Efficient parallel solution of linear systems
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Fast parallel orthogonalization
ACM SIGACT News
Complexity of parallel matrix computations
Theoretical Computer Science
Processor efficient parallel solution of linear systems over an abstract field
SPAA '91 Proceedings of the third annual ACM symposium on Parallel algorithms and architectures
An introduction to parallel algorithms
An introduction to parallel algorithms
Space and time efficient implementations of parallel nested dissection
SPAA '92 Proceedings of the fourth annual ACM symposium on Parallel algorithms and architectures
Fast and efficient parallel solution of sparse linear systems
SIAM Journal on Computing
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Synthesis of Parallel Algorithms
Synthesis of Parallel Algorithms
On Wiedemann's Method of Solving Sparse Linear Systems
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Approximate algorithms to derive exact solutions to systems of linear equations
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Fast and Efficient Parallel Algorithms for the Exact Inversion of Integer Matrices
Proceedings of the Fifth Conference on Foundations of Software Technology and Theoretical Computer Science
Work efficient parallel solution of Toeplitz systems and polynomial GCD
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
On the parallel complexity of matrix factorization algorithms
Proceedings of the ninth annual ACM symposium on Parallel algorithms and architectures
Work-efficient matrix inversion in polylogarithmic time
Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures
Hi-index | 0.00 |
Known polylog parallel algorithms for the solution of linear systems and related problems require computation of the characteristic polynomial or related forms, which are known to be highly unstable in practice. However, matrix factorizations of various types, bypassing computation of the characteristic polynomial, are used extensively in sequential numerical computations and are essential in many applications.This paper gives new parallel methods for various exact factorizations of several classes of matrices. We assume the input matrices are n × n with either integer entries of size ≤ 2n&ogr;(1). We make no other assumption on the input. We assume the arithmetic PRAM model of parallel computation. Our main result is a reduction of the known parallel time bounds for these factorizations from O(log3n) to O(log2n). Our results are work efficient; we match the best known work bounds of parallel algorithms with polylog time bounds, and are within a log n factor of the work bounds for the best known sequential algorithms for the same problems.The exact factorizations we compute for symmetric positive definite matrices include:1. recursive factorization sequences and trees,2. LU factorizations,3. QR factorizations, and4. reduction to upper Hessenberg form.The classes of matrices for which we can efficiently compute these factorizations include:1. dense matrices, in time O(log2n) with processor bound P(n) (the number of processors needed to multiply two n × n matrices in O(log n time),,2. block diagonal matrices, in time O(log2b with P(b)n/b processors,3. sparse matrices which are s(n)-separable (recursive factorizations only), in time O(log2n) with P(s(n)) processors where s(n) is of the form n&ngr; for 0 4. banded matrices, in parallel time O((logn) log b) with P(b)n/b processors.Our factorizations also provide us similarly efficient algorithms for exact computation (given arbitrary rational matrices that need not be symmetric positive definite) of the following:1. solution of the corresponding linear systems,2. the determinant,3. the inverse.Thus our results provide the first known efficient parallel algorithms for exact solution of these matrix problems, that avoids computation of the characteristic polynomial or related forms. Instead we use a construction which modifies the input matrix, which may initially have arbitrary condition, so as to have condition nearly 1, and then applies a multilevel, pipelined Newton iteration, followed by a similar multilevel, pipelined Hensel Lifting.