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Error bounds for Gauss-Turán quadrature formulae of analytic functions

Authors: Gradimir V. Milovanovic and Miodrag M. Spalevic
Journal: Math. Comp. 72 (2003), 1855-1872
MSC (2000): Primary 41A55; Secondary 65D30, 65D32
Published electronically: May 30, 2003
MathSciNet review: 1986808
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Abstract: We study the kernels of the remainder term $R_{n,s}(f)$ of Gauss-Turán quadrature formulas

\begin{displaymath}\int_{-1}^1f(t)w(t)\,dt=\sum_{\nu=1}^n \sum_{i=0}^{2s}A_{i,\n... ...au_\nu) +R_{n,s}(f)\qquad(n\in \mathbb{N};\, s\in\mathbb{N}_0)\end{displaymath}

for classes of analytic functions on elliptical contours with foci at $\pm1$, when the weight $w$ is one of the special Jacobi weights $w^{(\alpha,\beta)}(t)=(1-t)^\alpha(1+t)^\beta$ $(\alpha=\beta=-1/2$; $\alpha=\beta=1/2+s$; $\alpha=-1/2$, $\beta=1/2+s$; $\alpha=1/2+s$, $\beta=-1/2)$. We investigate the location on the contour where the modulus of the kernel attains its maximum value. Some numerical examples are included.

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  • 1. S. Bernstein, Sur les polynomes orthogonaux relatifs à un segment fini, J. Math. Pures Appl. 9 (1930), 127-177.
  • 2. L. Chakalov, Über eine allgemeine Quadraturformel, C.R. Acad. Bulgar. Sci. 1 (1948), 9-12.
  • 3. L. Chakalov, General quadrature formulae of Gaussian type, Bulgar. Akad. Nauk Izv. Mat. Inst. 1 (1954), 67-84 (Bulgarian) [English transl. East J. Approx. 1 (1995), 261-276]. MR 97b:41001
  • 4. L. Chakalov, Formules générales de quadrature mécanique du type de Gauss, Colloq. Math. 5 (1957), 69-73.
  • 5. J. D. Donaldson and D. Elliott, A unified approach to quadrature rules with asymptotic estimates of their remainders, SIAM J. Numer. Anal. 9 (1972), 573-602. MR 47:6069
  • 6. W. Gautschi, On the remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadratures, Rocky Mountain J. Math. 21 (1991) 209-226. MR 93a:41071b
  • 7. W. Gautschi and S. Li, The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple points, J. Comput. Appl. Math. 33 (1990), 315-329. MR 92a:65078
  • 8. W. Gautschi and G. V. Milovanovic, $S$-orthogonality and construction of Gauss-Turán type quadrature formulae, J. Comput. Appl. Math. 86 (1997), 205-218. MR 99a:65030
  • 9. W. Gautschi and S. E. Notaris, Gauss-Kronrod quadrature formulae for weight function of Bernstein-Szego type, J. Comput. Appl. Math. 25 (1989), 199-224. MR 90m:65055
  • 10. W. Gautschi and R. S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal. 20 (1983), 1170-1186. MR 85j:65010
  • 11. W. Gautschi, E. Tychopoulos and R. S. Varga, A note on the contour integral representation of the remainder term for a Gauss-Chebyshev quadrature rule, SIAM J. Numer. Anal. 27 (1990), 219-224. MR 91d:65044
  • 12. A. Ghizzetti and A. Ossicini, Quadrature formulae, Akademie - Verlag, Berlin, 1970. MR 42:4012
  • 13. A. Ghizzetti and A. Ossicini, Sull' esistenza e unicità delle formule di quadratura gaussiane, Rend. Mat. (6) 8 (1975), 1-15. MR 52:1116
  • 14. G. H. Golub and J. Kautsky, Calculation of Gauss quadratures with multiple free and fixed knots, Numer. Math. 41 (1983), 147-163. MR 84i:65030
  • 15. L. Gori Nicolò-Amati, On the behaviour of the zeros of some s-orthogonal polynomials, in Orthogonal Polynomials and Their Applications, 2nd Int. Symp. (Segovia, 1986), Monogr. Acad. Cienc. Exactas, Fis., Quim., Nat., Zaragoza, 1988, pp. 71-85.
  • 16. L. Gori and C. A. Micchelli, On weight functions which admit explicit Gauss-Turán quadrature formulas, Math. Comp. 65 (1996), 1567-1581. MR 97c:41034
  • 17. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Sixth Edition (A. Jeffrey and D. Zwillinger, eds.), Academic Press, San Diego, 2000. MR 2001c:00002
  • 18. D. Hunter and G. Nikolov, On the error term of symmetric Gauss-Lobatto quadrature formulae for analytic functions, Math. Comp. 69 (1999), 269-282. MR 2001a:65030
  • 19. D. V. Ionescu, Restes des formules de quadrature de Gauss et de Turán, Acta Math. Acad. Sci. Hungar. 18 (1967), 283-295. MR 35:7057
  • 20. S. Karlin and A. Pinkus, Gaussian quadrature with multiple nodes, in Studies in Spline Functions and Approximation Theory (S. Karlin, C. A. Micchelli, A. Pinkus, and I. J. Schoenberg, eds.), Academic Press, New York, 1976, pp. 113-141. MR 57:17094
  • 21. G. V. Milovanovic, Construction of $s$-orthogonal polynomials and Turán quadrature formulae, in Numerical Methods and Approximation Theory. III (Nis, 1987), (G. V. Milovanovic, ed.), Univ. Nis, Nis, 1988, pp. 311-328. MR 89g:65023
  • 22. G. V. Milovanovic, Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation, in Numerical Analysis in 20th Century, Vol. 5 (W. Gautschi, F. Marcellán, and L. Reichel, eds.), J. Comput. Appl. Math. 127 (2001), 267-286. MR 2002e:65039
  • 23. G. V. Milovanovic and M. M. Spalevic, A numerical procedure for coefficients in generalized Gauss-Turán quadratures, Filomat (Nis) 9(1) (1995), 1-8. MR 97d:65010
  • 24. G. V. Milovanovic and M. M. Spalevic, Construction of Chakalov-Popoviciu's type quadrature formulae, Rend. Circ. Mat. Palermo 52 (1998), 625-636. MR 99f:41035
  • 25. G. V. Milovanovic and M. M. Spalevic, Quadrature formulae connected to $\sigma$-orthogonal polynomials, J. Comput. Appl. Math. 140 (2002), 619-637.
  • 26. A. Ossicini, Costruzione di formule di quadratura di tipo Gaussiano, Ann. Mat. Pura Appl. (4) 72 (1966), 213-237. MR 34:2180
  • 27. A. Ossicini, Le funzioni di influenza nel problema di Gauss sulle formule di quadratura, Matematiche (Catania) 23 (1968), 7-30. MR 40:3719
  • 28. A. Ossicini, M. R. Martinelli, and F. Rosati, Funzioni caratteristiche e polinomi $s$-ortogonali, Rend. Mat. 14 (1994), 355-366. MR 95m:33010
  • 29. A. Ossicini and F. Rosati, Funzioni caratteristiche nelle formule di quadratura gaussiane con nodi multipli, Boll. Un. Mat. Ital. (4) 11 (1975), 224-237. MR 53:11977
  • 30. A. Ossicini and F. Rosati, Comparison theorems for the zeros of $s$-orthogonal polynomials, Calcolo 16 (1979), 371-381 (Italian). MR 81m:33009
  • 31. A. Ossicini and F. Rosati, $s$-orthogonal Jacobi polynomials, Rend. Mat. Appl. (7) 12 (1992), 399-403 (Italian). MR 94b:33010
  • 32. P. Pavel, On the remainder of some Gaussian formulae, Studia Univ. Babes-Bolyai Ser. Math.-Phys. 12 (1967), 65-70. MR 38:3674
  • 33. P. Pavel, On some quadrature formulae of Gaussian type, Studia Univ. Babes-Bolyai Ser. Math.-Phys. 13 (1968), 51-58 (Romanian). MR 37:2445
  • 34. P. Pavel, On the remainder of certain quadrature formulae of Gauss-Christoffel type, Studia Univ. Babes-Bolyai Ser. Math.-Phys. 13 (1968), 67-72 (Romanian). MR 39:5061
  • 35. F. Peherstorfer, On the remainder of Gaussian quadrature formulas for Bernstein-Szego weight functions, Math. Comp. 60 (1993), 317-325. MR 93d:65030
  • 36. T. Popoviciu, Sur une généralisation de la formule d'integration numérique de Gauss, Acad. R. P. Romîne Fil. Iasi Stud. Cerc. Sti. 6 (1955), 29-57. MR 19:64h
  • 37. T. Schira, The remainder term for analytic functions of Gauss-Lobatto quadratures, J. Comput. Appl. Math. 76 (1996), 171-193. MR 97m:41033
  • 38. T. Schira, The remainder term for analytic functions of symmetric Gaussian quadratures, Math. Comp. 66 (1997), 297-310. MR 97c:65050
  • 39. M. M. Spalevic, Product of Turán quadratures for cube, simplex, surface of the sphere, ${\overline E}_n^r$, $E_n^{r^2}$, J. Comput. Appl. Math. 106 (1999), 99-115. MR 2000f:65018
  • 40. M. M. Spalevic, Calculation of Chakalov-Popoviciu's quadratures of Radau and Lobatto type, ANZIAM J. (formerly as J. Aust. Math. Soc. B) 43 (2002), 429-447.
  • 41. D. D. Stancu, On a class of orthogonal polynomials and on some general quadrature formulas with minimum number of terms, Bull. Math. Soc. Sci. Math. Phys. R.P. Romîne (N.S) 1 (49) (1957), 479-498. MR 21:3700
  • 42. D. D. Stancu, On certain general numerical integration formulas, Acad. R.P. Romîne. Stud. Cerc. Mat. 9 (1958), 209-216 (Romanian). MR 20:4917
  • 43. A. H. Stroud and D. D. Stancu, Quadrature formulas with multiple Gaussian nodes, J. SIAM Numer. Anal. Ser. B 2 (1965), 129-143. MR 31:4177
  • 44. P. Turán, On the theory of the mechanical quadrature, Acta Sci. Math. Szeged 12(1950), 30-37. MR 12:164b
  • 45. G. Vincenti, On the computation of the coefficients of $s$-orthogonal polynomials, SIAM J. Numer. Anal 23 (1986), 1290-1294. MR 88b:65029

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Additional Information

Gradimir V. Milovanovic
Affiliation: Department of Mathematics, University of Niš, Faculty of Electronic Engineering, P.O. Box 73, 18000 Niš, Serbia

Miodrag M. Spalevic
Affiliation: Faculty of Science, Department of Mathematics and Informatics, P.O. Box 60, 34000 Kragujevac, Serbia

Keywords: Gauss-Tur\'an quadrature, $s$-orthogonality, zeros, multiple nodes, weight, measure, degree of exactness, remainder term for analytic functions, error estimate, contour integral representation, kernel function
Received by editor(s): February 7, 2002
Received by editor(s) in revised form: April 21, 2002
Published electronically: May 30, 2003
Additional Notes: The authors were supported in part by the Serbian Ministry of Science, Technology and Development (Project: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods).
Dedicated: This paper is dedicated to Professor Walter Gautschi on the occasion of his 75th birthday
Article copyright: © Copyright 2003 American Mathematical Society

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