{\bf (Revision of CS-1994-08.)} Work Efficient Parallel Solution of Toeplitz and Other Structured Linear Systems
Authors: H J Reif
Venue:  (1995)

A matrix $A = [a_{ij}]$ is {\em Toeplitz} if $a_{i,j} = a_{i+k,j+k}$ for each $k$ where the matrix elements are defined. We define %(this is a slight simplification of standard definitions) an $n \times n$ matrix to have {\it displacement rank} $\delta$ if it can be written as the sum of $\delta$ terms, where each term is either (i) the product of a lower triangular Toeplitz matrix and an upper triangular Toeplitz matrix or (ii) the product of an upper triangular Toeplitz matrix and a lower triangular Toeplitz matrix. There are known efficient sequential algorithms \cite{BA80,BGY80} (see report bibliography) for inverse, determinant, linear system solution, factorization, and finding the rank for the case of Toeplitz matrices and matrices of bounded displacement rank, but there are no such results for efficient parallel algorithms. %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 In this paper, we show that certain structured linear systems can be solved exactly and efficiently in parallel, dropping these processor bounds to linear, without significant slowdown. We give much improved parallel algorithms for the exact solution and factorization, determinant, inverse, and finding rank of various structured matrices: in particular Toeplitz and matrices of bounded displacement rank. % and their generalizations. We apply this result to efficient parallel algorithms for the following problems in the same parallel time $O(\log^{2} n)$ and $n \log^{\omega} n$ processor bounds: \begin{enumerate} \item polynomial greatest common divisors (GCD) and extended GCD, \item polynomial resultant, \item Pad\''e approximants of rational functions \cite{P1892,G72} (see report bibliography), and \item shift register synthesis and BCH decoding problems. \end{enumerate} %and with a factor of $O(\log n)$ time increase and the same processor %bounds, % we also solve: %\begin{enumerate} %\item the real root problem: %finding all the roots of a polynomial with only real roots, %\item the symmetric tridiagonal eigenvalue problem: % finding all the eigenvalue of a symmetric tridiagonal $n \times n$ matrix. %\end{enumerate} Previously, the best parallel algorithms \cite{PR87,P88,Pa90} (see report bibliography) for these problems required $\Omega(\log^2 n)$ time with $n^2$ processors (or $O(\log^{O(1)} n)$ time using at least $\Omega(n^2/\log^{O(1)} n)$ processors), whereas the best sequential time was $O(n \log^2 n)$. Our results {\bf drop by a linear factor the best previous processor bounds} for polylog time parallel algorithms for all these problems, and our results are within a polylog factor of work compaired to the best sequential work bounds of 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