Skip to main content
SpringerLink
Log in

Classifications of Recurrence Relations via Subclasses of (H, m)-quasiseparable Matrices

  • Chapter
  • First Online:
Numerical Linear Algebra in Signals, Systems and Control

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 80))

Abstract

The results on characterization of orthogonal polynomials and Szegö polynomials via tridiagonal matrices and unitary Hessenberg matrices, respectively, are classical. In a recent paper we observed that tridiagonal matrices and unitary Hessenberg matrices both belong to a wider class of \((H,1)\)-quasiseparable matrices and derived a complete characterization of the latter class via polynomials satisfying certain EGO-type recurrence relations. We also established a characterization of polynomials satisfying three-term recurrence relations via \((H,1)\)-well-free matrices and of polynomials satisfying the Szegö-type two-term recurrence relations via \((H,1)\)-semiseparable matrices. In this paper we generalize all of these results from \(scalar\) (H,1) to the block (H, m) case. Specifically, we provide a complete characterization of \((H,\,m)\)-quasiseparable matrices via polynomials satisfying \(block\) EGO-type two-term recurrence relations. Further, \((H,\,m)\)-semiseparable matrices are completely characterized by the polynomials obeying \(block\) Szegö-type recurrence relations. Finally, we completely characterize polynomials satisfying m-term recurrence relations via a new class of matrices called \((H,\,m)\)-well-free matrices.

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

Access this chapter

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 EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
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

Institutional subscriptions

Notes

  1. 1.

    More details on the meaning of these numbers will be provided in Sect. 2.2.1.

  2. 2.

    \(A_{12}=A(1:k,k+1:n), k=1,\ldots, n-1\) in the MATLAB notation.

  3. 3.

    The parameters \(\mu_k\) associated with the Szegö polynomials are defined by \(\mu_k=\sqrt{1-|\rho_k|^2}\) for \(0\leq|\rho_k|<1\) and \(\mu_k=1\) for \(|\rho_k|=1,\) and since \(|\rho_k|\leq1\) for all \(k,\) we always have \(\mu_k\neq0.\)

  4. 4.

    Every \(i\)-th row of B equals the row number \((i-1)\) times \( \rho_{i-1}^{*} / \rho_{i-2}^{*} \mu_{i-1}. \)

  5. 5.

    The invertibility of \(b_k\) implies that all \(b_k\) are square \(m\times m\) matrices.

References

  1. Ammar G, Calvetti D, Reichel L (1993) Computing the poles of autoregressive models from the reflection coefficients. In: Proceedings of the Thirty-First Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, October, pp 255–264

    Google Scholar 

  2. Ammar G, He C (1995) On an inverse eigenvalue problem for unitary Hessenberg matrices. Linear Algebra Appl 218:263–271

    Article  MathSciNet  MATH  Google Scholar 

  3. Bakonyi M, Constantinescu T (1992) Schur’s algorithm and several applications. Pitman Research Notes in Mathematics Series, vol 61, Longman Scientific and Technical, Harlow

    Google Scholar 

  4. Barnett S (1981) Congenial matrices. Linear Algebra Appl 41:277–298

    Article  MathSciNet  MATH  Google Scholar 

  5. Bella T, Eidelman Y, Gohberg I, Koltracht I, Olshevsky V (2007) A Björck-Pereyra-type algorithm for Szegö-Vandermonde matrices based on properties of unitary Hessenberg matrices. Linear Algebra and its Applications (to appear)

    Google Scholar 

  6. Bella T, Eidelman Y, Gohberg I, Koltracht I, Olshevsky V. A fast Björck-Pereyra-like algorithm for solving Hessenberg-quasiseparable-Vandermonde systems (submitted)

    Google Scholar 

  7. Bella T, Eidelman Y, Gohberg I, Olshevsky V, Tyrtyshnikov E. Fast inversion of Hessenberg-quasiseparable Vandermonde matrices (submitted)

    Google Scholar 

  8. Bella T, Eidelman Y, Gohberg I, Olshevsky V. Classifications of three-term and two-term recurrence relations and digital filter structures via subclasses of quasiseparable matrices. SIAM J Matrix Anal (SIMAX) (submitted)

    Google Scholar 

  9. Bunse-Gerstner A, He CY (1995) On a Sturm sequence of polynomials for unitary Hessenberg matrices. (English summary) SIAM J Matrix Anal Appl 16(4):1043–1055

    Article  MathSciNet  MATH  Google Scholar 

  10. Bruckstein A, Kailath T (1986) Some matrix factorization identities for discrete inverse scattering. Linear Algebra Appl 74:157–172

    Article  MathSciNet  MATH  Google Scholar 

  11. Bruckstein A, Kailath T (1987) Inverse scattering for discrete transmission line models. SIAM Rev 29:359–389

    Article  MathSciNet  MATH  Google Scholar 

  12. Bruckstein A, Kailath T (1987) An inverse scattering framework for several problems in signal processing. IEEE ASSP Mag 4(1):6–20

    Article  Google Scholar 

  13. Eidelman Y, Gohberg I (1999) On a new class of structured matrices. Integr Equ Oper Theory 34:293–324

    Article  MathSciNet  MATH  Google Scholar 

  14. Eidelman Y, Gohberg I (1999) Linear complexity inversion algorithms for a class of structured matrices. Integr Equ Oper Theory 35:28–52

    Article  MathSciNet  MATH  Google Scholar 

  15. Eidelman Y, Gohberg I (2002) A modification of the Dewilde-van der Veen method for inversion of finitestructured matrices. Linear Algebra Appl 343–344:419–450

    Article  MathSciNet  Google Scholar 

  16. Eidelman Y, Gohberg I (2005) On generators of quasiseparable finite block matrices. Calcolo 42:187–214

    Article  MathSciNet  MATH  Google Scholar 

  17. Eidelman Y, Gohberg I, Olshevsky V (2005) Eigenstructure of Order-One-Quasiseparable Matrices. Three-term and two-term Recurrence Relations. Linear Algebra Appl 405:1–40

    Article  MathSciNet  MATH  Google Scholar 

  18. Geronimus LY (1954) Polynomials orthogonal on a circle and their applications. Amer Math Transl 3:1–78 (Russian original 1948)

    Google Scholar 

  19. Gragg WB (1982) Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle (in Russian). In : Nikolaev ES (ed) Numerical methods in Linear Algebra. pp. 16–32, Moskow University Press, 1982; English translation: (1993) J Comput Appl Math 46:183–198

    Google Scholar 

  20. Grenader U, Szegö G (1958) Toeplitz forms and Applications. University of California Press, Berkeley

    Google Scholar 

  21. Kailath T, Porat B (1983) State-space generators for orthogonal polynomials. Prediction theory and harmonic analysis, pp 131–163, North-Holland, Amsterdam-New York

    Google Scholar 

  22. Lev-Ari H, Kailath T (1984) Lattice filter parameterization and modeling of nonstationary processes. IEEE Trans Inf Theory 30:2–16

    Article  MathSciNet  MATH  Google Scholar 

  23. Lev-Ari H, Kailath T (1986) Triangular factorization of structured Hermitian matrices. In: Gohberg I (ed) Operator Theory: Advances and Applications, vol. 18. Birkhäuser, Boston, pp 301–324

    Google Scholar 

  24. Maroulas J, Barnett S (1979) Polynomials with respect to a general basis. I. Theory. J Math Anal Appl 72:177–194

    Article  MathSciNet  MATH  Google Scholar 

  25. Olshevsky V (1998) Eigenvector computation for almost unitary Hessenberg matrices and inversion of Szego-Vandermonde matrices via discrete transmission lines. Linear Algebra Appl 285:37–67

    Article  MathSciNet  MATH  Google Scholar 

  26. Olshevsky V (2001) Associated polynomials, unitary Hessenberg matrices and fast generalized Parker-Traub and Bjorck-Pereyra algorithms for Szego-Vandermonde matrices. In: Bini D, Tyrtyshnikov E, Yalamov P (eds) Structured Matrices: Recent Developments in Theory and Computation. NOVA Science Publishers, USA, pp 67–78

    Google Scholar 

  27. Regalia PA (1995) Adaptive IIR filtering in signal processing and control. Marcel Dekker, New York

    MATH  Google Scholar 

  28. Schur I (1917) Uber potenzreihen, die in Innern des Einheitskrises Beschrnkt Sind. J Reine Angew Math 147:205–232; English translation: Schur I (1986) Methods in Operator Theory and Signal Processing, Gohberg I (ed) Birkhuser, pp 31–89

    Google Scholar 

  29. Simon B (2005) Orthogonal polynomials on the unit circle. Part 2. Spectral theory. American Mathematical Society Colloquium Publications, 54, Parts 1 & 2. American Mathematical Society, Providence, RI

    Google Scholar 

  30. Stoer J, Bulirsch R (1992) Introduction to numerical analysis. Springer-Verlag, New York, pp 277–301

    Google Scholar 

  31. Teplyaev AV (1992) The pure point spectrum of random orthogonal polynomials on the circle. Soviet Math Dokl 44:407–411

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Rhode Island, Kingston, RI, 02881-0816, USA

    T. Bella

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    V. Olshevsky & P. Zhlobich

Authors
  1. T. Bella
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Olshevsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Zhlobich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Bella .

Editor information

Editors and Affiliations

  1. CESAME, Batiment Euler, Universite Catholique de Louvain, 4, avenue Georges Lemaitre, Louvain la Neuve, 1348, Belgium

    Paul Van Dooren

  2. Department of Electrical and Computer, Engineering, Texas A&M University, Zachry Engineering Center, College Station, 77843-3128, Texas, USA

    Shankar P. Bhattacharyya

  3. Dept. Mathematics, Chinese University Hong Kong, Shatin, New Territories, Hong Kong/PR China

    Raymond H. Chan

  4. Dept. Mathematics, University of Connecticut, Auditorium Road 196, Storrs, 06269-3009, Connecticut, USA

    Vadim Olshevsky

  5. , Electrical Engineering, Indian Institute of Technology, Kharagpur, 721302, India

    Aurobinda Routray

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Bella, T., Olshevsky, V., Zhlobich, P. (2011). Classifications of Recurrence Relations via Subclasses of (H, m)-quasiseparable Matrices. In: Van Dooren, P., Bhattacharyya, S., Chan, R., Olshevsky, V., Routray, A. (eds) Numerical Linear Algebra in Signals, Systems and Control. Lecture Notes in Electrical Engineering, vol 80. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0602-6_2

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-0602-6_2

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-0601-9

  • Online ISBN: 978-94-007-0602-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation