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arXiv · 1703.10440

Efficient implementations of the modified Gram-Schmidt orthogonalization with a non-standard inner product

Akira Imakura·Yusaku Yamamoto
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Abstract

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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Akira Imakura, Yusaku Yamamoto. 2017-03-30. Efficient implementations of the modified Gram-Schmidt orthogonalization with a non-standard inner product. https://arxiv.org/abs/1703.10440

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