On computing the determinant in small parallel time using a small number of processors
Information Processing Letters
How to multiply matrices faster
How to multiply matrices faster
The algebraic eigenvalue problem
The algebraic eigenvalue problem
ACM Transactions on Mathematical Software (TOMS)
Circuit simulation on the connection machine
DAC '87 Proceedings of the 24th ACM/IEEE Design Automation Conference
Using smoothness to achieve parallelism
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
A Strassen-Newton algorithm for high-speed parallelizable matrix inversion
Proceedings of the 1988 ACM/IEEE conference on Supercomputing
Separators in two and three dimensions
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
On-line learning of linear functions
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
SODA '91 Proceedings of the second annual ACM-SIAM symposium on Discrete algorithms
Space and time efficient implementations of parallel nested dissection
SPAA '92 Proceedings of the fourth annual ACM symposium on Parallel algorithms and architectures
A deterministic linear time algorithm for geometric separators and its applications
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
O(log2 n) time efficient parallel factorization of dense, sparse separable, and banded matrices
SPAA '94 Proceedings of the sixth annual ACM symposium on Parallel algorithms and architectures
Work efficient parallel solution of Toeplitz systems and polynomial GCD
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Highly Scalable Parallel Algorithms for Sparse Matrix Factorization
IEEE Transactions on Parallel and Distributed Systems
IEEE Transactions on Parallel and Distributed Systems
Moments of inertia and graph separators
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Parallel preprocessing and postprocessing in finite-element analysis on a multiprocessor computer
ACM '86 Proceedings of 1986 ACM Fall joint computer conference
Parallel Matrix Multiplication on a Linear Array with a Reconfigurable Pipelined Bus System
IEEE Transactions on Computers
An Overlaying Technique for Solving Linear Equations in Real-Time Computing
IEEE Transactions on Computers
IPDPS '01 Proceedings of the 15th International Parallel & Distributed Processing Symposium
Effectiveness of approximate inverse preconditioning by using the MR algorithm on an origin 2400
ICECT'03 Proceedings of the third international conference on Engineering computational technology
Fast and Scalable Parallel Matrix Computations on Distributed Memory Systems
IPDPS '05 Proceedings of the 19th IEEE International Parallel and Distributed Processing Symposium (IPDPS'05) - Papers - Volume 01
IEEE Transactions on Parallel and Distributed Systems
Formula dissection: A parallel algorithm for constraint satisfaction
Computers & Mathematics with Applications
Some polynomial and Toeplitz matrix computations
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
A parallel algorithm for finding a separator in planar graphs
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
The Journal of Supercomputing
Space-round tradeoffs for MapReduce computations
Proceedings of the 26th ACM international conference on Supercomputing
A Deterministic Linear Time Algorithm For Geometric Separators And Its Applications
Fundamenta Informaticae
Work-efficient matrix inversion in polylogarithmic time
Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures
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The most efficient known parallel algorithms for inversion of a nonsingular n × n matrix A or solving a linear system Ax = b over the rationals require &Ogr;(log n)2 time and M(n)n0.5 processors (where M(n) is the number of processors required in order to multiply two n × n rational matrices in time &Ogr;(log n).) Furthermore, all known polylog time algorithms for those problems are unstable: they require the calculation to be done with perfect precision; otherwise they give no results at all.This paper describes parallel algorithms that have good numerical stability and remain efficient as n grows large. In particular, we describe a quadratically convergent iterative method that gives the inverse (within the relative precision 2-nO(1)) of an n × n rational matrix A with condition ≤ n0(1) in &Ogr;(log n)2 time using M(n) processors. This is the optimum processor bound and the factor n0.5 improvement of known processor bounds for polylog time matrix inversion. It is the first known polylog time algorithm that is numerically stable. The algorithm relies on our method of computing an approximate inverse of A that involves &Ogr;(log n) parallel steps and n2 processors.Also, we give a parallel algorithm for solution of a linear system Ax = b with a sparse n × n symmetric positive definite matrix A. If the graph G(A) (which has n vertices and has an edge for each nonzero entry of A) is s(n)-separable, then our algorithm requires only &Ogr;((log n)(log s(n))2) time and |E| + M(s(n)) processors. The algorithm computes a recursive factorization of A so that the solution of any other linear system Ax = b′ with the same matrix A requires only &Ogr;(log n log s(n)) time and |E| + s(n)2 processors.