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On the condition number of some gram matrices arising from least squares approximation in the complex plane
 Youcef Saad
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This paper analyses the growth of the condition number of a class of Gram matrices that arise when computing least squares polynomials in polygons of the complex plane. It is shown that if the polygon is inserted between two ellipses then the condition number of the (n+1)×(n+1) Gram matrix is bounded from above by 4m(n+1)^{2}(k)^{2n } wherem is the number of edges of the polygon, andk≥1 is a known ratio which is close to one if the two ellipses are close to each other.
This work was supported in part by the U.S. Office of Naval Research under grant N0001482K0184, in part by Dept. Of Energy under Grant AC0281ER10996, and in part by Army Research Office under contract DAAG830177
 Davis, P.J. (1963) Interpolation and Approximation. Blaisdell, Waltham, MA
 Gautschi, W. (1982) On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput. 3: pp. 289317
 Gautschi, W. (1968) Construction of GaussChristoffel quadrature formulas. Math. Comput. 22: pp. 251270
 Lorentz, G.G. (1966) Approximation of functions. Holt, Rinehart & Winston, New York
 Manteuffel, T.A.: An iterative method for solving nonsymmetric linear systems with dynamic estimation of parameters. Technical Report UIUCDCS75758, University of Illinois at UrbanaChampaign (Ph. D. dissertation) 1975
 Rivlin, T.J. (1976) The Chebyshev Polynomials. Wiley, New York
 Saad, Y.: Least squares polynomials in the complex plane and their use for solving sparse nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. (Submitted)
 Saad, Y.: Least squares polynomials in the complex plane with applications to solving sparse nonsymmetric matrix problems. Technical Report 276, Yale University, Computer Science Dept. 1983
 Saad, Y. (1985) Practical use of polynomial preconditionings for the Conjugate Gradient Method. Technical Report YALEU/DCS/RR282, Yale University, 1983. SIAM J. Sci. Stat. Comput. 6: pp. 865881
 Sack, R.A., Donovan, A.F. (1971) An algorithm for Gaussian quadrature given modified moments. Numer. Math. 18: pp. 465478
 Smolarski, D.C.: Optimum semiiterative methods for the solution of any linear algebraic system with a square matrix. Technical Report UIUCDCSR811077, University of Illinois at UrbanaChampaign (Ph. D. Thesis) 1981
 Smolarski, D.C., Saylor, P.E.: Optimum parameters for the solution of linear equations by Richardson iteration, 1982 (To appear)
 Title
 On the condition number of some gram matrices arising from least squares approximation in the complex plane
 Journal

Numerische Mathematik
Volume 48, Issue 3 , pp 337347
 Cover Date
 19860501
 DOI
 10.1007/BF01389479
 Print ISSN
 0029599X
 Online ISSN
 09453245
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 AMS(MOS): 65D99
 CR: G1.2
 Industry Sectors
 Authors

 Youcef Saad ^{(1)}
 Author Affiliations

 1. Research Center for Scientific Computation, Yale University, Yale Station, P.O. Box 2158, 06520, New Haven, CT, USA