# A Traub-like Algorithm for Hessenbergquasiseparable- Vandermonde Matrices of Arbitrary Order

## Abstract

Although Gaussian elimination uses *O*(*n* ^{3}) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only *O*(*n* _{2}) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modification of it yields a high accuracy in practice. The Traub algorithm has been extended from Vandermonde matrices involving monomials to polynomial-Vandermonde matrices involving real orthogonal polynomials, and the Szegö polynomials.

In this paper we consider a new more general class of polynomials that we suggest to call Hessenberg order *m* quasisseparable polynomials, or (*H, m*)-quasiseparable polynomials. The new class is wide enough to include all of the above important special cases, e.g., monomials, real orthogonal polynomials and the Szcgö polynomials, as well as new subclasses. We derive a fast *O*(*n* ^{2}) Traub-like algorithm to invert the associated (*H, m*)-*quasisseparable-Vandermonde matrices*.

The class of *quasiseparable matrices* is garnering a lot of attention recently; it has been found to be useful in designing a number fo fast algorithms. The derivation of our new Traub-like algorithm is also based on exploiting quasiseparable structure of the corresponding Hessenberg matrices. Preliminary numerical experiments are presented comparing the algorithm to standard structure ignoring methods.

This paper extends our recent results in [6] from the (*H*,0)-and (*H*,1)-quasiseparable cases to the more general (*H, m*)-quasiseparable case.

## Mathematics Subject Classification (2000)

15A09 15–04 15B05## Keywords

Orthogonal polynomials Szego polynomials quasiseparable matrices Vandermonde matrices Hessenberg matrices inversion fast algorithm## Preview

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## References

- [1]M. Bakonyi and T. Constantinescu,
*Schur’s algorithm and several applications*, in Pitman Research Notes in Mathematics Series, vol. 61, Longman Scientific and Technical, Harlow, 1992.Google Scholar - [2]A. Björck and V. Pereyra,
*Solution of Vandermonde Systems of Equations*,*Math. Comp.*,**24**(1970), 893–903.CrossRefMathSciNetGoogle 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 and Applications, Volume**420**, Issues 2–3 pp. 634–647, 2007.zbMATHCrossRefMathSciNetGoogle Scholar - [4]T. Bella, Y. Eidelman, I. Gohberg, I. Koltracht and V. Olshevsky
*A fast Bjorck-Pereyra like algorithm for solving Hessenberg-quasiseparable-Vandermonde systems*, submitted to SIAM Journal of Matrix Analysis (SIMAX), 2007.Google Scholar - [5]T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky,
*Classifications of three-term and two-term recurrence relations and digital filter structures via subclasses of quasiseparable matrices*, submitted to SIAM Journal of Matrix Analysis (SIMAX), 2007.Google Scholar - [6]T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky, E. Tyrtyshnikov
*Fast Traub-like inversion algorithm for Hessenberg order one quasiseparable Vandermonde matrices*, submitted to*Journal of Complexity*, 2007.Google Scholar - [7]Calvetti, D. and Reichel, L., Fast inversion of Vandermonde-like matrices involving orthogonal polynomials,
*BIT*, 1993.Google Scholar - [8]Y. Eidelman and I. Gohberg,
*On a new class of structured matrices*, Integral Equations and Operator Theory,**34**(1999), 293–324.zbMATHCrossRefMathSciNetGoogle Scholar - [9]Y. Eidelman and I. Gohberg,
*Linear complexity inversion algorithms for a class of structured matrices*, Integral Equations and Operator Theory,**35**(1999), 28–52.zbMATHCrossRefMathSciNetGoogle Scholar - [10]Y. Eidelman and I. Gohberg,
*A modification of the Dewilde-van der Veen method for inversion of finite-structured matrices*, Linear Algebra Appl.,**343–344**(2002), 419–450.CrossRefMathSciNetGoogle Scholar - [11]Y. Eidelman, I. Gohberg and V. Olshevsky,
*Eigenstructure of Order-One-Quasiseparable Matrices. Three-term and Two-term Recurrence Relations*, Linear Algebra and its Applications, Volume**405**, 1 August 2005, Pages 1–40.zbMATHCrossRefMathSciNetGoogle Scholar - [12]G. Forney,
*Concatenated codes*, The M.I.T. Press, 1966, Cambridge.Google Scholar - [13]L.Y. Geronimus,
*Polynomials orthogonal on a circle and their applications*, Amer. Math. Translations, 3 p. 1–78, 1954 (Russian original 1948).Google Scholar - [14]I. Gohberg and V. Olshevsky,
*Fast inversion of Chebyshev-Vandermonde matrices*, Numerische Mathematik,**67, No. 1**(1994), 71–92.zbMATHCrossRefMathSciNetGoogle Scholar - [15]I. Gohberg and V. Olshevsky,
*A fast generalized Parker-Traub algorithm for inversion of Vandermonde and related matrices*, Journal of Complexity,**13(2)**(1997), 208–234. A short version in Proceedings in Communications, Computation, Control and Signal Processing: A tribute to Thomas Kailath, Eds. A. Paulraj, V. Roychowdhury and C. Shaper, Kluwer Academic Publishing, 1996, 205-221.zbMATHCrossRefMathSciNetGoogle Scholar - [16]I. Gohberg and V. Olshevsky,
*The fast generalized Parker-Traub algorithm for inversion of Vandermonde and related matrices*, J. of Complexity,**13(2)**(1997), 208–234.zbMATHCrossRefMathSciNetGoogle Scholar - [17]U. Grenader and G. Szegö,
*Toeplitz forms and Applications*, University of California Press, 1958.Google Scholar - [18]W.G. Horner,
*A new method of solving numerical equations of all orders by continuous approximation*, Philos. Trans. Roy. Soc. London, (1819), 308–335.Google Scholar - [19]N.J. Higham,
*Stability analysis of algorithms for solving confluent Vandermonde-like systems*, SIAM J. Matrix Anal. Appl.,**11(1)**(1990), 23–41.zbMATHCrossRefMathSciNetGoogle Scholar - [20]T. Kailath and V. Olshevsky,
*Displacement structure approach to polynomial Vandermonde and related matrices*,*Linear Algebra and Its Applications*,**261**(1997), 49–90.zbMATHCrossRefMathSciNetGoogle Scholar - [21]J. Maroulas and S. Barnett, Polynomials with respect to a general basis. I. Theory, J. of Math. Analysis and Appl.,
**72**(1979), 177–194.zbMATHCrossRefMathSciNetGoogle Scholar - [22]V. Olshevsky,
*Eigenvector computation for almost unitary Hessenberg matrices and inversion of Szegö-Vandermonde matrices via Discrete Transmission lines*. Linear Algebra and Its Applications, 285 (1998), 37–67.zbMATHCrossRefMathSciNetGoogle Scholar - [23]V. Olshevsky,
*Associated polynomials, unitary Hessenberg matrices and fast generalized Parker-Traub and Bjorck-Pereyra algorithms for Szegö-Vandermonde matrices*invited chapter in the book “Structured Matrices: Recent Developments in Theory and Computation,” 67–78, (D. Bini, E. Tyrtyshnikov, P. Yalamov., Eds.), 2001, NOVA Science Publ., USA.Google Scholar - [24]V. Olshevsky,
*Pivoting for structured matrices and rational tangential interpolation*, in Fast Algorithms for Structured Matrices: Theory and Applications, CONM/323, 1–75, AMS publications, May 2003.Google Scholar - [25]F. Parker,
*Inverses of Vandermonde matrices*, Amer. Math. Monthly,**71**(1964), 410–411.zbMATHCrossRefMathSciNetGoogle Scholar - [26]L. Reichel and G. Opfer,
*Chebyshev-Vandermonde systems*, Math. of Comp.,**57**(1991), 703–721.zbMATHCrossRefMathSciNetGoogle Scholar - [27]J. Stoer, R. Bulirsch,
*Introduction to Numerical Analysis*, Springer-Verlag, 1992. 277–301.Google Scholar - [28]J. Traub,
*Associated polynomials and uniform methods for the solution of linear problems*, SIAM Review,**8**, No. 3 (1966), 277–301.zbMATHCrossRefMathSciNetGoogle Scholar