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)


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.


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



  1. 1.
    Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)CrossRefzbMATHGoogle Scholar
  2. 2.
    Dubrulle, A.A.: Retooling the method of block conjugate gradients. ETNA 12, 216–233 (2001)MathSciNetzbMATHGoogle Scholar
  3. 3.
    ELSES matrix library. Accessed 15 Apr 2018
  4. 4.
    Essai, A.: Weighted FOM and GMRES for solving nonsymmetric linear systems. Numer. Algorithms 18, 277–292 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gulliksson, M.: On the modified Gram–Schmidt algorithm for weighted and constrained linear least squares problems. BIT Numer. Math. 35, 453–468 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)CrossRefzbMATHGoogle Scholar
  7. 7.
    Imakura, A., Du, L., Tadano, H.: A weighted block GMRES method for solving linear systems with multiple right-hand sides. JSIAM Lett. 5, 65–68 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Imakura, A., Sakurai, T.: Block Krylov-type complex moment-based eigensolvers for solving generalized eigenvalue problems. Numer. Algorithm 75, 413–433 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Knyazev, A.V.: Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 517–541 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lowery, B.R., Langou, J.: Stability analysis of QR factorization in an oblique inner product. arXiv:1401.5171 [math.NA] (2014)
  11. 11.
    Rozložník, M., Tůma, M., Smoktunowicz, A., Kopal, J.: Numerical stability of orthogonalization methods with a non-standard inner product. BIT 52, 1035–1058 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Smoktunowicz, A., Barlow, J.L., Langou, J.: A note on the error analysis of classical Gram–Schmidt. Numer. Math. 105, 299–313 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Stewart, G.W.: Matrix Algorithms Volume II: Eigensysmtems. SIAM, Philadelphia (2001)CrossRefGoogle Scholar
  14. 14.
    Trefethen, L.N.: Householder triangularization of a quasimatrix. IMA J. Numer. Anal. 30, 887–897 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Trefethen, L.N., Bau III, D.: Numerical Linear Algebra. SIAM, Philadelphia (1997)CrossRefzbMATHGoogle Scholar
  16. 16.
    Yamamoto, Y., Nakatsukasa, Y., Yanagisawa, Y., Fukaya, T.: Roundoff error analysis of the CholeskyQR2 algorithm in an oblique inner product. JSIAM Lett. 8, 5–8 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yanagisawa, Y., Nakatsukasa, Y., Fukaya, T.: Cholesky-QR and Householder-QR factorizations in nonstandard inner product spaces. International Workshop on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2014) (2014)Google Scholar
  18. 18.
    Zhao, J.Q.: S-Orthogonal QR decomposition algorithms on multicore systems. ProQuest Dissertations Publishing, University of California, Davis (2013)Google Scholar

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© 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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