Skip to main content
SpringerLink
Log in

Accurate computations for eigenvalues of products of Cauchy-polynomial-Vandermonde matrices

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we consider the product eigenvalue problem for the class of Cauchy-polynomial-Vandermonde (CPV) matrices arising in a rational interpolation problem. We present the explicit expressions of minors of CPV matrices. An algorithm is designed to accurately compute the bidiagonal decomposition for strictly totally positive CPV matrices and their additive inverses. We then illustrate the sign regularity of the bidiagonal decomposition to show that all the eigenvalues of a product involving such matrices are computed to high relative accuracy. Numerical experiments are given to confirm the claimed high relative accuracy.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Boros, T., Kailath, T., Olshevsky, V.: Fast Algorithms for Solving Vandemonde and Chebyshev-Vandermonde Systems, Reprint, Information Systems Laboratory, Department of Electrical Engineering, Stanford University, Stanford CA (1994)

  2. Dailey, M., Dopico, F. M., Ye, Q.: Relative perturbation theory for diagonally dominant matrices. SIAM J. Matrix Anal. Appl. 35, 1303–1328 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dailey, M., Dopico, F. M., Ye, Q.: A new perturbation bound for the LDU factorization of diagonally dominant matrices. SIAM J. Matrix Anal. Appl. 35, 904–930 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Delgado, J., Peña, J. M.: Accurate computations with collocation matrices of rational bases. Appl. Math. Comput. 219, 4354–4364 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Delgado, J., Peña, J. M.: Accurate computations with collocation matrices of q-Bernstein polynomials. SIAM J. Matrix Anal. Appl. 36, 880–893 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Demmel, J., Kahan, W.: Accurate singular values of bidiagonal matrices. SIAM J. Sci Stat. Comp. 11, 873–912 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Demmel, J., Gu, M., Eisenstat, S., Slapničar, I., Veselić, K., Drmač, Z.: Computing the singular value decomposition with high relative accuracy. Linear Algebra Appl. 299, 21–80 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Demmel, J., Koev, P.: Accurate SVDs of weakly diagonally dominant M-matrices. Numer. Math. 98, 99–104 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Demmel, J., Koev, P.: The accurate and efficient solution of a totally positive generalized Vandermonde linear system. SIAM J. Matrix Anal. Appl. 27, 142–152 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Demmel, J., Koev, P.: Accurate and efficient evaluation of Schur and Jack functions. Math. Comput. 75, 223–239 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Demmel, J., Dumitriu, I., Holtz, O., Koev, P.: Accurate and efficient expression evaluation and linear algebra. Acta Numer. 17, 87–145 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fallat, S. M., Johnson, C. R.: Totally Nonnegative Matrices. Princeton University Press, Princeton (2011)

    Book  MATH  Google Scholar 

  13. Dopico, F. M., Koev, P.: Accurate symmetric rank revealing and eigendecompositions of symmetric structured matrices. SIAM J. Matrix Anal. Appl. 28, 1126–1156 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dopico, F. M., Koev, P.: Perturbation theory for the LDU factorization and accurate computations for diagonally dominant matrices. Numer. Math. 119, 337 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gasca, M., Martinez, J. J., Mühlbach, G.: Computation of rational interpolants with prescribed poles. J. Comput. Appl. Math 26, 297–309 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gasca, M., Peña, J. M.: Total positivity and Neville elimination. Linear Algebra Appl. 165, 25–44 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gasca, M., Peña, J. M.: Total positivity, QR factorization and Neville elimination. SIAM J. Matrix Anal. 14, 1132–1140 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gasca, M., Peña, J. M.: A matricial description of Neville elimination with applications to total positivity. Linear Algebra Appl. 202, 33–53 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gasca, M., Peña, J. M.: On factorizations of totally positive matrices, in Total positivity and its applications, Springer, Dordrecht, pp. 109–130 (1996)

  20. Higham, N. J.: Fast solution of Vandermonde-like systems involving orthogonal polynomials. IMA J. Numer. Anal. 8, 473–486 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. Higham, N. J.: Stability analysis of algorithms for solving confluent Vandermonde-like systems. SIAM J. Matrix Anal. Appl. 11, 23–41 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huang, R.: A periodic qd-Type reduction for computing eigenvalues of structured matrix products to high relative accuracy. J. Sci. Comput. 75, 1229–1261 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Junghanns, P., Oestreich, D.: Numerische Lösung des Staudammproblems mit Drainage. Z. Angew. Math. Mech. 69, 83–92 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kailath, T., Olshevsky, V.: Displacement structure approach to Chebyshev Vandermonde and related matrices. Integral Equ. Oper. Theory 22, 65–92 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Koev, P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27, 1–23 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29, 731–751 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Koev, P., Dopico, F. M.: Accurate eigenvalues of certain sign regular matrices. Linear Algebra Appl. 424, 435–447 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Macdonald, I. G.: Symmetric function and hall polynomials, 2nd edn. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  29. Mainar, E., Peña, J. M.: Accurate computations with collocation matrices of a general class of bases. Numer. Linear Algebra Appl. 25, e2184 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Marco, A., Martínez, J. J.: Accurate computations with Said-Ball-Vandermonde matrices. Linear Algebra Appl. 432, 2894–2908 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Marco, A., Martínez, J. J.: Accurate computations with totally positive Bernstein-Vandermonde matrices. Electron. J. Linear Algebra. 26, 357–380 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Marco, A., Martínez, J. J.: Bidiagonal decomposition of rectangular totally positive Said-Ball-Vandermonde matrices: error analysis, perturbation theory and applications. Linear Algebra Appl. 495, 90–107 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Marco, A., Martínez, J. J., Peña, J. M.: Accurate bidiagonal decomposition of totally positive Cauchy-Vandermonde matrices and applications. Linear Algebra Appl. 517, 63–84 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Martínez, J. J., Peña, J. M.: Factorizations of Cauchy-Vandermonde matrices. Linear Algebra Appl. 284, 229–237 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pan, V. Y.: Structured matrices and polynomials: unified superfast algorithms springer science and business media (2012)

  36. Phillips, G. M.: Interpolation and approximation by polynomials, Springer Science and Business Media (2003)

  37. Reichel, L., Opfer, G.: Chebyshev-Vandermonde systems. Math. Comp. 57, 703–721 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  38. Verde-Star, L.: Inverses of generalized Vandermonde matrices. J. Math. Anal. Appl. 131, 341–353 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  39. Weideman, J. A. C., Laurie, D. P.: Quadrature rules based on partial fraction expansions. Numer. Algorithms 24, 159–178 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the Editor and the anonymous referees for their valuable comments and suggestions which have helped to improve the overall presentation of the paper.

Funding

This research was supported by the National Natural Science Foundations of China (Grants No. 11871020 and 11471279), the Natural Science Foundation for Distinguished Young Scholars of Hunan Province (Grant No. 2017JJ1025), and the Research Foundation of Education Bureau of Hunan Province (Grant No. 18A198).

Author information

Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, China

    Zhao Yang & Wei Zhu

  2. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, 411201, Hunan, China

    Rong Huang

Authors
  1. Zhao Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rong Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rong Huang.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, Z., Huang, R. & Zhu, W. Accurate computations for eigenvalues of products of Cauchy-polynomial-Vandermonde matrices. Numer Algor 85, 329–351 (2020). https://doi.org/10.1007/s11075-019-00816-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00816-5

Keywords

Search

Navigation