Symmetric Subresultants and Applications

Schur's transforms of a polynomial are used to count its roots in the unit disk. These are generalized them by introducing the sequence of symmetric sub-resultants of two polynomials. Although they do have a determinantal definition, we show that they satisfy a structure theorem which allows us to compute them with a type of Euclidean division. As a consequence, a fast algorithm based on a dichotomic process and FFT is designed. We prove also that these symmetric sub-resultants have a deep link with Toeplitz matrices. Finally, we propose a new algorithm of inversion for such matrices. It has the same cost as those already known, however it is fraction-free and consequently well adapted to computer algebra

A CDCL-style calculus for solving non-linear constraints

In this paper we propose a novel approach for checking satisfiability of non-linear constraints over the reals, called ksmt. The procedure is based on conflict resolution in CDCL style calculus, using a composition of symbolical and numerical methods. To deal with the non-linear components in case of conflicts we use numerically constructed restricted linearisations. This approach covers a large number of computable non-linear real functions such as polynomials, rational or trigonometrical functions and beyond. A prototypical implementation has been evaluated on several non-linear SMT-LIB examples and the results have been compared with state-of-the-art SMT solvers.Comment: 17 pages, 3 figures; accepted at FroCoS 2019; software available at <http://informatik.uni-trier.de/~brausse/ksmt/

Mathematical model and implementation of rational processing

Precision in computations is a considerable challenge to adequately addressing many current scientific or engineering problems. The way in which the numbers are represented constitutes the first step to compute them and determines the validity of the results. The aim of this research is to provide a formal framework and a set of computational primitives to address high precision problems of mathematical calculation in engineering and numerical simulation. The main contribution of this research is a mathematical model to build an exact arithmetical unit able to represent without error rational numbers in positional notation system. The functions under consideration are addition and multiplication because they form an algebraic commutative ring which contains a multiplicative inverse for every non-zero element. This paper reviews other specialized arithmetic units based on existing formats to show ways to make high precision computing. It is proposed a formal framework of the whole arithmetic architecture in which the operators are based. Then, the design of the addition operation is detailed and its hardware implementation is described. Finally, extensive evaluation of this operator is performed to prove its ability for exact processing

Exact Numerical Processing

Paper submitted to Euromicro Symposium on Digital Systems Design (DSD), Belek-Antalya, Turkey, 2003.A model of an exact arithmetic processing is presented. We describe a representation format that gives us a greater expressive capability and covers a wider numerical set. The rational numbers are represented by means of fractional notation and explicit codification of its periodic part. We also give a brief description of exact arithmetic operations on the proposed format. This model constitutes a good alternative for the symbolic arithmetic, in special when numerical exact values are required. As an example, we show an application of the exact numerical processing to calculate the perpendicular vector to another one for aerospace purposes.This work is being backed by grant DPI2002-04434-C04-01 from the Ministerio de Ciencia y Tecnología of the Spanish Government