An overview of the computational aspects of Kronrod quadrature rules

Numerical Algorithms

Volumn 26, Issue 2, 2001, Pages 173-196

  Monegato, Giovanni ^a  

^a POLITECNICO DI TORINO   (Italy)

Author keywords

Algorithms; Automatic routines; Error evaluation; Quadrature rules

Indexed keywords

EID: 0035614738     PISSN: 10171398     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1016640617732     Document Type: Review

References (71)

1. G.S. Ammar, D. Calvetti and L. Reichel, Computation of Gauss-Kronrod quadrature rules with non-positive weights, Electron. Trans. Numer. Anal. 9 (1999) 26-38.
2. B. Baillaud and H. Bourget, Correspondance d'Hermite et de Stieltjes I,II (Gauthier-Villars, Paris, 1905).
3. P. Baratella, Un' estensione ottimale della formula di quadratura di Radau, Rend. Sem. Mat. Univ. Politecn. Torino 37 (1979) 147-158.
4. P. Barrucand, Intégration numérique, abscisse de Kronrod-Patterson et polynômes de Szego, C. R. Acad. Sci. Paris Ser. A 270 (1970) 147-158.
5. A. Begumisa and I. Robinson, Suboptimal Kronrod extension formulas for numerical quadrature, Numer. Math. 58(8) (1991) 807-818.
6. J. Berntsen and T.O. Espelid, Error estimation in automatic quadrature rules, ACM Trans. Math. Software 7 (1991) 233-252.
7. H. Brass, Quadraturverfahren (Vandenhoeck und Ruprecht, Göttingen, 1977).
8. H. Brass and K.-J. Förster, On the estimation of linear functionals, Analysis 7 (1987) 237-258.
9. C. Brezinski, Error estimate in Padé approximation, in: Orthogonal Polynomials and their Applications, eds. M. Alfaro et al., Lecture Notes in Mathematics, Vol. 1329 (Springer, Heidelberg, 1988) pp. 1-19.
10. C. Brezinski, Procedures for estimating the error in Padé approximation, Math. Comp. 53 (1989) 639-648.
11. F. Caliò, W. Gautschi and E. Marchetti, On computing Gauss-Kronrod formulae, Math. Comp. 47 (1986) 639-650.
12. F. Caliò and E. Marchetti, Complex Gauss-Kronrod integration rules for certain Cauchy principal value integrals, Computing 50 (1993) 165-173.
13. D. Calvetti, G.H. Golub, W.B. Gragg and L. Reichel, Computation of Gauss-Kronrod quadrature rules, Math. Comp. 69 (2000) 1035-1052.
14. S. Ehrich, On the construction of Gaussian quadrature formulae containing preassigned nodes, in: Numerical Integration IV, eds. H. Brass and G. Hämmerlin, International Series of Numerical Mathematics, Vol. 112 (Birkhäuser, Basel, 1993) pp. 67-76.
15. S. Ehrich, Error bounds for Gauss-Kronrod quadrature formulae, Math. Comp. 62 (1994) 295-304.
16. S. Ehrich, Asymptotic properties of Stieltjes polynomials and Gauss-Kronrod quadrature formulae, J. Approx. Theory 82 (1995) 287-303.
17. S. Ehrich, Asymptotic behaviour of Stieltjes polynomials for ultraspherical weight functions, J. Comput. Appl. Math. 65 (1995) 135-144.
18. S. Ehrich, A note on Peano constants of Gauss-Kronrod quadrature schemes, J. Comput. Appl. Math. 66 (1996) 177-183.
19. S. Ehrich, Practical error estimates for the Gauss-Kronrod quadrature formula, in: Numerical Methods and Error Bounds, eds. G. Alefeld and J. Herzberger (Akademie-Verlag, Berlin, 1996) pp. 74-79.
20. S. Ehrich, On product integration with Gauss-Kronrod nodes, SIAM. J. Numer. Anal. 35 (1998) 78-92.
21. S. Ehrich, Stieltjes polynomials and the error of Gauss-Kronrod quadrature formulas, in: Applications and Computation of Orthogonal Polynomials, eds. W. Gautschi, G. Golub and G. Opfer, International Series of Numerical Mathematics, Vol. 131 (Birkhäuser, Basel, 1999) pp. 57-77.
22. S. Ehrich, Stopping functionals for Gaussian quadrature formulas, J. Comput. Appl. Math., to appear.
23. S. Ehrich and G. Mastroianni, Stieltjes polynomials and Lagrange interpolation, Math. Comp. 66 (1997) 311-331.
24. S. Ehrich and G. Mastroianni, Weighted convergence of Lagrange interpolation based on Gauss-Kronrod nodes, J. Comput. Anal. Appl. 2(2) (2000) 129-157.
25. S. Elhay and J. Kautsky, Generalized Kronrod-Patterson-type embedded quadratures, Appl. Mat. 37 (1992) 81-103.
26. P. Favati, G. Lotti and F. Romani, Interpolatory integration formulas for optimal composition, ACM Trans. Math. Software 17 (1991) 207-217.
27. P. Favati, G. Lotti and F. Romani, Algorithm 691: Improving Quadpack automatic integration routines, ACM Trans. Math. Software 17 (1991) 218-232.
28. W. Gautschi, Gauss-Kronrod quadrature - a survey, in: Numerical Methods of Approximation Theory III, ed. G.V. Milovanović (Univ. Niš, 1988) pp. 39-66.
29. W. Gautschi, The use of rational functions in numerical quadrature, J. Comput. Appl. Math., to appear.
30. W. Gautschi and S.E. Notaris, Newton's method and Gauss-Kronrod quadrature, in: Numerical Integration III, eds. H. Brass and G. Hämmerlin, International Series of Numerical Mathematics, Vol. 85 (Birkhäuser, Basel, 1988) pp. 60-71.
31. E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary Differential Equations I (Springer, Heidelberg, 1987).
32. D.K. Kahaner, J. Waldvogel and L.W. Fullerton, Addition of points to Gauss-Laguerre quadrature formulas, SIAM J. Sci. Statist. Comput. 5 (1984) 42-55.
33. J. Kautsky and S. Elhay, Gauss quadratures and Jacobi matrices for weight functions not of one sign, Mat. Comp. 43 (1984) 543-550.
34. F. Krogh and W. Van Snyder, Algorithm 699: A new representation of Patterson's quadrature formulae, ACM Trans. Math. Software 17(4) (1991) 457-461.
35. A.S. Kronrod, Integration with control of accuracy, Soviet Phys. Dokl. 9(1) (1964) 17-19.
36. A.S. Kronrod, Nodes and Weights for Quadrature Formulae. Sixteen Place Tables (Nauka, Moscow, 1964; Translation by Consultants Bureau, New York, 1965).
37. D.P. Laurie, Practical error estimation in numerical integration, J. Comput. Appl. Math. 12/13 (1985) 425-431.
38. D.P. Laurie, Stratified sequences of nested quadrature formulas, Quaestiones Math. 15 (1992) 365-384.
39. D.P. Laurie, Null rules and orthogonal expansions, in: Approximation and Computation, a Festschrift in Honor of Walter Gautschi, ed. R.V.M. Zahar, International Series of Numerical Mathematics, Vol. 119 (Birkhäuser, Basel, 1994) pp. 359-370.
40. D.P. Laurie, Anti-Gaussian quadrature formulas, Math. Comp. 65 (1996) 739-747.
41. D.P. Laurie, Calculation of Gauss-Kronrod quadrature rules, Math. Comp. 66 (1997) 1133-1145.
42. D.P. Laurie, Computation of Gauss-type quadrature formulas, J. Comput. Appl. Math., to appear.
43. S. Li, Kronrod extension of Turán formula, Studia Sci. Math. Hungar. 29 (1994) 71-83.
44. S. Li, Kronrod extension of generalized Gauss-Radau and Gauss-Lobatto formulae, Rocky Mountain J. Math. 26 (1996) 1455-1472.
45. G. Monegato, Positivity of weights of extended Gauss-Legendre quadrature rules, Math. Comp. 32 (1978) 243-245.
46. G. Monegato, Some remarks on the construction of Kronrod rules, Math. Comp. 32 (1978) 812-817.
47. G. Monegato, An overview of results and questions related to Kronrod schemes, in: Numerische Integration, Proc. of Conf. Oberwolfach, ed. G. Hämmerlin, International Series of Numerical Mathematics, Vol. 45 (Birkhäuser, Basel, 1979) pp. 231-240.
48. G. Monegato, Stieltjes polynomials and related quadrature rules, SIAM Rev. 24 (1982) 137-158.
49. I.P. Mysovskih, A special case of quadrature formulae containing preassigned nodes (in Russian), Vesci Akad. Navuk BSSR Ser. Fiz.-Tehn. Navuk 4 (1964) 125-127.
50. S.E. Notaris, An overview of results on the existence or nonexistence and the error term of Gauss-Kronrod quadrature formulae, in: Approximation and Computation, a Festschrift in Honor of Walter Gautschi, ed. R.V.M. Zahar, International Series of Numerical Mathematics, Vol. 119 (Birkhäuser, Basel, 1994) pp. 485-496.
51. T.N.L. Patterson, On some Gauss and Lobatto based integration formulae. Math. Comp. 22 (1968) 877-881.
52. T.N.L. Patterson, The optimum addition of points to quadrature formulae, Math. Comp. 22 (1968) 847-856 (Microfiche C1-C11; Errata, ibid. 23 (1969) 892).
53. T.N.L. Patterson, Algorithm 468: Algorithm for numerical integration over a finite interval, Comm. ACM 16 (1973) 694-699.
54. T.N.L. Patterson, An algorithm for generating interpolatory quadrature-rules of the highest degree of precision with preassigned nodes for general weight-functions, ACM Trans. Math. Software 15(2) (1989) 123-136.
55. T.N.L. Patterson, Modified optimal quadrature extensions, Numer. Math. 64 (1993) 511-520.
56. T.N.L. Patterson, Stratified nested and related quadrature rules, J. Comput. Appl. Math. 112 (1999) 243-251.
57. F. Peherstorfer, On Stieltjes polynomials and Gauss-Kronrod quadrature, Math. Comp. 55 (1990) 649-664.
58. F. Peherstorfer, Weight functions admitting repeated positive Kronrod quadrature, BIT 30 (1990) 145-151.
59. F. Peherstorfer, On the asymptotic behaviour of functions of the second kind and Stieltjes polynomials and on Gauss-Kronrod quadrature formulas, J. Approx. Theory 70 (1992) 156-190.
60. F. Peherstorfer and K. Petras, Ultraspherical Gauss-Kronrod quadrature is not possible for λ > 3, SIAM J. Numer. Anal. 37 (2000) 927-948.
61. R. Piessens, New quadrature formulas for the numerical inversion of the Laplace transform, BIT 9 (1969) 351-361.
62. R. Piessens, An algorithm for automatic integration, Angew. Inform. 9 (1973) 399-401.
63. R. Piessens and M. Branders, A note on the optimum addition of abscissas to quadrature formulas of Gauss and Lobatto type, Math. Comp. 28 (1974) 135-139 and 344-347.
64. R. Piessens, E. de Doncker, C. Überhuber and D.K. Kahaner, QUADPACK - A Subroutine Package for Automatic Integration, Springer Series in Computational Mathematics, Vol. 1 (Springer, Berlin, 1983).
65. P. Rabinowitz, The exact degree of precision of generalized Gauss-Kronrod integration rules, Math. Comp. 35 (1980) 1275-1283.
66. P. Rabinowitz, Gauss-Kronrod integration rules for Cauchy principal value integrals, Math. Comp. 41 (1983) 63-78 (Corrigenda: ibid. 45 (1985) 277).
67. P. Rabinowitz, J. Kautsky and S. Elhay, Empirical mathematics: the first Patterson extension of Gauss-Kronrod rules, Internat. J. Comput. Math. 36 (1990) 119-129.
68. Y.G. Shi, Generalised Gaussian Kronrod-Turán quadrature formulas, Acta Sci. Math. (Szeged) 62 (1996) 175-185.
69. G. Szego, Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören, Math. Ann. 110 (1934) 501-513 (cf. also R. Askey, ed., Collected Works, Vol. 2, pp. 545- 558).
70. G. Szego, Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören, Math. Ann. 110 (1934) 501-513 (cf. also R. Askey, ed., Collected Works, Vol. 2, pp. 545-558).
71. G. Szego, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. 23 (Amer. Math. Soc., Providence, RI, 1975).