimension at least 17.

Pattern matching is a powerful tool which is part of many functional programming languages as well as computer algebra systems such as Mathematica. Among the existing systems, Mathematica offers the most expressive pattern matching. Unfortunately, no open source alternative has comparable pattern matching capabilities. Notably, these features include support for associative and/or commutative function symbols and sequence variables. While those features have individually been subject of previous research, their comprehensive combination has not yet been investigated. Furthermore, in many applications, a fixed set of patterns is matched repeatedly against different subjects. This many-to-one matching can be sped up by exploiting similarities between patterns. Discrimination nets are the state-of-the-art solution for many-to-one matching. In this thesis, a generalized discrimination net which supports the full feature set is presented. All algorithms have been implemented as an open-source library for Python. In experiments on real world examples, significant speedups of many-to-one over one-to-one matching have been observed.

We consider problems originating in economics that may be solved automatically using mathematical software. We present and make freely available a new benchmark set of such problems. The problems have been shown to fall within the framework of non-linear real arithmetic, and so are in theory soluble via Quantifier Elimination (QE) technology as usually implemented in computer algebra systems. Further, they all can be phrased in prenex normal form with only existential quantifiers and so are also admissible to those Satisfiability Module Theory (SMT) solvers that support the QF_NRA. There is a great body of work considering QE and SMT application in science and engineering, but we demonstrate here that there is potential for this technology also in the social sciences.

The optimal calculation order of a computational graph can be represented by a set of algebraic expressions. Computational graph and algebraic expression both have close relations and significant differences, this paper looks into these relations and differences, making plain their interconvertibility. By revealing different types of multiplication relations in algebraic expressions and their elimination dependencies in line-graph, we establish a theoretical limit on the efficiency of face elimination.

Algorithms of numeric (in exact arithmetic) deduction of analytical expressions, proposed and described by Shevchenko and Vasiliev (1993), are developed and implemented in a computer algebra code. This code is built as a superstructure for the computer algebra package by Shevchenko and Sokolsky (1993a) for normalization of Hamiltonian systems of ordinary differential equations, in order that high complexity problems of normalization could be solved. As an example, a resonant normal form of a Hamiltonian describing the hyperboloidal precession of a dynamically symmetric satellite is derived by means of the numeric deduction technique. The technique provides a considerable economy, about 30 times in this particular application, in computer memory consumption. It is naturally parallelizable. Thus the economy of memory consumption is convertible into a gain in computation speed.

Gaussian elimination with no pivoting and block Gaussian elimination are attractive alternatives to the customary but communication intensive Gaussian elimination with partial pivoting (hereafter we use the acronyms GENP, BGE, and GEPP} provided that the computations proceed safely and numerically safely}, that is, run into neither division by 0 nor numerical problems. Empirically, safety and numerical safety of GENP have been consistently observed in a number of papers where an input matrix was pre-processed with various structured multipliers chosen ad hoc. Our present paper provides missing formal support for this empirical observation and explains why it was elusive so far. Namely we prove that GENP is numerically unsafe for a specific class of input matrices in spite of its pre-processing with some well-known and well-tested structured multipliers, but we also prove that GENP and BGE are safe and numerically safe for the average input matrix pre-processed with any nonsingular and well-conditioned multiplier. This should embolden search for sparse and structured multipliers, and we list and test some new classes of them. We also seek randomized pre-processing that universally (that is, for all input matrices) supports (i) safe GENP and BGE with probability 1 and/or (ii) numerically safe GENP and BGE with a probability close to 1.We achieve goal (i) with a Gaussian structured multiplier and goal (ii) with a Gaussian unstructured multiplier and alternatively with Gaussian structured augmentation. We consistently confirm all these formal results with our tests of GENP for benchmark inputs. We have extended our approach to other fundamental matrix computations and keep working on further extensions.

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{ö}bner bases taking into account the valuation of K .Because of the use of the valuation, the theory of tropical Gr{ö}bner bases has proved to provide settings for computations over polynomial rings over a p -adic field that are more stable than that of classical Gr{ö}bner bases.Beforehand, these strategies were only available for homogeneous polynomials. In this article, we extend the F5 strategy to a new definition of tropical Gr{ö}bner bases in an affine setting.We provide numerical examples to illustrate time-complexity and p -adic stability of this tropical F5 algorithm.We also illustrate its merits as a first step before an FGLM algorithm to compute (classical) lex bases over p -adics.