Parallel Solution of Narrow Banded Diagonally Dominant Linear Systems

Mikkelsen, Carl Christian Kjelgaard; Kågström, Bo

doi:10.1007/978-3-642-28145-7_28

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2012

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Cited by 2 publications

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“…and this estimate is tight [7]. This general estimate carries to the ScaLAPACK case, but it does not reflect the banded structure of the odd numbered diagonal blocks.…”

Section: The Algorithmmentioning

confidence: 76%

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Incomplete Cyclic Reduction of Banded and Strictly Diagonally Dominant Linear Systems

Mikkelsen

Kågström

2012

Parallel Processing and Applied Mathematics

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Abstract. The ScaLAPACK library contains a pair of routines for solving banded linear systems which are strictly diagonally dominant by rows. Mathematically, the algorithm is complete block cyclic reduction corresponding to a particular block partitioning of the system. In this paper we extend Heller's analysis of incomplete cyclic reduction for block tridiagonal systems to the ScaLAPACK case. We obtain a tight estimate on the significance of the off diagonal blocks of the tridiagonal linear systems generated by the cyclic reduction algorithm. Numerical experiments illustrate the advantage of omitting all but the first reduction step for a class of matrices related to high order approximations of the Laplace operator.

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“…and this estimate is tight [7]. This general estimate carries to the ScaLAPACK case, but it does not reflect the banded structure of the odd numbered diagonal blocks.…”

Section: The Algorithmmentioning

confidence: 76%

“…The new results are derived in Section 3. We have already proved Theorem 1 for tridiagonal matrices (k = 1) in a previous paper [7]. In this paper, we derive Theorem 2 and use it to prove Theorem 1 in the general case.…”

Section: Introductionmentioning

confidence: 81%