Quadrature formulas based on rational interpolation

Van Assche, Walter; Vanherwegen, Ingrid

doi:10.1090/S0025-5718-1993-1195424-6

Quadrature formulas based on rational interpolation
HTML articles powered by AMS MathViewer

by Walter Van Assche and Ingrid Vanherwegen PDF

Math. Comp. 61 (1993), 765-783 Request permission

Abstract:

We consider quadrature formulas based on interpolation using the basis functions $1/(1 + {t_k}x)\quad (k = 1,2,3, \ldots )$ on $[ - 1,1]$, where ${t_k}$ are parameters on the interval $( - 1,1)$. We investigate two types of quadratures: quadrature formulas of maximum accuracy which correctly integrate as many basis functions as possible (Gaussian quadrature), and quadrature formulas whose nodes are the zeros of the orthogonal functions obtained by orthogonalizing the system of basis functions (orthogonal quadrature). We show that both approaches involve orthogonal polynomials with modified weights which depend on the number of quadrature nodes. The asymptotic distribution of the nodes is obtained as well as various interlacing properties and monotonicity results for the nodes.

References

N. I. Achieser, Theory of approximation, Frederick Ungar Publishing Co., New York, 1956. Translated by Charles J. Hyman. MR 0095369
David L. Barrow, On multiple node Gaussian quadrature formulae, Math. Comp. 32 (1978), no. 142, 431–439. MR 482257, DOI 10.1090/S0025-5718-1978-0482257-0
Borislav D. Bojanov, Dietrich Braess, and Nira Dyn, Generalized Gaussian quadrature formulas, J. Approx. Theory 48 (1986), no. 4, 335–353. MR 865436, DOI 10.1016/0021-9045(86)90008-0
Dietrich Braess, Nonlinear approximation theory, Springer Series in Computational Mathematics, vol. 7, Springer-Verlag, Berlin, 1986. MR 866667, DOI 10.1007/978-3-642-61609-9

A FORTRAN multiple-precision arithmetic package

4

E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1963. MR 0157156
Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
Walter Gautschi, Gauss-type quadrature rules for rational functions, Numerical integration, IV (Oberwolfach, 1992) Internat. Ser. Numer. Math., vol. 112, Birkhäuser, Basel, 1993, pp. 111–130. MR 1248398, DOI 10.1007/978-3-0348-6338-4_{9}
Walter Gautschi, On the computation of generalized Fermi-Dirac and Bose-Einstein integrals, Comput. Phys. Comm. 74 (1993), no. 2, 233–238. MR 1202296, DOI 10.1016/0010-4655(93)90093-R
A. A. Gonchar and E. A. Rakhmanov, The equilibrium measure and distribution of zeros of extremal polynomials, Mat. Sb. (N.S.) 125(167) (1984), no. 1(9), 117–127 (Russian). MR 760416
Samuel Karlin and Allan Pinkus, Gaussian quadrature formulae with multiple nodes, Studies in spline functions and approximation theory, Academic Press, New York, 1976, pp. 113–141. MR 0477575
Samuel Karlin and Allan Pinkus, An extremal property of multiple Gaussian nodes, Studies in spline functions and approximation theory, Academic Press, New York, 1976, pp. 143–162. MR 0481763
Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0204922

The ideas of P.L. Chebysheff and A.A. Markov in the theory of limiting values of integrals and their further developments

12

G. L. Lopes, Comparative asymptotics for polynomials that are orthogonal on the real axis, Mat. Sb. (N.S.) 137(179) (1988), no. 4, 500–525, 57 (Russian); English transl., Math. USSR-Sb. 65 (1990), no. 2, 505–529. MR 981523, DOI 10.1070/SM1990v065n02ABEH002078
Guillermo López Lagomasino, Asymptotics of polynomials orthogonal with respect to varying measures, Constr. Approx. 5 (1989), no. 2, 199–219. MR 989673, DOI 10.1007/BF01889607

A note on generalized quadrature formulas of Gauss-Jacobi type

Sobre los métodos interpolatorios de integración numérica y su conexión con la aproximación racional

8

Szegő’s theorem for polynomials orthogonal with respect to varying measures

G. López and E. A. Rakhmanov, Rational approximations, orthogonal polynomials and equilibrium distributions, Orthogonal polynomials and their applications (Segovia, 1986) Lecture Notes in Math., vol. 1329, Springer, Berlin, 1988, pp. 125–157. MR 973424, DOI 10.1007/BFb0083356
H. N. Mhaskar and E. B. Saff, Weighted polynomials on finite and infinite intervals: a unified approach, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 351–354. MR 752796, DOI 10.1090/S0273-0979-1984-15303-5
H. N. Mhaskar and E. B. Saff, Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials), Constr. Approx. 1 (1985), no. 1, 71–91. MR 766096, DOI 10.1007/BF01890023
H. N. Mhaskar and E. B. Saff, Where does the $L^p$-norm of a weighted polynomial live?, Trans. Amer. Math. Soc. 303 (1987), no. 1, 109–124. MR 896010, DOI 10.1090/S0002-9947-1987-0896010-9
Allan Pinkus and Zvi Ziegler, Interlacing properties of the zeros of the error functions in best $L^{p}$-approximations, J. Approx. Theory 27 (1979), no. 1, 1–18. MR 554112, DOI 10.1016/0021-9045(79)90093-5

Orthogonal polynomials

Quadrature formulas based on interpolation using

polynomials

On zeros of orthogonal polynomials

4

Mathematica^extsctm , A system for doing mathematics by computer