Advertisement

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

  • Zhao Yang
  • Rong HuangEmail author
  • Wei Zhu
Original Paper
  • 17 Downloads

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.

Keywords

Product eigenvalue problems Cauchy-polynomial-Vandermonde matrices Bidiagonal decompositions High relative accuracy 

Notes

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 information

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).

References

  1. 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)Google Scholar
  2. 2.
    Dailey, M., Dopico, F. M., Ye, Q.: Relative perturbation theory for diagonally dominant matrices. SIAM J. Matrix Anal. Appl. 35, 1303–1328 (2014)MathSciNetCrossRefGoogle Scholar
  3. 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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Delgado, J., Peña, J. M.: Accurate computations with collocation matrices of rational bases. Appl. Math. Comput. 219, 4354–4364 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Delgado, J., Peña, J. M.: Accurate computations with collocation matrices of q-Bernstein polynomials. SIAM J. Matrix Anal. Appl. 36, 880–893 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Demmel, J., Kahan, W.: Accurate singular values of bidiagonal matrices. SIAM J. Sci Stat. Comp. 11, 873–912 (1990)MathSciNetCrossRefGoogle Scholar
  7. 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)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Demmel, J., Koev, P.: Accurate SVDs of weakly diagonally dominant M-matrices. Numer. Math. 98, 99–104 (2004)MathSciNetCrossRefGoogle Scholar
  9. 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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Demmel, J., Koev, P.: Accurate and efficient evaluation of Schur and Jack functions. Math. Comput. 75, 223–239 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Demmel, J., Dumitriu, I., Holtz, O., Koev, P.: Accurate and efficient expression evaluation and linear algebra. Acta Numer. 17, 87–145 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fallat, S. M., Johnson, C. R.: Totally Nonnegative Matrices. Princeton University Press, Princeton (2011)CrossRefGoogle Scholar
  13. 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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dopico, F. M., Koev, P.: Perturbation theory for the LDU factorization and accurate computations for diagonally dominant matrices. Numer. Math. 119, 337 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gasca, M., Martinez, J. J., Mühlbach, G.: Computation of rational interpolants with prescribed poles. J. Comput. Appl. Math 26, 297–309 (1989)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gasca, M., Peña, J. M.: Total positivity and Neville elimination. Linear Algebra Appl. 165, 25–44 (1992)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gasca, M., Peña, J. M.: Total positivity, QR factorization and Neville elimination. SIAM J. Matrix Anal. 14, 1132–1140 (1993)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gasca, M., Peña, J. M.: A matricial description of Neville elimination with applications to total positivity. Linear Algebra Appl. 202, 33–53 (1994)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gasca, M., Peña, J. M.: On factorizations of totally positive matrices, in Total positivity and its applications, Springer, Dordrecht, pp. 109–130 (1996)CrossRefGoogle Scholar
  20. 20.
    Higham, N. J.: Fast solution of Vandermonde-like systems involving orthogonal polynomials. IMA J. Numer. Anal. 8, 473–486 (1988)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Higham, N. J.: Stability analysis of algorithms for solving confluent Vandermonde-like systems. SIAM J. Matrix Anal. Appl. 11, 23–41 (1990)MathSciNetCrossRefGoogle Scholar
  22. 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)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Junghanns, P., Oestreich, D.: Numerische Lösung des Staudammproblems mit Drainage. Z. Angew. Math. Mech. 69, 83–92 (1989)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kailath, T., Olshevsky, V.: Displacement structure approach to Chebyshev Vandermonde and related matrices. Integral Equ. Oper. Theory 22, 65–92 (1995)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Koev, P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27, 1–23 (2005)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29, 731–751 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Koev, P., Dopico, F. M.: Accurate eigenvalues of certain sign regular matrices. Linear Algebra Appl. 424, 435–447 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Macdonald, I. G.: Symmetric function and hall polynomials, 2nd edn. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  29. 29.
    Mainar, E., Peña, J. M.: Accurate computations with collocation matrices of a general class of bases. Numer. Linear Algebra Appl. 25, e2184 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Marco, A., Martínez, J. J.: Accurate computations with Said-Ball-Vandermonde matrices. Linear Algebra Appl. 432, 2894–2908 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Marco, A., Martínez, J. J.: Accurate computations with totally positive Bernstein-Vandermonde matrices. Electron. J. Linear Algebra. 26, 357–380 (2013)MathSciNetCrossRefGoogle Scholar
  32. 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)MathSciNetCrossRefGoogle Scholar
  33. 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)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Martínez, J. J., Peña, J. M.: Factorizations of Cauchy-Vandermonde matrices. Linear Algebra Appl. 284, 229–237 (1998)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Pan, V. Y.: Structured matrices and polynomials: unified superfast algorithms springer science and business media (2012)Google Scholar
  36. 36.
    Phillips, G. M.: Interpolation and approximation by polynomials, Springer Science and Business Media (2003)Google Scholar
  37. 37.
    Reichel, L., Opfer, G.: Chebyshev-Vandermonde systems. Math. Comp. 57, 703–721 (1991)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Verde-Star, L.: Inverses of generalized Vandermonde matrices. J. Math. Anal. Appl. 131, 341–353 (1988)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Weideman, J. A. C., Laurie, D. P.: Quadrature rules based on partial fraction expansions. Numer. Algorithms 24, 159–178 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and Computational ScienceXiangtan UniversityXiangtanChina
  2. 2.School of Mathematics and Computational ScienceHunan University of Science and TechnologyXiangtanChina

Personalised recommendations