{\bf (Revision of CS-1994-07.)} $0$($log^2 n$) Time Efficient Parallel Factorization of Dense, Sparse Separable, and Banded Matrices
Authors: H J Reif
Venue:  (1995)

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. %begin resubmit SICOMP edit 0.1 We assume the input matrices are $n \times n$ with either integer entries of magnitude $\le 2^{\beta}$ or rational entries expressible as a ratio of integers of magnitude $\le 2^{\beta},$ where $\beta \le n^{O(1)}.$ We make no other assumption on the input. We assume the arithmetic PRAM model of parallel computation. %end resubmit SICOMP edit 0.1 Our main result is a reduction of the known parallel time bounds for these factorizations from $\Omega(\log^3 n)$ to $O(\log^2 n)$. 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. Our computations require bit precision $O(n(\beta+\log n)),$ which is the {\bf asymptotically optimal bit precision} for $\beta \ge \log n$ since the determinant, exact $LU$ factorization and matrix inverse requires bit precision at least $\Omega(n \beta)$. The exact factorizations we compute for symmetric positive definite matrices include: recursive factorization sequences and trees, $LU$ factorizations, $QR$ factorizations, and reduction into upper Hessenberg form. The classes of matrices for which we can efficiently compute these factorizations include: \begin{enumerate} \item dense matrices, in time $O(\log^2 n)$ with processor bound $P(n)$ (the number of processors needed to multiply two $n \times n $ matrices in $O(\log n)$ time), \item block diagonal matrices, in time $O(\log^2 b)$ with $P(b)n/b$ processors, \item sparse matrices which are $s(n)$-separable (recursive factorizations only), in time $O(\log^2 n)$ with $P(s(n))$ processors where $s(n)$ is of the form $n^{\gamma}$ for $0> \gamma >1$, and \item banded matrices, in parallel time $O((\log n)\log b)$ with $P(b)n/b$ processors. \end{enumerate} 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: solution of the corresponding linear systems, the determinant, 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 appl The ACM Portal is published by the Association for Computing Machinery. Copyright © 2010 ACM, Inc. Terms of Usage Privacy Policy Code of Ethics Contact Us Useful downloads: Adobe Acrobat QuickTime Windows Media Player Real Player