Advertisement

Accuracy Improvement of the Shifted Block BiCGGR Method for Linear Systems with Multiple Shifts and Multiple Right-Hand Sides

Conference paper
First Online:
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 117)

Abstract

We consider solving linear systems with multiple shifts and multiple right-hand sides. In order to solve these linear systems efficiently, we develop the Shifted Block BiCGGR method. This method is based on the shift-invariance property of Block Krylov subspaces. Using this property, the Shifted systems can be solved in the process of solving the Seed system without matrix-vector multiplications. The Shifted Block BiCGGR method can generate high accuracy approximate solutions of the Seed system. However, the accuracy of the approximate solutions of the Shifted systems may deteriorate due to the error of the matrix multiplications appearing in the algorithm. In this paper, we improve the accuracy of the approximate solutions of the Shifted systems generated by the Shifted Block BiCGGR method.

This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

This work was partly supported by JSPS KAKENHI Grant Numbers 15K15996, 25286097, 25870099, and University of Tsukuba Basic Research Support Program Type A.

References

  1. 1.
  2. 2.
    Davis, T.: The University of Florida Sparse Matrix Collection. http://www.cise.ufl.edu/research/sparse/matrices/
  3. 3.
    Dubrulle, A.A.: Retooling the method of block conjugate gradients. Electron. Trans. Numer. Anal. 12, 216–233 (2001)MathSciNetzbMATHGoogle Scholar
  4. 4.
    El Guennouni, A., Jbilou, K., Sadok, H.: A block version of BiCGSTAB for linear systems with multiple right-hand sides. Electron. Trans. Numer. Anal. 16, 129–142 (2003)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Frommer, A.: BiCGSTAB(ℓ) for families of shifted linear systems. Computing 70, 87–109 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Frommer, A., Glassner, U.: Restarted GMRES for shifted linear systems. SIAM J. Sci. Comput. 19(1), 15–26 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    O’Leary, D.P.: The block conjugate gradient algorithm and related methods. Linear Algebra Appl. 29, 293–322 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159, 119–128 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Soodhalter, K.M.: Block Krylov subspace recycling for shifted systems with unrelated right-hand sides. SIAM J. Sci. Comput. 38(1), A302–A324 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Tadano, H., Sakurai, T., Kuramashi, Y.: Block BiCGGR: a new block Krylov subspace method for computing high accuracy solutions. JSIAM Lett. 1, 44–47 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of TsukubaTsukubaJapan

Personalised recommendations

Cite paper