Advertisement

Calcolo

, Volume 54, Issue 1, pp 455–470 | Cite as

Aitken’s method for estimating bilinear forms arising in applications

  • Paraskevi Fika
  • Marilena MitrouliEmail author
Article

Abstract

In the present work we study how the Aitken’s method can be applied to the sequence of moments \(\{c_k=(x, A^{k}x),~k \in {\mathbb {Z}}\},\) of a given symmetric positive definite matrix \(A \in {\mathbb {R}}^{p \times p},\) for the prediction of possible unknown terms of this sequence. Direct estimation of \(c_k\) leads to approximating bilinear quantities i.e. \((y, A^{k}x)\) as well. By employing Taylor series expansion for appropriate f,  prediction of \(y^Tf(A)x\) can be achieved. Estimates for many useful linear algebra quantities can be derived by appropriately selecting the vectors y and x. Numerical examples concerning such applications are presented. The estimates are illustrated through numerical examples executed on the high-performance computing system ARIS.

Keywords

Prediction Aitken’s process Moments Bilinear form 

Mathematics Subject Classification

65F15 65F30 65B05 15A63 

Notes

Acknowledgments

The authors would like to thank the reviewers of this paper for their helpful comments that resulted in improving the presentation of the manuscript. Also, the authors are thankful to Prof. Claude Brezinski for constructive discussions and for bringing [6] to their attention. Sincere thanks also to Prof. Giuseppe Rodriguez for providing them with the adjacency matrices tested in numerical examples, that represent real-world networks. In addition, the authors acknowledge GRNET S.A. for awarding them access to the Greek national high performance systems, ARIS. This research is being funded by the Department of Mathematics and the Social Account of Research Grants of the National and Kapodistrian University of Athens, Code Number 70/3/13297.

References

  1. 1.
    Atkinson, K.: An introduction to numerical analysis, 2nd edn. Wiley, New York (1989)Google Scholar
  2. 2.
    Bai, Z., Fahey, M., Golub, G.: Some large scale computation problems. J. Comput. Appl. Math. 74, 71–89 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bellalij, M., Reichel, L., Rodriguez, G., Sadok, H.: Bounding matrix functionals via partial global block Lanczos decomposition. Appl. Numer. Math. 94, 127–139 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benzi, M., Boito, P.: Quadrature rule-based bounds for functions of adjacency matrices. Linear Algebra Appl. 433, 637–652 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benzi, M., Razouk, N.: Decay bounds and O(n) algorithms for approximating functions of sparse matrices. Electron. Trans. Numer. Anal. 28, 16–39 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brezinski, C.: Prediction properties of some extrapolation methods. Appl. Numer. Math. 61, 457–462 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brezinski, C.: Error estimates for the solution of linear systems. SIAM J. Sci. Comput. 21, 764–781 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brezinski, C., Fika, P., Mitrouli, M.: Estimations of the trace of powers of positive self-adjoint operators by extrapolation of the moments. Electron. Trans. Numer. Anal. 39, 144–155 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Brezinski, C., Raydan, M.: Cauchy–Schwartz and Kantorovich type inequalities for scalar and matrix moment sequences. Adv. Comput. Math. 26, 71–80 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Estrada, E., Rodríguez-Velázquez, J.A.: Subgraph centrality in complex networks. Phys. Rev. E 71, 056103 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fenu, C., Martin, D., Reichel, L., Rodriguez, G.: Block Gauss and anti-Gauss quadrature with application to networks. SIAM J. Matrix Anal. Appl. 34, 1655–1684 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fika, P., Mitrouli, M., Roupa, P.: Estimates for the bilinear form \({\rm x}^T{\rm A}^{-1}{\rm y}\) with applications to linear algebra problems. Electron. Trans. Numer. Anal. 43, 70–89 (2014)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fika, P., Mitrouli, M.: Estimation of the bilinear form \({\rm y}^{*} {\rm f(A)} {\rm x}\) for Hermitian matrices. Linear Algebra Appl. doi: 10.1016/j.laa.2015.08.033
  14. 14.
    The University of Florida Sparse Matrix Collection, http://www.cise.ufl.edu/research/sparse/matrices/
  15. 15.
    Golub, G.H., Meurant, G.: Matrices, Moments and Quadrature with Applications. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar
  16. 16.
    Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)CrossRefzbMATHGoogle Scholar
  17. 17.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefzbMATHGoogle Scholar
  18. 18.
    Taylor, A., Higham, D.J.: CONTEST: toolbox files and documentation, http://www.mathstat.strath.ac.uk/research/groups/numerical_analysis/contest/toolbox

Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece

Personalised recommendations