Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-25T21:58:09.880Z Has data issue: false hasContentIssue false
  • English
  • Français

Orthogonal polynomials: applications and computation

Published online by Cambridge University Press:  07 November 2008

Walter Gautschi
Walter Gautschi
Affiliation:
Department of Computer SciencesPurdue UniversityWest Lafayette, IN 47907–1398, USA E-mail: wxg@cs.purdue.edu
Get access Rights & Permissions [Opens in a new window]

Extract

We give examples of problem areas in interpolation, approximation, and quadrature, that call for orthogonal polynomials not of the classical kind. We then discuss numerical methods of computing the respective Gauss-type quadrature rules and orthogonal polynomials. The basic task is to compute the coefficients in the three-term recurrence relation for the orthogonal polynomials. This can be done by methods relying either on moment information or on discretization procedures. The effect on the recurrence coefficients of multiplying the weight function by a rational function is also discussed. Similar methods are applicable to computing Sobolev orthogonal polynomials, although their recurrence relations are more complicated. The paper concludes with a brief account of available software.

Type
Research Article
Information
Acta Numerica , Volume 5 , January 1996 , pp. 45 - 119
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

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.Google Scholar
Althammer, P. (1962), ‘Eine Erweiterung des Orthogonalitätsbegriffes bei Polynomen und deren Anwendung auf die beste Approximation’, J. Reine Angew. Math. 211, 192–204.CrossRefGoogle Scholar
Askey, R. and Ismail, M. (1984), ‘Recurrence relations, continued fractions and orthogonal polynomials’, Memoirs AMS 49, no. 300, Amer. Math. Soc, Providence, RI.Google Scholar
Bellen, A. (1981), ‘A note on mean convergence of Lagrange interpolation’, J. Approx. Theory 33, 85–95.CrossRefGoogle Scholar
Bellen, A. (1988), ‘Alcuni problemi aperti sulla convergenza in media dell'interpolazione Lagrangiana estesa’, Rend. Istit. Mat. Univ. Trieste 20, Fasc. suppl., 1–9.Google Scholar
Boersma, J. and Dempsey, J. P. (1992), ‘On the numerical evaluation of Legendre's chi-function’, Math. Comp. 59, 157–163.Google Scholar
Boley, D. and Golub, G. H. (1987), ‘A survey of matrix inverse eigenvalue problems’, Inverse Problems 3, 595–622.CrossRefGoogle Scholar
Borges, C. F. (1994), ‘On a class of Gauss-like quadrature rules’, Numer. Math. 67, 271–288.CrossRefGoogle Scholar
Bowers, K. and Lund, J., eds (1989), Computation and Control, Progress in Systems and Control Theory 1, Birkhäuser, Boston.CrossRefGoogle Scholar
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.Google Scholar
Buhmann, M. D. and Iserles, A. (1992), ‘On orthogonal polynomials transformed by the QR algorithm’, J. Comput. Appl. Math. 43, 117–134.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Calder, A. C., Laframboise, J. G. and Stauffer, A. D. (1983), ‘Optimum step-function approximation of the Maxwell distribution’, unpublished manuscript.Google Scholar
Caliò, F., Gautschi, W. and Marchetti, E. (1986), ‘On computing Gauss–Kronrod quadrature formulae’, Math. Comp. 47, 639–650.CrossRefGoogle Scholar
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.]Google Scholar
Chihara, T. S. (1978), An Introduction to Orthogonal Polynomials, Gordon and Breach, New York.Google Scholar
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.]Google Scholar
Criscuolo, G., Mastroianni, G. and Nevai, P. (1993), ‘Mean convergence of derivatives of extended Lagrange interpolation with additional nodes’, Math. Nachr. 163, 73–92.CrossRefGoogle Scholar
Criscuolo, G., Mastroianni, G. and Occorsio, D. (1990), ‘Convergence of extended Lagrange interpolation’, Math. Comp. 55, 197–212.CrossRefGoogle Scholar
Criscuolo, G., Mastroianni, G. and Occorsio, D. (1991), ‘Uniform convergence of derivatives of extended Lagrange interpolation’, Numer. Math. 60, 195–218.CrossRefGoogle Scholar
Criscuolo, G., Mastroianni, G. and Vértesi, P. (1992), ‘Pointwise simultaneous convergence of extended Lagrange interpolation with additional knots’, Math. Comp. 59, 515–531.CrossRefGoogle Scholar
Davis, P. J. (1993), Spirals: From Theodorus to Chaos, A K Peters, Wellesley, MA.Google Scholar
Davis, P. J. and Rabinowitz, P. (1956), ‘Abscissas and weights for Gaussian quadratures of high order’, J. Res. Nat. Bur. Standards 56, 35–37.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Dombrowski, J. and Nevai, P. (1986), ‘Orthogonal polynomials, measures and recurrence relations’, SIAM J. Math. Anal. 17, 752–759.CrossRefGoogle Scholar
Egecioglu, Ö. and Koç, C. K. (1989), ‘A fast algorithm for rational interpolation via orthogonal polynomials’, Math. Comp. 53, 249–264.CrossRefGoogle Scholar
Erdős, P. and Turán, P. (1937), ‘On interpolation I: quadrature- and mean-convergence in the Lagrange-interpolation’, Ann. of Math. 38, 142–155.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Fejér, L. (1933), ‘Mechanische Quadraturen mit positiven Cotesschen Zahlen’, Math. Z. 37, 287–309.CrossRefGoogle Scholar
Fischer, H.-J. (1996), ‘On the condition of orthogonal polynomials via modified moments’, Z. Anal. Anwendungen, to appear.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Freud, G. (1971), Orthogonal Polynomials, Pergamon, New York.Google Scholar
Freund, R. W., Golub, G. H. and Nachtigal, N. M. (1991), ‘Iterative solution of linear systems’, Acta Numerica, Cambridge University Press, 57–100.Google Scholar
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.Google Scholar
Frontini, M., Gautschi, W. and Milovanović, G. V. (1987), ‘Moment-preserving spline approximation on finite intervals’, Numer. Math. 50, 503–518.CrossRefGoogle Scholar
Galant, D. (1971), ‘An implementation of Christoffel's theorem in the theory of orthogonal polynomials’, Math. Comp. 25, 111–113.Google Scholar
Galant, D. (1992), ‘Algebraic methods for modified orthogonal polynomials’, Math. Comp. 59, 541–546.CrossRefGoogle Scholar
Gautschi, W. (1963), ‘On inverses of Vandermonde and confluent Vandermonde matrices. II’, Numer. Math. 5, 425–430.CrossRefGoogle Scholar
Gautschi, W. (1967), ‘Numerical quadrature in the presence of a singularity’, SIAM J. Numer. Anal. 4, 357–362.CrossRefGoogle Scholar
Gautschi, W. (1968), ‘Construction of Gauss–Christoffel quadrature formulas’, Math. Comp. 22, 251–270.CrossRefGoogle Scholar
Gautschi, W. (1970), ‘On the construction of Gaussian quadrature rules from modified moments’, Math. Comp. 24, 245–260.Google Scholar
Gautschi, W. (1979), ‘On generating Gaussian quadrature rules’, in Numerische Integration (Hämmerlin, G., ed.), ISNM 45, Birkhäuser, Basel, 147–154.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Gautschi, W. (1981 b), ‘Minimal solutions of three-term recurrence relations and orthogonal polynomials’, Math. Comp. 36, 547–554.CrossRefGoogle Scholar
Gautschi, W. (1982 a), ‘On generating orthogonal polynomials’, SIAM J. Sci. Statist. Comput. 3, 289–317.CrossRefGoogle Scholar
Gautschi, W. (1982 b), ‘An algorithmic implementation of the generalized Christoffel theorem’, in Numerical Integration (Hämmerlin, G., ed.), ISNM 57, Birkhäuser, Basel, 89–106.CrossRefGoogle Scholar
Gautschi, W. (1983), ‘How and how not to check Gaussian quadrature formulae’, BIT 23, 209–216.CrossRefGoogle Scholar
Gautschi, W. (1984 a), ‘On some orthogonal polynomials of interest in theoretical chemistry’, BIT 24, 473–483.CrossRefGoogle Scholar
Gautschi, W. (1984 b), ‘Discrete approximations to spherically symmetric distributions’, Numer. Math. 44, 53–60.CrossRefGoogle Scholar
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.Google Scholar
Gautschi, W. (1985), ‘Orthogonal polynomials – constructive theory and applications’, J. Comput. Appl. Math. 12/13, 61–76.CrossRefGoogle Scholar
Gautschi, W. (1986 a), ‘On the sensitivity of orthogonal polynomials to perturbations in the moments’, Numer. Math. 48, 369–382.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
Gautschi, W. (1991 a), ‘A class of slowly convergent series and their summation by Gaussian quadrature’, Math. Comp. 57, 309–324.CrossRefGoogle Scholar
Gautschi, W. (1991 b), ‘On certain slowly convergent series occurring in plate contact problems’, Math. Comp. 57, 325–338.CrossRefGoogle Scholar
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.Google Scholar
Gautschi, W. (1992), ‘On mean convergence of extended Lagrange interpolation’, J. Appl. Comput. Math. 43, 19–35.CrossRefGoogle Scholar
Gautschi, W. (1993 a), ‘Is the recurrence relation for orthogonal polynomials always stable?’, BIT 33, 277–284.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Gautschi, W. (1993 c), ‘On the computation of generalized Fermi–Dirac and Bose–Einstein integrals’, Comput. Phys. Comm. 74, 233–238.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Gautschi, W. (1996 a), ‘On the computation of special Sobolev-type orthogonal polynomials’, Ann. Numer. Math., to appear.Google Scholar
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.Google Scholar
Gautschi, W. (1996 c), ‘Moments in quadrature problems’, Comput. Math. Appl., Ser. B, to appear.Google Scholar
Gautschi, W. and Li, S. (1993), ‘A set of orthogonal polynomials induced by a given orthogonal polynomial’, Aequationes Math. 46, 174–198.CrossRefGoogle Scholar
Gautschi, W. and Li, S. (1996), ‘On quadrature convergence of extended Lagrange interpolation’, Math. Comp., to appear.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Gautschi, W. and Milovanović, G. V. (1986), ‘Spline approximations to spherically symmetric distributions’, Numer. Math. 49, 111–121.CrossRefGoogle Scholar
Gautschi, W. and Zhang, M. (1995), ‘Computing orthogonal polynomials in Sobolev spaces’, Numer. Math. 71, 159–183.CrossRefGoogle Scholar
Golub, G. H. (1973), ‘Some modified matrix eigenvalue problems’, SIAM Rev. 15, 318–334.CrossRefGoogle Scholar
Golub, G. H. and Van Loan, C. F. (1989), Matrix Computations, 2nd edn, The Johns Hopkins University Press, Baltimore.Google Scholar
Golub, G. H. and Welsch, J. H. (1969), ‘Calculation of Gauss quadrature rules’, Math. Comp. 23, 221–230.CrossRefGoogle Scholar
Gordon, R. G. (1968), ‘Error bounds in equilibrium statistical mechanics’, J. Mathematical Phys. 9, 655–663.CrossRefGoogle Scholar
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.Google Scholar
Gori, L. and Santi, E. (1992), ‘Moment-preserving approximations: a monospline approach’, Rend. Mat. (7) 12, 1031–1044.Google Scholar
Gragg, W. B. and Harrod, W. J. (1984), ‘The numerically stable reconstruction of Jacobi matrices from spectral data’, Numer. Math. 44, 317–335.CrossRefGoogle Scholar
Graves-Morris, P. R. and Hopkins, T. R. (1981), ‘Reliable rational interpolation’, Numer. Math. 36, 111–128.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Hageman, L. A. and Young, D. M. (1981), Applied Iterative Methods, Academic Press, New York.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Jacobi, C. G. J. (1826), ‘Ueber Gaußs neue Methode, die Werthe der Integrale näherungsweise zu finden’, J. Reine Angew. Math. 1, 301–308.Google Scholar
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.]Google Scholar
Kautsky, J. and Elhay, S. (1984), ‘Gauss quadratures and Jacobi matrices for weight functions not of one sign’, Math. Comp. 43, 543–550.Google Scholar
Kautsky, J. and Golub, G. H. (1983), ‘On the calculation of Jacobi matrices’, Linear Algebra Appl. 52/53, 439–455.CrossRefGoogle Scholar
Kovačević, M. A. and Milovanović, G. V. (1996), ‘Spline approximation and generalized Turán quadratures’, Portugal. Math., to appear.Google Scholar
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.Google Scholar
Laurie, D. P. (1996), ‘Calculation of Gauss–Kronrod quadrature rules’, Math. Comp., to appear.Google Scholar
Lewanowicz, S. (1994), ‘A simple approach to the summation of certain slowly convergent series’, Math. Comp. 63, 741–745.CrossRefGoogle Scholar
Lewis, D. C. (1947), ‘Polynomial least square approximations’, Amer. J. Math. 69, 273–278.CrossRefGoogle Scholar
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.Google Scholar
Lin, J.-C. (1988), ‘Rational L 2-approximation with interpolation’, Ph.D. thesis, Purdue University.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
Mastroianni, G. and V, P.értesi (1993), ‘Mean convergence of Lagrange interpolation on arbitrary systems of nodes’, Acta Sci. Math. (Szeged) 57, 429–441.Google Scholar
Meijer, H. G. (1993), ‘Coherent pairs and zeros of Sobolev-type orthogonal polynomials’, Indag. Math. (N.S.) 4, 163–176.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Milovanović, G. V. (1994), ‘Summation of series and Gaussian quadratures’, in Approximation and Computation (Zahar, R. V. M., ed.), Birkhäuser, Boston, 459–475.Google Scholar
Milovanović, G. V (1995), ‘Summation of series and Gaussian quadratures, II’, Numer. Algorithms 10, 127–136.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Milovanović, G. V. and Kovačević, M. A. (1992), ‘Moment-preserving spline approximation and quadrature’, Facta Univ. Ser. Math. Inform. 7, 85–98.Google Scholar
Monegato, G. (1982), ‘Stieltjes polynomials and related quadrature rules’, SIAM Rev. 24, 137–158.CrossRefGoogle Scholar
Moszyński, K. (1992), ‘Remarks on polynomial methods for solving systems of linear algebraic equations’, Appl. Math. 37, 419–436.CrossRefGoogle Scholar
NAG Fortran Library Manual, Mark 15, Vol. 10, NAG Inc., 1400 Opus Place, Suite 200, Downers Grove, IL 60515–5702.Google Scholar
Natanson, I. P. (1964/1965), Constructive Function Theory, Vols 1–III, Ungar Publ. Co., New York.Google Scholar
Newbery, A. C. R. (unpublished), ‘Gaussian principles applied to summation,’ unpublished manuscript.Google Scholar
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.Google Scholar
Parlett, B. N. (1980), The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
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.CrossRefGoogle Scholar
Pólya, G. (1933), ‘Über die Konvergenz von Quadraturverfahren’, Math. Z. 37, 264–286.CrossRefGoogle Scholar
Rabinowitz, P. (1960), ‘Abscissas and weights for Lobatto quadrature of high order’, Math. Comp. 14, 47–52.CrossRefGoogle Scholar
Rabinowitz, P. and Weiss, G. (1959), ‘Tables of abscissas and weights for numerical evaluation of integrals of the form ’, Math, Tables Aids Comp. 13, 285–294.CrossRefGoogle Scholar
Russon, A. E. and Blair, J. M. (1969), ‘Rational function minimax approximations for the Bessel functions K 0(x) and K 1(x)’, Rep. AECL–3461, Atomic Energy of Canada Limited, Chalk River, Ontario.Google Scholar
Sack, R. A. and Donovan, A. F. (1969), ‘An algorithm for Gaussian quadrature given generalized moments’, Dept. Math., Univ. of Salford, Salford, UK.Google Scholar
Sack, R. A. and Donovan, A. F. (1971/1972), ‘An algorithm for Gaussian quadrature given modified moments’, Numer. Math. 18, 465–478.CrossRefGoogle Scholar
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.Google Scholar
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.]CrossRefGoogle Scholar
Stoer, J. and Bulirsch, R. (1980), Introduction to Numerical Analysis, 2nd edn, Springer, New York.CrossRefGoogle Scholar
Szeg, G.ő (1975), Orthogonal Polynomials, Colloq. Publ. 23, 4th edn, Amer. Math. Soc, Providence, RI.Google Scholar
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.Google Scholar
Tricomi, F. G. (1954), Funzioni Ipergeometriche Confluenti, Edizioni Cremonese, Rome.Google Scholar
Uvarov, V. B. (1959), ‘Relation between polynomials orthogonal with different weights’ (Russian), Dokl. Akad. Nauk SSSR 126, 33–36.Google Scholar
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.]Google Scholar
Van Assche, W. and Vanherwegen, I. (1993), ‘Quadrature formulas based on rational interpolation’, Math. Comp. 61, 765–783.CrossRefGoogle Scholar
Walsh, J. L. (1969), Interpolation and Approximation by Rational Functions in the Complex Domain, Colloq. Publ. 20, 5th edn, Amer. Math. Soc, Providence, RI.Google Scholar
Wheeler, J. C. (1974), ‘Modified moments and Gaussian quadrature’, Rocky Mountain J. Math. 4, 287–296.CrossRefGoogle Scholar
Wheeler, J. C. (1984), ‘Modified moments and continued fraction coefficients for the diatomic linear chain’, J. Chem. Phys. 80, 472–476.CrossRefGoogle Scholar
Widder, D. V. (1941), The Laplace Transform, Princeton University Press.Google Scholar
Wilf, H. S. (1962), Mathematics for the Physical Sciences, Wiley, New York.Google Scholar
Wilf, H. S. (1980), personal communication.Google Scholar
Wong, R. (1982), ‘Quadrature formulas for oscillatory integral transforms’, Numer. Math. 39, 351–360.CrossRefGoogle Scholar
Wong, R. (1989), Asymptotic Approximations of Integrals, Academic Press, Boston.Google Scholar
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.Google Scholar