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Orthogonal polyanalytic polynomials and normal matrices

Author: Marko Huhtanen
Journal: Math. Comp. 72 (2003), 355-373
MSC (2000): Primary 42C05; Secondary 15A57
Published electronically: February 22, 2002
MathSciNet review: 1933825
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Abstract: The Hermitian Lanczos method for Hermitian matrices has a well-known connection with a 3-term recurrence for polynomials orthogonal on a discrete subset of $\mathbb{R}$. This connection disappears for normal matrices with the Arnoldi method. In this paper we consider an iterative method that is more faithful to the normality than the Arnoldi iteration. The approach is based on enlarging the set of polynomials to the set of polyanalytic polynomials. Denoting by $d$ the number of elements computed so far, the arising scheme yields a recurrence of length bounded by $\sqrt{8d}$ for polyanalytic polynomials orthogonal on a discrete subset of $\mathbb{C}$. Like this slowly growing length of the recurrence, the method preserves, at least partially, the properties of the Hermitian Lanczos method. We employ the algorithm in least squares approximation and bivariate Lagrange interpolation.

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  • 1. W.E. Arnoldi, The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quart. Apll. Math., 9, $(1951)$, 17-29. MR 13:163e
  • 2. M.B. Balk, Polyanalytic functions, Wiley$/$VCH, Weinh., 1991. MR 93k:30076
  • 3. C. de Boor, Polynomial interpolation in several variables, Studies in Computer Science, R. DeMillo and J. R. Rice (eds.), Plenum Press, New York, (1994), pp. 87-119.
  • 4. C. de Boor and A. Ron, Computational aspects of polynomial interpolation in several variables, Math. Comp., 58, (1992), 705-727. MR 92i:65022
  • 5. P. Borwein, The arc length of the lemniscate $\{\vert p(z)\vert=1\}$, Proc. Amer. Math. Soc., 123, (1995), 797-799. MR 95d:31001
  • 6. C. Brezinski, Formal orthogonality on an algebraic curve, Annals of Numer. Math., 2, (1995), 21-33. MR 97a:42017
  • 7. A. Bultheel and M. Van Barel, Vector orthogonal polynomials and least squares approximation, SIAM J. Matrix Anal. Appl., 16, (1995), 863-885. MR 96h:65060
  • 8. J. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997. MR 98m:65001
  • 9. S. Elhay, G. Golub and J. Kautsky, Updating and downdating of orthogonal polynomials with data fitting applications, SIAM J. Matrix Anal. Appl., 12, 1991, 327-353. MR 91m:65054
  • 10. L. Elsner and KH.D. Ikramov, On a condensed form for normal matrices under finite sequence of elementary similarities, Lin. Alg. Appl., 254, $(1997)$, 79-98. MR 98b:15011
  • 11. W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Comp., 3 $(1982)$, 289-317. MR 84e:65022
  • 12. W. Gautschi, Orthogonal Polynomials: Applications and Computations, Acta Numerica 1996, Cambridge University Press, $($1996$)$, 45-119. MR 99i:65019
  • 13. W. Gautschi and G. Inglese, Lower bounds for the condition number of Vandermonde matrices, Numer. Math., 52 $(1988)$, 241-250. MR 89b:65108
  • 14. G. H. Golub and C. F. Van Loan, Matrix Computations, The John Hopkins University Press, Baltimore and London, the $3^{rd}$ ed., 1996. MR 97g:65006
  • 15. R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, 1991. MR 92e:15003
  • 16. A.S. Householder, Lectures on Numerical Algebra, Mathematical Association of America, Buffalo, N.Y., 1972. MR 53:11952
  • 17. M. Huhtanen, A stratification of the set of normal matrices, SIAM J. Matrix Anal. Appl., to appear.
  • 18. M. Huhtanen, A Hermitian Lanczos method for normal matrices, SIAM J. Matrix Anal. Appl., to appear.
  • 19. M. Huhtanen and R. M. Larsen, Exclusion and inclusion regions for the eigenvalues of a normal matrix, Stanford University, SCCM-01-02 report, 2001.
  • 20. G. G. Lorentz and R. A. Lorentz, Bivariate Hermite interpolation and applications to algebraic geometry, Numer. Math. 57, (1990), 669-680. MR 92f:41004
  • 21. O. Nevanlinna, Convergence of Iterations for Linear Equations, Lectures in Mathematics ETH Zürich, Birkäuser Verlag, Basel, 1993. MR 94h:65055
  • 22. L. Reichel, Fast QR decomposition of Vandermonde-like matrices and polynomial least squares approximation, SIAM J. Matrix Anal. Appl., 12, (1991), 552-564. MR 92a:65126
  • 23. Y. Saad, Numerical Methods for Large Eigenvalue Problems, Halstead Press, NY, 1992. MR 93h:65052
  • 24. L. A. Santalo, Integral geometry and geometric probability, Addison-Wesley, Reading, MA, 1976. MR 55:6340
  • 25. T. Sauer, Polynomial interpolation of minimal degree, Numer. Math., 78, (1997), 59-85. MR 99f:41042
  • 26. P. K. Suetin, Orthogonal polynomials in two variables, Anal. Meth. and Spec. Funct., 3, Gordon and Breach Sci. Publ., Amsterdam, 1999. MR 2000h:42020
  • 27. I. G. Sprinkhuizen-Kuyper, Orthogonal polynomials in two variables. A further analysis of the polynomials orthogonal over a region bounded by two lines and a parabola, SIAM J. Math. Anal., 7, (1976), 501-518. MR 54:3278
  • 28. M.J. Zygmunt, Recurrence formula for polynomials of two variables, orthogonal with respect to a rotation invariant measure, Constr. Approx., 15 $($1999$)$, 301-309. MR 2000b:33008

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Additional Information

Marko Huhtanen
Affiliation: SCCM program, Computer Science Department, Stanford University, Stanford, California 94305
Address at time of publication: Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 01239

Keywords: Normal matrix, orthogonal polynanalytic polynomial, slowly growing length of the recurrence, least squares approximation, bivariate Lagrange interpolation
Received by editor(s): September 12, 2000
Received by editor(s) in revised form: March 1, 2001
Published electronically: February 22, 2002
Additional Notes: This work was supported by the Academy of Finland and the Alfred Kordelin Foundation
Article copyright: © Copyright 2002 American Mathematical Society

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