Abstract: "Although Gaussian elimination with partial pivoting is a robust algorithm to solve unsymmetric sparse linear systems of equations, it is difficult to implement efficiently on parallel machines, because of its dynamic and somewhat unpredictable way of generating work and intermediate results at run time. In this paper, we present an efficient parallel algorithm that overcomes this difficulty. The high performance of our algorithm is achieved through (1) using a graph reduction technique and a supernode-panel computational kernel for high single processor utilization, and (2) scheduling two types of parallel tasks for a high level of concurrency. One such task is factoring the independent panels on the disjoint subtrees in the column elimination tree of A. Another task is updating a panel by previously computed supernodes. A scheduler assigns tasks to free processors dynamically and facilitates the smooth transition between the two types of parallel tasks. No global synchronization is used in the algorithm. The algorithm is well suited for shared memory machines (SMP) with a modest number of processors. We demonstrate 4-7 fold speedups on a range of 8 processor SMPs, and more on larger SMPs. One realistic problem arising from a 3-D flow calculation achieves factorization rates of 1.0, 2.5, 0.8 and 0.8 Gigaflops, on the 12 processor Power Challenge, 8 processor Cray J90, 16 processor Cray J90, and 8 processor AlphaServer 8400, respectively."

Large sparse linear systems of equations are ubiquitous in science, engineering and beyond. This open access monograph focuses on factorization algorithms for solving such systems. It presents classical techniques for complete factorizations that are used in sparse direct methods and discusses the computation of approximate direct and inverse factorizations that are key to constructing general-purpose algebraic preconditioners for iterative solvers. A unified framework is used that emphasizes the underlying sparsity structures and highlights the importance of understanding sparse direct methods when developing algebraic preconditioners. Theoretical results are complemented by sparse matrix algorithm outlines. This monograph is aimed at students of applied mathematics and scientific computing, as well as computational scientists and software developers who are interested in understanding the theory and algorithms needed to tackle sparse systems. It is assumed that the reader has completed a basic course in linear algebra and numerical mathematics.

Abstract: "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 supernode-panel updates to achieve better data reuse in cache and floating-point registers. A further refinement, a two-dimensional matrix partitioning scheme, enhances performance for large matrices or machines with small caches. We conduct extensive performance evaluations on several recent superscalar architectures, such as the IBM RS/6000-590, MIPS R8000 and DEC Alpha 21164, and show that our algorithm is much faster than its predecessors. The advantage is particularly evident for large problems. In addition, we develop a detailed model to systematically choose a set of blocking parameters in the algorithm. The second part focuses on the design, implementation and performance analysis of a shared memory parallel algorithm based on our new serial algorithm. We parallelize the computation along the column dimension of the matrix, assigning one block of columns (a panel) to a processor. The parallel algorithm retains the serial algorithm's ability to reuse cached data. We develop a dynamic scheduling mechanism to schedule tasks onto available processors. One merit of this approach is the ability to balance work load automatically. The algorithm attempts to schedule independent tasks to different processors. When this is not possible in the later stage of factorization, a pipeline approach is used to coordinate dependent computations. We demonstrate that the new parallel algorithm is very efficient on shared memory machines with modest numbers of processors, such as the SGI Power Challenge, DEC AlphaServer 8400, and Cray C90/J90. We also develop performance models to study available concurrency and identify performance bottlenecks."

The subject of sparse matrices has its root in such diverse fields as management science, power systems analysis, surveying, circuit theory, and structural analysis. Efficient use of sparsity is a key to solving large problems in many fields. This book provides both insight and answers for those attempting to solve these problems.

Mathematics of Computing -- General.

Presents the fundamentals of sparse matrix algorithms to provide the requisite background. The book includes CSparse, a concise downloadable sparse matrix package that illustrates the algorithms and theorems presented in the book and equips readers with the tools necessary to understand larger and more complex software packages.

This volume is the first in a self-contained five-volume series devoted to matrix algorithms. It focuses on the computation of matrix decompositions--that is, the factorization of matrices into products of similar ones. The first two chapters provide the required background from mathematics and computer science needed to work effectively in matrix computations. The remaining chapters are devoted to the LU and QR decompositions--their computation and applications. The singular value decomposition is also treated, although algorithms for its computation will appear in the second volume of the series. The present volume contains 65 algorithms formally presented in pseudocode. Other volumes in the series will treat eigensystems, iterative methods, sparse matrices, and structured problems. The series is aimed at the nonspecialist who needs more than black-box proficiency with matrix computations. To give the series focus, the emphasis is on algorithms, their derivation, and their analysis. The reader is assumed to have a knowledge of elementary analysis and linear algebra and a reasonable amount of programming experience, typically that of the beginning graduate engineer or the undergraduate in an honors program. Strictly speaking, the individual volumes are not textbooks, although they are intended to teach, the guiding principle being that if something is worth explaining, it is worth explaining fully. This has necessarily restricted the scope of the series, but the selection of topics should give the reader a sound basis for further study.

Good,No Highlights,No Markup,all pages are intact, Slight Shelfwear,may have the corners slightly dented, may have slight color changes/slightly damaged spine.

Comprises 10 contributions that summarize the state of the art in the areas of high performance solutions of structured linear systems and structured eigenvalue and singular-value problems. Topics covered range from parallel solvers for sparse or banded linear systems to parallel computation of eigenvalues and singular values of tridiagonal and bidiagonal matrices. Specific paper topics include: the stable parallel solution of general narrow banded linear systems; efficient algorithms for reducing banded matrices to bidiagonal and tridiagonal form; a numerical comparison of look-ahead Levinson and Schur algorithms for non-Hermitian Toeplitz systems; and parallel CG-methods automatically optimized for PC and workstation clusters. Annotation copyrighted by Book News, Inc., Portland, OR