

Preprint 47/2004
2-matrix arithmetics in linear complexity
Steffen Börm
Contact the author: Please use for correspondence this email.
Submission date: 22. Jul. 2004 (revised version: September 2005)
Pages: 32
published in: Computing, 77 (2006) 1, p. 1-28
DOI number (of the published article): 10.1007/s00607-005-0146-y
Bibtex
MSC-Numbers: 65F30
Keywords and phrases: hierarchical matrices, formatted matrix operations
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Abstract:
For hierarchical matrices, approximations of the matrix-matrix sum and product
can be computed in almost linear complexity, and using these matrix operations
it is possible to construct the matrix inverse, efficient preconditioners or
solutions of certain matrix equations.
-matrices are a variant of hierarchical matrices that allow us
to perform certain operations, like the matrix-vector product, in ``true''
linear complexity, but until now it was not clear whether matrix arithmetic
operations could also reach this, in some sense optimal, complexity.
We present algorithms that compute the best-approximation of the sum and
product of two -matrices in a prescribed
-matrix format, and we prove that this computation can
be accomplished in linear complexity.