We give an overview on Paclib, a library for parallel algebraic computation on shared memory multiprocessors. Paclib is essentially a package of C functions that provide the basic objects and methods of computer algebra in a parallel context. The Paclib programming model supports concurrency, shared memory communication, non-determinism and speculative parallelism. The system is based on a heap management kernel with parallelized garbage collection that is portable among most Unix machines. We present the successful application of paclib for the parallelization of several algebraic algorithms and discuss the achieved results. 1 Introduction Scientific computing is a rich source of challenging problems such as the solution of systems of partial differential equations. Classical numerical methods operate with efficient finite-precision (floating point) arithmetic and thus quickly yield approximative solutions. However, often one is also interested in certain qualitative aspects like s...

. This paper describes the runtime kernel of Paclib, a new system for parallel algebraic computation on shared memory computers. Paclib has been developed as a professional tool for the simple design and efficient implementation of parallel algorithms in computer algebra and related areas. It provides concurrency, shared memory communication, non-determinism, speculative parallelism, streams and a parallelized garbage collection. We explain the main design decisions as motivated by the special demands of algebraic computation and give several benchmarks that demonstrate the performance of the system. Paclib has been implemented on a Sequent Symmetry multiprocessor and is portable to other shared memory machines and workstations. 1 Introduction Computer algebra is that branch of computer science that aims to provide exact solutions of scientific problems. Research results of this area are e.g. algorithms for symbolic integration, polynomial factorization or the exact solution of algeb...

We describe the design of a parallel algorithm for the exact solution of linear equation systems with integer coefficients and the implementation of this algorithm on a shared memory multiprocessor. An efficient solution of the original problem is difficult since the coefficients grow during the computation and arithmetic becomes very time-consuming. Therefore we transform the problem into a problem of determinant computation and apply a modular approach: the system is mapped into several finite fields where the determinants can be efficiently computed. The subresults are combined to yield the original determinants and to compute the solutions of the system. Several parallel versions of this algorithm have been developed and implemented on a shared memory multiprocessor. The programs are applied to equation systems of different characteristics and the results are analyzed and compared. Keywords: Parallel algorithms, scientific computing, computer algebra, shared memory machines. Fund...

. In this article techniques borrowed from Computer Algebra (Grobner Bases) are applied to deal with Medical Appropriateness Criteria including uncertainty. The knowledge was provided in the format of a table. A previous translation of the table into the format of a "Rule Based System" (denoted RBS) based on a three-valued logic is required beforehand to apply these techniques. Once the RBS has been obtained, we apply a Computer Algebra based inference engine, both to detect anomalies and to infer new knowledge. A specific set of criteria for coronary artery surgery (originally presented in the form of a table) is analyzed in detail. Keywords. Verification. Inference Engines. RBSs in Medicine. Grobner Bases. Top i cs : Integration of Logical Reasoning and Computer Algebra. Symbolic Computation for Expert Systems and Machine Learning. 1 Introduction "Appropriateness criteria" are ratings of the appropriateness for a given diagnostic or therapeutic procedure. Whereas other policies such...

This article, along with [Elkadi and Mourrain 1996], explain the correlation between residue theory and the Dixon matrix, which yields an alternative method for studying and approximating all common solutions. In 1916, Macaulay [1916] constructed a matrix whose determinant is a multiple of the classical resultant for n homogeneous polynomials in n variables. The Macaulay matrix si16 multaneously generalizes the Sylvester matrix and the coefficient matrix of a system of linear equations [Kapur and Lakshman Y. N. 1992]. As the Dixon formulation, the Macaulay determinant is a multiple of the resultant. Macaulay, however, proved that a certain minor of his matrix divides the matrix determinant so as to yield the exact resultant in the case of generic homogeneous polynomials. Canny [1990] has invented a general method that perturbs any polynomial system and extracts a non-trivial projection operator. Using recent results pertaining to sparse polynomial systems [Gelfand et al. 1994, Sturmfels 1991], a matrix formula for computing the sparse resultant of n + 1 polynomials in n variables was given by Canny and Emiris [1993] and consequently improved in [Canny and Pedersen 1993, Emiris and Canny 1995]. The determinant of the sparse resultant matrix, like the Macaulay and Dixon matrices, only yields a projection operation, not the exact resultant. Here, sparsity means that only certain monomials in each of the n + 1 polynomials have non-zero coefficients. Sparsity is measured in geometric terms, namely, by the Newton polytope