Skip to main content
SpringerLink
Log in
Book cover

Numerical Methods for Structured Matrices and Applications pp 203–220Cite as

A Priori Estimates on the Structured Conditioning of Cauchy and Vandermonde Matrices

  • Chapter

  • 1020 Accesses

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 199))

Abstract

We analyze the componentwise and normwise sensitivity of inverses of Cauchy, Vandermonde, and Cauchy-Vandermonde matrices, with respect to relative componentwise perturbations in the nodes defining these matrices. We obtain a priori, easily computable upper bounds for these condition numbers. In particular, we improve known estimates for Vandermonde matrices with generic real nodes; twe consider in detail Vandermonde matrices with nonnegative or symmetric nodes; and we extend the analysis to the class of complex Cauchy-Vandermonde matrices.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S.G. Bartels and D.J. Higham; The structured sensitivity of Vandermonde-like systems. Numer. Math. 62 (1992), 17–33.

    Article  MATH  MathSciNet  Google Scholar 

  2. B. Beckermann; The condition number of real Vandermonde, Krylov and positive definite Hankel matrices. Numer. Math. 85 (2000), 553–577.

    Article  MATH  MathSciNet  Google Scholar 

  3. T. Bella, Y. Eidelman, I. Gohberg, I. Koltracht and V. Olshevsky; A Björck-Pereyratype algorithm for Szegö-Vandermonde matrices based on properties of unitary Hessenberg matrices. Linear Algebra Appl. 420 (2007), 634–647.

    Article  MATH  MathSciNet  Google Scholar 

  4. T. Boros, T. Kailath and V. Olshevsky; A fast parallel Björck-Pereyra-type algorithm for solving Cauchy linear equations. Linear Algebra Appl. 302/303 (1999), 265–293.

    Article  MathSciNet  Google Scholar 

  5. T. Boros, T. Kailath and V. Olshevsky; Pivoting and backward stability of fast algorithms for solving Cauchy linear equations. Linear Algebra Appl. 343/344 (2002), 63–99.

    Article  MathSciNet  Google Scholar 

  6. E. Bozzo, D. Fasino and O. Menchi; The componentwise conditioning of the DFT. Calcolo 39 (2002), 181–187.

    Article  MATH  MathSciNet  Google Scholar 

  7. F. Cucker and H. Diao; Mixed and componentwise condition numbers for rectangular structured matrices. Calcolo 44 (2007), 89–115.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Fasino and V. Olshevsky; How bad are symmetric Pick matrices?, in: Structured Matrices in Operator Theory, Numerical Analysis, Control, Signal and Image Processing (V. Olshevsky, Ed.) AMS Series on Contemporary Mathematics, 280 (2001), 301–311.

    Google Scholar 

  9. T. Finck, G. Heinig and K. Rost; An inversion formula and fast algorithms for Cauchy-Vandermonde matrices. Linear Algebra Appl. 183 (1993), 179–191.

    Article  MATH  MathSciNet  Google Scholar 

  10. I. Gohberg and I. Koltracht; On the inversion of Cauchy matrices, in: M.A. Kaashoek, J.H. van Schuppen, A.C.M. Ran (Eds.), Signal processing, scattering and operator theory, and numerical methods (Proceedings of MTNS-98), pp. 381–392; Birkhäuser, 1990.

    Google Scholar 

  11. I. Gohberg and I. Koltracht; Mixed, componentwise, and structured condition numbers. SIAM J. Matrix Anal. Appl. 14 (1993), 688–704.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Heinig and K. Rost; Recursive solution of Cauchy-Vandermonde systems of equations. Linear Algebra Appl. 218 (1995), 59–72.

    Article  MATH  MathSciNet  Google Scholar 

  13. G. Heinig and K. Rost; Representations of inverses of real Toeplitz-plus-Hankel matrices using trigonometric transformations, in: Large-Scale Scientific Computation of Engineering and Environmental Problems II (M. Griebel, S. Margenov and P. Yalamov, Eds.), Vieweg, 73 (2000) 80–86.

    Google Scholar 

  14. D.J. Higham and N.J. Higham; Backward error and condition of structured linear systems. SIAM J. Matrix Anal. Appl. 13 (1992), 162–175.

    Article  MATH  MathSciNet  Google Scholar 

  15. N.J. Higham; Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems. Numer. Math. 50 (1987), 613–632.

    Article  MATH  MathSciNet  Google Scholar 

  16. R.-C. Li; Asymptotically optimal lower bounds for the condition number of a real Vandermonde matrix. SIAM J. Matrix Anal. Appl. 28 (2006), 829–844.

    Article  MATH  MathSciNet  Google Scholar 

  17. V. Olshevsky and V. Pan; Polynomial and rational evaluation and interpolation (with structured matrices). ICALP99 Proceedings, Springer, LNCS 1644 (1999), 585–594.

    MathSciNet  Google Scholar 

  18. G. Mühlbach; Interpolation by Cauchy-Vandermonde systems and applications. J. Comput. Appl. Math. 122 (2000), 203–222.

    Article  MATH  MathSciNet  Google Scholar 

  19. G. Mühlbach; On Hermite interpolation by Cauchy-Vandermonde systems: the Lagrange formula, the adjoint and the inverse of a Cauchy-Vandermonde matrix. J. Comput. Appl. Math. 67 (1996), 147–159.

    Article  MATH  MathSciNet  Google Scholar 

  20. T. Penzl; Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case. Systems Control Lett. 40 (2000), 139–144.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Serra Capizzano; An elementary proof of the exponential conditioning of real Vandermonde matrices. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 10 (2007), 761–768.

    MATH  MathSciNet  Google Scholar 

  22. J.-G. Sun; Bounds for the structured backward errors of Vandermonde systems. SIAM J. Matrix Anal. Appl. 20 (1999), 45–59.

    Article  MathSciNet  Google Scholar 

  23. E.E. Tyrtyshnikov; Singular values of Cauchy-Toeplitz matrices. Linear Algebra Appl. 161 (1992), 99–116.

    Article  MATH  MathSciNet  Google Scholar 

  24. E.E. Tyrtyshnikov; How bad are Hankel matrices? Numer. Math. 67 (1994), 261–269.

    Article  MATH  MathSciNet  Google Scholar 

  25. H. Xiang and Y. Wei; Structured mixed and componentwise condition numbers of some structured matrices. J. Comput. Appl. Math. 202 (2007), 217–229.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze, 208, I-33100, Udine, Italy

    Enrico Bozzo & Dario Fasino

Authors
  1. Enrico Bozzo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dario Fasino
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, 56127, Pisa, Italy

    Dario Andrea Bini

  2. Institut für Mathematik, Technische Universität Berlin, Straβe des 17. Juni 136, 10623, Berlin, Germany

    Volker Mehrmann

  3. Department of Mathematics, University of Connecticut, 196 Auditorium Road, U-9, Storrs, CT, 06269, USA

    Vadim Olshevsky

  4. Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina Street, 8, Moscow, 119991, Russia

    Eugene E. Tyrtyshnikov

  5. Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001, Leuven (Heverlee), Belgium

    Marc van Barel

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Bozzo, E., Fasino, D. (2010). A Priori Estimates on the Structured Conditioning of Cauchy and Vandermonde Matrices. In: Bini, D.A., Mehrmann, V., Olshevsky, V., Tyrtyshnikov, E.E., van Barel, M. (eds) Numerical Methods for Structured Matrices and Applications. Operator Theory: Advances and Applications, vol 199. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8996-3_9

Download citation

Publish with us

Policies and ethics

Search

Navigation