Efficient implementations of the modified Gram-Schmidt orthogonalization with a non-standard inner product
The modified Gram-Schmidt (MGS) orthogonalization is one of the most well-used algorithms for computing the thin QR factorization. MGS can be straightforwardly extended to a non-standard inner product with respect to a symmetric positive definite matrix $A$. For the thin QR factorization of an $m \t...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
30.03.2017
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Subjects | |
Online Access | Get full text |
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Summary: | The modified Gram-Schmidt (MGS) orthogonalization is one of the most
well-used algorithms for computing the thin QR factorization. MGS can be
straightforwardly extended to a non-standard inner product with respect to a
symmetric positive definite matrix $A$. For the thin QR factorization of an $m
\times n$ matrix with the non-standard inner product, a naive implementation of
MGS requires $2n$ matrix-vector multiplications (MV) with respect to $A$. In
this paper, we propose $n$-MV implementations: a high accuracy (HA) type and a
high performance (HP) type, of MGS. We also provide error bounds of the HA-type
implementation. Numerical experiments and analysis indicate that the proposed
implementations have competitive advantages over the naive implementation in
terms of both computational cost and accuracy. |
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DOI: | 10.48550/arxiv.1703.10440 |