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

  • Akira ImakuraEmail author
  • Yusaku Yamamoto
Special Feature: Original Paper International Workshop on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2018)
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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 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.

Keywords

Modified Gram–Schmidt orthogonalization Non-standard inner product Efficient implementations Error bounds 

Notes

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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Engineering, Information and SystemsUniversity of TsukubaTsukubaJapan
  2. 2.Department of Communication Engineering and InformaticsThe University of Electro-CommunicationsTokyoJapan

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