NDLI: Parallel Solution of Certain Toeplitz Linear Systems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Bini, Dario
Copyright Year	1984
Abstract	Using the concept of approximate algorithm it is shown that $6\log n + 6$ parallel steps and $2n$ processors suffice to approximate, with any precision, the solution of a linear system with an $n \times n$ triangular Toeplitz matrix A. Moreover, $7\log n + 7$ steps are sufficient for an exact computation, whereas the number of processors is increased to $( \frac{5}{2} )n^2 $. If A is also banded and k is its bandwidth, the number of processors is reduced to $( \frac{5}{2} )n(k + 1)$. Two applications are shown. It is proved that if B is any matrix belonging to the algebra generated over the complex field by a given $n \times n$ matrix, then the system $Bx = b$ can be solved with no more than $9\log n + 4$ steps with $O(n^2 )$ processors. It is proved that, given a Toeplitz matrix $A = (a_{i,j} )$ such that $a_{i,j} = 0$ if $i - j > k$ or $j - i > h$, $a_{k,1} \ne 0$, then $13\log n + O(\log ^2 k)$ steps and $\max ( (\frac{5}{2} )n(k + h), n(n + 1)/ 2 )$ processors are sufficient to solve the system $Ax = b$. Such algorithms work under the sole condition $\det A \ne 0$.
Starting Page	268
Ending Page	276
Page Count	9
File Format	PDF
ISSN	00975397
DOI	10.1137/0213019
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	2
Volume Number	13
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-13
Access Restriction	Subscribed
Subject Keyword	approximate algorithms band matrices parallel matrix inversion Toeplitz matrices
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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