Orthogonal polynomials: Applications and computation

Acta Numerica

Volumn 5, Issue suppl., 1996, Pages 45-119

  Gautschi, Walter ^a  

^a Purdue University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85014479701     PISSN: 09624929     EISSN: 14740508     Source Type: Journal    
DOI: 10.1017/S0962492900002622     Document Type: Article

References (139)

1. Abramowitz M. and Stegun I. A., eds (1964), Handbook of Mathematical Functions, NBS Appl. Math. Ser., 55, U.S. Government Printing Office, Washington, D.C.
2. Althammer P. (1962), ‘Eine Erweiterung des Orthogonalitätsbegriffes bei Polynomen und deren Anwendung auf die beste Approximation’, J. Reine Angew. Math. 211, 192–204.
3. Askey R. and Ismail M. (1984), ‘Recurrence relations, continued fractions and orthogonal polynomials’, Memoirs AMS 49, no. 300, Amer. Math. Soc, Providence, RI.
4. Bellen A. (1981), ‘A note on mean convergence of Lagrange interpolation’, J. Approx. Theory 33, 85–95.
5. Bellen A. (1988), ‘Alcuni problemi aperti sulla convergenza in media dell'interpolazione Lagrangiana estesa’, Rend. Istit. Mat. Univ. Trieste 20, Fasc. suppl., 1–9.
6. Boersma J. and Dempsey J. P. (1992), ‘On the numerical evaluation of Legendre's chi-function’, Math. Comp. 59, 157–163.
7. Boley D. and Golub G. H. (1987), ‘A survey of matrix inverse eigenvalue problems’, Inverse Problems 3, 595–622.
8. Borges C. F. (1994), ‘On a class of Gauss-like quadrature rules’, Numer. Math. 67, 271–288.
9. Bowers K. and Lund J., eds (1989), Computation and Control, Progress in Systems and Control Theory 1, Birkhäuser, Boston.
10. de Bruin M. G. and Meijer H. G. (1995), ‘Zeros of orthogonal polynomials in a non-discrete Sobolev space’, Ann. Numer. Math. 2, 233–246.
11. Buhmann M. D. and Iserles A. (1992), ‘On orthogonal polynomials transformed by the QR algorithm’, J. Comput. Appl. Math. 43, 117–134.
12. Calder A. C. and Laframboise J. G. (1986), ‘Multiple-water-bag simulation of in-homogeneous plasma motion near an electrode’, J. Comput. Phys. 65, 18–45.
13. Calder A. C., Laframboise J. G. and Stauffer A. D. (1983), ‘Optimum step-function approximation of the Maxwell distribution’, unpublished manuscript.
14. Caliò F., Gautschi W. and Marchetti E. (1986), ‘On computing Gauss–Kronrod quadrature formulae’, Math. Comp. 47, 639–650.
15. Chebyshev P. L. (1859), ‘Sur l'interpolation par la méthode des moindres carrés’, Mém. Acad. Impér. Sci. St. Petersbourg (7) 1, No. 15, 1–24. [Œuvres I, 473–498.]
16. Chihara T. S. (1978), An Introduction to Orthogonal Polynomials, Gordon and Breach, New York.
17. Christoffel E. B. (1858), ‘Über die Gaußische Quadratur und eine Verallgemeinerung derselben’, J. Reine Angew. Math. 55, 61–82. [Ges. Math. Abhandlungen I, 42–50.]
18. Criscuolo G., Mastroianni G. and Nevai P. (1993), ‘Mean convergence of derivatives of extended Lagrange interpolation with additional nodes’, Math. Nachr. 163, 73–92.
19. Criscuolo G., Mastroianni G. and Occorsio D. (1990), ‘Convergence of extended Lagrange interpolation’, Math. Comp. 55, 197–212.
20. Criscuolo G., Mastroianni G. and Occorsio D. (1991), ‘Uniform convergence of derivatives of extended Lagrange interpolation’, Numer. Math. 60, 195–218.
21. Criscuolo G., Mastroianni G. and Vértesi P. (1992), ‘Pointwise simultaneous convergence of extended Lagrange interpolation with additional knots’, Math. Comp. 59, 515–531.
22. Davis P. J. (1993), Spirals: From Theodorus to Chaos, A K Peters, Wellesley, MA.
23. Davis P. J. and Rabinowitz P. (1956), ‘Abscissas and weights for Gaussian quadratures of high order’, J. Res. Nat. Bur. Standards 56, 35–37.
24. Davis P. J. and Rabinowitz P. (1958), ’Additional abscissas and weights for Gaussian quadratures of high order. Values for n = 64, 80, and 96’, J. Res. Nat. Bur. Standards 60, 613–614.
25. Dombrowski J. and Nevai P. (1986), ‘Orthogonal polynomials, measures and recurrence relations’, SIAM J. Math. Anal. 17, 752–759.
26. Egecioglu Ö. and Koç C. K. (1989), ‘A fast algorithm for rational interpolation via orthogonal polynomials’, Math. Comp. 53, 249–264.
27. Erdős P. and Turán P. (1937), ‘On interpolation I: quadrature- and mean-convergence in the Lagrange-interpolation’, Ann. of Math. 38, 142–155.
28. Evans W. D., Littlejohn L. L., Marcellán F., Markett C. and Ronveaux A. (1995), ‘On recurrence relations for Sobolev orthogonal polynomials’, SIAM J. Math. Anal. 26, 446–467.
29. Fejér L. (1933), ‘Mechanische Quadraturen mit positiven Cotesschen Zahlen’, Math. Z. 37, 287–309.
30. Fischer H.-J. (1996), ‘On the condition of orthogonal polynomials via modified moments’, Z. Anal. Anwendungen, to appear.
31. Forsythe G. E. (1957), ‘Generation and use of orthogonal polynomials for data-fitting with a digital computer’, J. Soc. Indust. Appl. Math. 5, 74–88.
32. Freud G. (1971), Orthogonal Polynomials, Pergamon, New York.
33. Freund R. W., Golub G. H. and Nachtigal N. M. (1991), ‘Iterative solution of linear systems’, Acta Numerica, Cambridge University Press, 57–100.
34. Frontini M. and Milovanović G. V. (1989), ‘Moment-preserving spline approximation on finite intervals and Turán quadratures’, Facta Univ. Ser. Math. Inform. 4, 45–56.
35. Frontini M., Gautschi W. and Milovanović G. V. (1987), ‘Moment-preserving spline approximation on finite intervals’, Numer. Math. 50, 503–518.
36. Galant D. (1971), ‘An implementation of Christoffel's theorem in the theory of orthogonal polynomials’, Math. Comp. 25, 111–113.
37. Galant D. (1992), ‘Algebraic methods for modified orthogonal polynomials’, Math. Comp. 59, 541–546.
38. Gautschi W. (1963), ‘On inverses of Vandermonde and confluent Vandermonde matrices. II’, Numer. Math. 5, 425–430.
39. Gautschi W. (1967), ‘Numerical quadrature in the presence of a singularity’, SIAM J. Numer. Anal. 4, 357–362.
40. Gautschi W. (1968), ‘Construction of Gauss–Christoffel quadrature formulas’, Math. Comp. 22, 251–270.
41. Gautschi W. (1970), ‘On the construction of Gaussian quadrature rules from modified moments’, Math. Comp. 24, 245–260.
42. Gautschi W. (1979), ‘On generating Gaussian quadrature rules’, in Numerische Integration (Hämmerlin G., ed.), ISNM 45, Birkhäuser, Basel, 147–154.
43. Gautschi W. (1981 a), ‘A survey of Gauss–Christoffel quadrature formulae’, in E. B. Christoffel: The Influence of his Work in Mathematics and the Physical Sciences (Butzer P. L. and Fehér F., eds), Birkhäuser, Basel, 72–147.
44. Gautschi W. (1981 b), ‘Minimal solutions of three-term recurrence relations and orthogonal polynomials’, Math. Comp. 36, 547–554.
45. Gautschi W. (1982 a), ‘On generating orthogonal polynomials’, SIAM J. Sci. Statist. Comput. 3, 289–317.
46. Gautschi W. (1982 b), ‘An algorithmic implementation of the generalized Christoffel theorem’, in Numerical Integration (Hämmerlin G., ed.), ISNM 57, Birkhäuser, Basel, 89–106.
47. Gautschi W. (1983), ‘How and how not to check Gaussian quadrature formulae’, BIT 23, 209–216.
48. Gautschi W. (1984 a), ‘On some orthogonal polynomials of interest in theoretical chemistry’, BIT 24, 473–483.
49. Gautschi W. (1984 b), ‘Discrete approximations to spherically symmetric distributions’, Numer. Math. 44, 53–60.
50. Gautschi W. (1984 c), ‘Questions of numerical condition related to polynomials’, in Studies in Numerical Analysis (Golub G. H., ed.), Studies in Mathematics 24, The Mathematical Association of America, 140–177.
51. Gautschi W. (1985), ‘Orthogonal polynomials – constructive theory and applications’, J. Comput. Appl. Math. 12/13, 61–76.
52. Gautschi W. (1986 a), ‘On the sensitivity of orthogonal polynomials to perturbations in the moments’, Numer. Math. 48, 369–382.
53. Gautschi W. (1986 b), ‘Reminiscences of my involvement in de Branges's proof of the Bieberbach conjecture’, in The Bieberbach Conjecture (Baernstein A. II, Drasin D., Duren P. and Marden A., eds), Math. Surveys Monographs, no. 21, Amer. Math. Soc, Providence, RI, 205–211.
54. Gautschi W. (1988), ‘Gauss–Kronrod quadrature – a survey’, in Numerical Methods and Approximation Theory III (Milovanović G. V., ed.), Faculty of Electronic Engineering, Univ. Niš, Niš, 39–66.
55. Gautschi W. (1989), ‘Orthogonality – conventional and unconventional – in numerical analysis’, in Computation and Control (Bowers K. and Lund J., eds), Progress in Systems and Control Theory, v. 1, Birkhäuser, Boston, 63–95.
56. Gautschi W. (1991 a), ‘A class of slowly convergent series and their summation by Gaussian quadrature’, Math. Comp. 57, 309–324.
57. Gautschi W. (1991 b), ‘On certain slowly convergent series occurring in plate contact problems’, Math. Comp. 57, 325–338.
58. Gautschi W. (1991 c), ‘Computational problems and applications of orthogonal polynomials’, in Orthogonal Polynomials and Their Applications (Brezinski C., Gori L. and Ronveaux A., eds), IMACS Annals Comput. Appl. Math. 9, Baltzer, Basel, 61–71.
59. Gautschi W. (1992), ‘On mean convergence of extended Lagrange interpolation’, J. Appl. Comput. Math. 43, 19–35.
60. Gautschi W. (1993 a), ‘Is the recurrence relation for orthogonal polynomials always stable?’, BIT 33, 277–284.
61. Gautschi W. (1993 b), ‘Gauss-type quadrature rules for rational functions’, in Numerical Integration IV (Brass H. and Hämmerlin G., eds), ISNM 112, Birkhäuser, Basel, 111–130.
62. Gautschi W. (1993 c), ‘On the computation of generalized Fermi–Dirac and Bose–Einstein integrals’, Comput. Phys. Comm. 74, 233–238.
63. Gautschi W. (1994), ‘Algorithm 726: ORTHPOL – a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules’, ACM Trans. Math. Software 20, 21–62.
64. Gautschi W. (1996 a), ‘On the computation of special Sobolev-type orthogonal polynomials’, Ann. Numer. Math., to appear.
65. Gautschi W. (1996 b), ‘The computation of special functions by linear difference equations’, in Proc. 2nd Internat. Conf. on Difference Equations and Applications (Elaydi S., Ladas G. and Gyóri I., eds), Gordon and Breach, Newark, NJ, to appear.
66. Gautschi W. (1996 c), ‘Moments in quadrature problems’, Comput. Math. Appl., Ser. B, to appear.
67. Gautschi W. and Li S. (1993), ‘A set of orthogonal polynomials induced by a given orthogonal polynomial’, Aequationes Math. 46, 174–198.
68. Gautschi W. and Li S. (1996), ‘On quadrature convergence of extended Lagrange interpolation’, Math. Comp., to appear.
69. Gautschi W. and Milovanović G. V. (1985), ‘Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series’, Math. Comp. 44, 177–190.
70. Gautschi W. and Milovanović G. V. (1986), ‘Spline approximations to spherically symmetric distributions’, Numer. Math. 49, 111–121.
71. Gautschi W. and Zhang M. (1995), ‘Computing orthogonal polynomials in Sobolev spaces’, Numer. Math. 71, 159–183.
72. Golub G. H. (1973), ‘Some modified matrix eigenvalue problems’, SIAM Rev. 15, 318–334.
73. Golub G. H. and Van Loan C. F. (1989), Matrix Computations, 2nd edn, The Johns Hopkins University Press, Baltimore.
74. Golub G. H. and Welsch J. H. (1969), ‘Calculation of Gauss quadrature rules’, Math. Comp. 23, 221–230.
75. Gordon R. G. (1968), ‘Error bounds in equilibrium statistical mechanics’, J. Mathematical Phys. 9, 655–663.
76. Gori Nicolò-Amati L. and Santi E. (1989), ‘On a method of approximation by means of spline functions’, Proc. Internat. Symp. Approx., Optim. and Computing,Dalian,China.
77. Gori L. and Santi E. (1992), ‘Moment-preserving approximations: a monospline approach’, Rend. Mat. (7) 12, 1031–1044.
78. Gragg W. B. and Harrod W. J. (1984), ‘The numerically stable reconstruction of Jacobi matrices from spectral data’, Numer. Math. 44, 317–335.
79. Graves-Morris P. R. and Hopkins T. R. (1981), ‘Reliable rational interpolation’, Numer. Math. 36, 111–128.
80. Gröbner W. (1967), ‘Orthogonale Polynomsysteme, die gleichzeitig mit f(x) auch deren Ableitung f′(x) approximieren’, in Funktionalanalysis, Approximationstheorie, Numerische Mathematik (Collatz L., Meinardus G. and Unger H., eds), ISNM 7, Birkhäuser, Basel, 24–32.
81. Hageman L. A. and Young D. M. (1981), Applied Iterative Methods, Academic Press, New York.
82. Iserles A., Koch P. E., Nørsett S. P. and Sanz-Serna J. M. (1990), ‘Orthogonality and approximation in a Sobolev space’, in Algorithms for Approximation II (Mason J. C. and Cox M. G., eds), Chapman and Hall, London, 117–124.
83. Iserles A., Koch P. E., Nørsett S. P. and Sanz-Serna J. M. (1991), ‘On polynomials orthogonal with respect to certain Sobolev inner products’, J. Approx. Theory 65, 151–175.
84. Jacobi C. G. J. (1826), ‘Ueber Gaußs neue Methode, die Werthe der Integrale näherungsweise zu finden’, J. Reine Angew. Math. 1, 301–308.
85. Jacobi C. G. J. (1846), ‘Über die Darstellung einer Reihe gegebener Werthe durch eine gebrochene rationale Funktion’, J. Reine Angew. Math. 30, 127–156. [Math. Werke, vol. 1, 287–316.]
86. Kautsky J. and Elhay S. (1984), ‘Gauss quadratures and Jacobi matrices for weight functions not of one sign’, Math. Comp. 43, 543–550.
87. Kautsky J. and Golub G. H. (1983), ‘On the calculation of Jacobi matrices’, Linear Algebra Appl. 52/53, 439–455.
88. Kovačević M. A. and Milovanović G. V. (1996), ‘Spline approximation and generalized Turán quadratures’, Portugal. Math., to appear.
89. Lanczos C. (1950), ‘An iteration method for the solution of the eigenvalue problem of linear differential and integral operators’, J. Res. Nat. Bur. Standards 45B, 225–280.
90. Laurie D. P. (1996), ‘Calculation of Gauss–Kronrod quadrature rules’, Math. Comp., to appear.
91. Lewanowicz S. (1994), ‘A simple approach to the summation of certain slowly convergent series’, Math. Comp. 63, 741–745.
92. Lewis D. C. (1947), ‘Polynomial least square approximations’, Amer. J. Math. 69, 273–278.
93. Li S. (1994), ‘On mean convergence of Lagrange–Kronrod interpolation’, in Approximation and Computation (Zahar R. V. M., ed.), ISNM 119, Birkhäuser, Boston, 383–396.
94. 2-approximation with interpolation’, Ph.D. thesis, Purdue University.
95. López Lagomasino G. and Illán J. (1984), ‘A note on generalized quadrature formulas of Gauss–Jacobi type’, in Constructive Theory of Functions, Publ. House Bulgarian Acad. Sci., Sofia, 513–518.
96. López Lagomasino G. and Illán Gonzalez J. (1987), ‘Sobre los métodos interpolatorios de integración numérica y su conexión con la aproximación racional’, Rev. Ciencias Matém. 8, no. 2, 31–44.
97. Marcellán F., Pérez T. E. and Piñar M. A. (1995), ‘Orthogonal polynomials on weighted Sobolev spaces: the semiclassical case’, Ann. Numer. Math. 2, 93–122.
98. Marcellán F. and Ronveaux A. (1990), ‘On a class of polynomials orthogonal with respect to a discrete Sobolev inner product’, Indag. Math. (N.S.) 1, 451–464.
99. Marcellán F. and Ronveaux A. (1995), ‘Orthogonal polynomials and Sobolev inner products: a bibliography’, Laboratoire de Physique Mathématique, Facultés Universitaires N.D. de la Paix, Namur, Belgium.
100. Marcellán F., Alfaro M. and Rezola M. L. (1993), ‘Orthogonal polynomials on Sobolev spaces: old and new directions’, J. Comput. Appl. Math. 48, 113–131.
101. Mastroianni G. (1994), ‘Approximation of functions by extended Lagrange interpolation’, in Approximation and Computation (Zahar R. V. M., ed.), ISNM 119, Birkhäuser, Boston, 409–420.
102. Mastroianni G. and V P. értesi (1993), ‘Mean convergence of Lagrange interpolation on arbitrary systems of nodes’, Acta Sci. Math. (Szeged) 57, 429–441.
103. Meijer H. G. (1993), ‘Coherent pairs and zeros of Sobolev-type orthogonal polynomials’, Indag. Math. (N.S.) 4, 163–176.
104. Micchelli C. A. (1988), ‘Monosplines and moment preserving spline approximation’, in Numerical Integration III (Brass H. and Hämmerlin G., eds), ISNM 85, Birkhäuser, Basel, 130–139.
105. Milovanović G. V. (1994), ‘Summation of series and Gaussian quadratures’, in Approximation and Computation (Zahar R. V. M., ed.), Birkhäuser, Boston, 459–475.
106. Milovanović G. V (1995), ‘Summation of series and Gaussian quadratures, II’, Numer. Algorithms 10, 127–136.
107. Milovanović G. V. and Kovačević M. A. (1988), ‘Moment-preserving spline approximations and Turán quadratures’, in Numerical Mathematics Singapore 1988 (Agarwal R. P., Chow Y. M. and Wilson S. J., eds), ISNM 86, Birkhäuser, Basel, 357–365.
108. Milovanović G. V. and Kovačević M. A. (1992), ‘Moment-preserving spline approximation and quadrature’, Facta Univ. Ser. Math. Inform. 7, 85–98.
109. Monegato G. (1982), ‘Stieltjes polynomials and related quadrature rules’, SIAM Rev. 24, 137–158.
110. Moszyński K. (1992), ‘Remarks on polynomial methods for solving systems of linear algebraic equations’, Appl. Math. 37, 419–436.
111. Natanson I. P. (1964/1965), Constructive Function Theory, Vols 1–III, Ungar Publ. Co., New York.
112. Newbery A. C. R. (unpublished), ‘Gaussian principles applied to summation,’ unpublished manuscript.
113. Notaris S. E. (1994), ‘An overview of results on the existence or nonexistence and the error term of Gauss–Kronrod quadrature formulae’, in Approximation and Computation (Zahar R. V. M., ed.), ISNM 119, Birkhäuser, Boston, 485–496.
114. Parlett B. N. (1980), The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ.
115. Piessens R., de Doncker-Kapenga E., Überhuber C. W. and Kahaner D. K. (1983), QUADPACK: A Subroutine Package for Automatic Integration, Springer Ser. Comput Math., vol. 1, Springer, Berlin.
116. Pólya G. (1933), ‘Über die Konvergenz von Quadraturverfahren’, Math. Z. 37, 264–286.
117. Rabinowitz P. (1960), ‘Abscissas and weights for Lobatto quadrature of high order’, Math. Comp. 14, 47–52.
118. Rabinowitz P. and Weiss G. (1959), ‘Tables of abscissas and weights for numerical evaluation of integrals of the form [formula omitted]’, Math, Tables Aids Comp. 13, 285–294.
119. 1(x)’, Rep. AECL–3461, Atomic Energy of Canada Limited, Chalk River, Ontario.
120. Sack R. A. and Donovan A. F. (1969), ‘An algorithm for Gaussian quadrature given generalized moments’, Dept. Math., Univ. of Salford, Salford, UK.
121. Sack R. A. and Donovan A. F. (1971/1972), ‘An algorithm for Gaussian quadrature given modified moments’, Numer. Math. 18, 465–478.
122. Stiefel E. L. (1958), ‘Kernel polynomials in linear algebra and their numerical applications’, in Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues, NBS Appl. Math. Ser., 49, U.S. Government Printing Office, Washington, D.C., 1–22.
123. Stieltjes T. J. (1884), ‘Quelques recherches sur la théorie des quadratures dites mécaniques’, Ann. Sci. École Norm. Paris, Sér. 3, 1, 409–426. [Œuvres I, 377–396.]
124. Stoer J. and Bulirsch R. (1980), Introduction to Numerical Analysis, 2nd edn, Springer, New York.
125. Szeg˝ G. (1975), Orthogonal Polynomials, Colloq. Publ. 23, 4th edn, Amer. Math. Soc, Providence, RI.
126. Todd J. (1954), ‘The condition of the finite segments of the Hilbert matrix’, NBS Appl. Math. Ser., v. 39, U.S. Government Printing Office, Washington, D.C., 109–116.
127. Tricomi F. G. (1954), Funzioni Ipergeometriche Confluenti, Edizioni Cremonese, Rome.
128. Uvarov V. B. (1959), ‘Relation between polynomials orthogonal with different weights’ (Russian), Dokl. Akad. Nauk SSSR 126, 33–36.
129. Uvarov V. B. (1969), ‘The connection between systems of polynomials that are orthogonal with respect to different distribution functions’ (Russian), Ž. Vyčisl. Mat. i Mat. Fiz. 9, 1253–1262. [English translation in U.S.S.R. Comput. Math. and Math. Phys. 9 (1969), No. 6, 25–36.]
130. Van Assche W. and Vanherwegen I. (1993), ‘Quadrature formulas based on rational interpolation’, Math. Comp. 61, 765–783.
131. Walsh J. L. (1969), Interpolation and Approximation by Rational Functions in the Complex Domain, Colloq. Publ. 20, 5th edn, Amer. Math. Soc, Providence, RI.
132. Wheeler J. C. (1974), ‘Modified moments and Gaussian quadrature’, Rocky Mountain J. Math. 4, 287–296.
133. Wheeler J. C. (1984), ‘Modified moments and continued fraction coefficients for the diatomic linear chain’, J. Chem. Phys. 80, 472–476.
134. Widder D. V. (1941), The Laplace Transform, Princeton University Press.
135. Wilf H. S. (1962), Mathematics for the Physical Sciences, Wiley, New York.
136. Wilf H. S. (1980), personal communication.
137. Wong R. (1982), ‘Quadrature formulas for oscillatory integral transforms’, Numer. Math. 39, 351–360.
138. Wong R. (1989), Asymptotic Approximations of Integrals, Academic Press, Boston.
139. Zhang M. (1994), ‘Sensitivity analysis for computing orthogonal polynomials of Sobolev type’, in Approximation and Computation (Zahar R. V. M., ed.), ISNM 119, Birkhäuser, Boston, 563–576.