This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much as possible intermediate computation, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of several polynomials. This powerful reduction mechanism is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear algebra used in Computer Algebra are reviewed together with other methods coming from the numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical comparison of a first implementation of this algorithm with other well known programs. This comparison pays careful attention to methodology issues. All the benchmarks and CPU times used in this paper are frequently updated and available on a Web page. Even though the new algorithm does not improve the worst case complexity it is several times faster than previous implementations both for integers and modulo computations. 1

In this paper, we describe a scalable parallel algorithm for sparse matrix factorization, analyze their performance and scalability, and present experimental results for up to 1024 processors on a Cray T3D parallel computer. Through our analysis and experimental results, we demonstrate that our algorithm substantially improves the state of the art in parallel direct solution of sparse linear systems—both in terms of scalability and overall performance. It is a well known fact that dense matrix factorization scales well and can be implemented efficiently on parallel computers. In this paper, we present the first algorithm to factor a wide class of sparse matrices (including those arising from two- and three-dimensional finite element problems) that is asymptotically as scalable as dense matrix factorization algorithms on a variety of parallel architectures. Our algorithm incurs less communication overhead and is more scalable than any previously known parallel formulation of sparse matrix factorization. Although, in this paper, we discuss Cholesky factorization of symmetric positive definite matrices, the algorithms can be adapted for solving sparse linear least squares problems and for Gaussian elimination of diagonally dominant matrices that are almost symmetric in structure. An implementation of our sparse Cholesky factorization algorithm delivers up to 20 GFlops on a Cray T3D for medium-size structural engineering and linear programming problems. To the best of our knowledge,

Sparse matrix factorization algorithms for general problems are typically characterized by irregular memory access patterns that limit their performance on parallel-vector supercomputers. For symmetric problems, methods such as the multifrontal method avoid indirect addressing in the innermost loops by using dense matrix kernels. However, no efficient LU factorization algorithm based primarily on dense matrix kernels exists for matrices whose pattern is very unsymmetric. We address this deficiency and present a new unsymmetric-pattern multifrontal method based on dense matrix kernels. As in the classical multifrontal method, advantage is taken of repetitive structure in the matrix by factorizing more than one pivot in each frontal matrix thus enabling the use of Level 2 and Level 3 BLAS. The performance is compared with the classical multifrontal method and other unsymmetric solvers on a CRAY YMP.

We consider the solution of both symmetric and unsymmetric systems of sparse linear equations. A new parallel distributed memory multifrontal approach is described. To handle numerical pivoting efficiently, a parallel asynchronous algorithm with dynamic scheduling of the computing tasks has been developed. We discuss some of the main algorithmic choices and compare both implementation issues and the performance of the LDL T and LU factorizations. Performance analysis on an IBM SP2 shows the efficiency and the potential of the method. The test problems used are from the Rutherford-Boeing collection and from the PARASOL end users.

Abstract. We introduce block versions of the multielimination incomplete LU (ILUM) factorization preconditioning technique for solving general sparse unstructured linear systems. These preconditioners have a multilevel structure and, for certain types of problems, may exhibit properties that are typically enjoyed by multigrid methods. Several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed. The advantages of block ILUM over point ILUM include increased robustness and efficiency. We compare several versions of the block ILUM, point ILUM, and the dual-threshold-based ILUT preconditioners. In particular, tests with some convection-diffusion problems show that it may be possible to obtain convergence that is nearly independent of the Reynolds number as well as of the grid size.

Standard preconditioning techniques based on incomplete LU (ILU) factorizations offer a limited degree of parallelism, in general. A few of the alternatives advocated so far consist of either using some form of polynomial preconditioning, or applying the usual ILU factorization to a matrix obtained from a multicolor ordering. In this paper we present an incomplete factorization technique based on independent set orderings and multicoloring. We note that in order to improve robustness, it is necessary to allow the preconditioner to have an arbitrarily high accuracy, as is done with ILUs based on threshold techniques. The ILUM factorization described in this paper is in this category. It can be viewed as a multifrontal version a Gaussian elimination procedure with threshold dropping which has a high degree of potential parallelism. The emphasis is on methods that deal specifically with general unstructured sparse matrices such as those arising from finite element methods on un...

In this paper, we consider the problem of designing a dynamic scheduling strategy that takes into account both workload and memory information in the context of the parallel multifrontal factorization. The originality of our approach is that we base our estimations (work and memory) on a static optimistic scenario during the analysis phase. This scenario is then used during the factorization phase to constrain the dynamic decisions. The task scheduler has been redesigned to take into account these new features. Moreover performance have been improved because the new constraints allow the new scheduler to make optimal decisions that were forbidden or too dangerous in unconstrained formulations. Performance analysis show that the memory estimation becomes much closer to the memory effectively used and that even in a constrained memory environment we decrease the factorization time with respect to the initial approach.

A new method for sparse LU factorization is presented that combines a column pre-ordering strategy with a right-looking unsymmetric-pattern multifrontal numerical factorization. The column ordering is selected to give a good a priori upper bound on fill-in and then refined during numerical factorization (while preserving the bound). Pivot rows are selected to maintain numerical stability and to preserve sparsity. The method analyzes the matrix and automatically selects one of three pre-ordering and pivoting strategies. The number of nonzeros in the LU factors computed by the method is typically less than or equal to those found by a wide range of unsymmetric sparse LU factorization methods, including left-looking methods and prior multifrontal methods.

This dissertation presents new techniques for solving large sparse unsymmetric linear systems on high performance computers, using Gaussian elimination with partial pivoting. The efficiencies of the new algorithms are demonstrated for matrices from various fields and for a variety of high performance machines. In the first part we discuss optimizations of a sequential algorithm to exploit the memory hierarchies that exist in most RISC-based superscalar computers. We begin with the left-looking supernode-column algorithm by Eisenstat, Gilbert and Liu, which includes Eisenstat and Liu's symmetric structural reduction for fast symbolic factorization. Our key contribution is to develop both numeric and symbolic schemes to perform supernodepanel updates to achieve better data reuse in cache and floating-point register...

This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g., payment of royalties). IBM Research Division Almaden \Delta Austin \Delta China \Delta Delhi \Delta Haifa \Delta Tokyo \Delta Watson \Delta Zurich Recent Advances in Direct Methods for Solving Unsymmetric Sparse Systems of Linear Equations Anshul Gupta IBM T.J. Watson Research Center During the past few years, algorithmic improve