Skip to main content
SpringerLink
Log in

A Survey of Gauss-Christoffel Quadrature Formulae

  • Chapter
E. B. Christoffel

Abstract

We present a historical survey of Gauss-Christoffel quadrature formulae, beginning with Gauss’ discovery of his well-known method of approximate integration and the early contributions of Jacobi and Christoffel, but emphasizing the more recent advances made after the emergence of powerful digital computing machinery. One group of inquiry concerns the development of the quadrature formula itself, e.g. the inclusion of preassigned nodes and the admission of multiple nodes, as well as other generalizations of the quadrature sum. Another is directed towards the widening of the class of integrals made accessible to Gauss-Christoffel quadrature. These include integrals with nonpositive measures of integration and singular principal value integrals. An account of the error and convergence theory will also be given, as well as a discussion of modern methods for generating Gauss-Christoffel formulae, and a survey of numerical tables.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 74.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • Afshar, R., Mueller, F.M. and Shaffer, J.C. [1973]: Hilbert transformation of densities of states using Hermite functions. J. Computational Phys. 11, 190–209.

    Google Scholar 

  • Aizentat, V.S., Krylov, V.I. and Metelsk, A.S. [1962]: Tables for the Numerical Laplace Transform and the Evaluation of Integrals of the Form xef (x)dx (Russian). Izdat. Akad. Nauk BSSR, Minsk.

    Google Scholar 

  • Akkerman, R.B. [1959]: Quadrature formulas of the type of Markov formulas (Russian). Trudy Mat. Inst. Steklov 53, 5–15.

    Google Scholar 

  • Bahvalov, N.S. and Vasil’eva, L.G. [1968]: The calculation of integrals of oscillatory functions by interpolation at the Gaussian nodes (Russian), Vyisl. Mat. i Mat. Fiz. 8, 175–181.

    Google Scholar 

  • Baillaud, B. and Bourget, H. [1905]: Correspondance d’Hermite et de Stieltjes I, I I. Gauthier-Villars, Paris.

    Google Scholar 

  • Bandemer, H. [1966]: Erweiterung der Stekloffschen Sätze über mechanische Quadraturverfahren. Wiss. Z. Techn. Hochsch. Karl-Marx-Stadt 8, 205–208.

    Google Scholar 

  • Bandemer, H. [1967]: Über verallgemeinerte Interpolationsquadratur. Math. Nachr. 34, 379–387.

    Google Scholar 

  • Baratella, P. [1979]: Un’estensione ottimale della formula di quadratura di Radau. Rend. Sem. Mat. Univ. e Politec. Torino 37, 147–158.

    Google Scholar 

  • Barnhill, R.E. [1968]: Asymptotic properties of minimum norm and optimal quadratures. Numer. Math. 12, 384–393.

    Google Scholar 

  • Barrett, W. [1960/61): Convergence properties of Gaussian quadrature formulae. Comput. J. 3, 272–277.

    Google Scholar 

  • Barrow, D.L. [1976]: Existence of Gauss interpolation formulas for the one-dimensional heat equation. Math. Comp. 30, 24–34.

    Google Scholar 

  • Barrow, D.L. [1977]: Gauss interpolation formulas and totally positive kernels. Math. Comp. 31, 984–993.

    Google Scholar 

  • Barrow, D.L. [1978]: On multiple node Gaussian quadrature formulae. Math. Comp. 32, 431–439.

    Google Scholar 

  • Barrow, D.L., Stroud, A.H. [1976]: Existence of Gauss harmonic interpolation formulas SIAM J. Numer. Anal. 13, 18–26.

    Google Scholar 

  • Berger, B.S. and Danson, R. [1968]: Tables of zeros and weights for Gauss -Laguerre quadrature. Math. Comp. 22, 458–459.

    Google Scholar 

  • Berger, B.S. and Danson, R., Carpenter, R. [1969]: Tables of zeros and weights for Gauss -Laguerre quadrature to 24S for N = 400, 500 and 600, and tables of zeros and weights for Gauss -Laguerre quadrature to 23S for N = 700, 800 and 900. Math. Comp. 23, Review 60, 882.

    Google Scholar 

  • Bernstein, S. [1918]: Quelques remarques sur l’interpolation. Math. Ann. 79, 1–12.

    Google Scholar 

  • Bernstein, S. [1930]: Sur les polynomes orthogonaux relatifs à un segment fini. J. Math. Pures Appl. (9) 9, 127–177.

    Google Scholar 

  • Bertova, E.I., Kuznecov, Ja.T., Natanson, I.P. and Caregradskii, H.A. [1953]: Approximate computation of definite integrals by means of the multiplicative method of isolating the singularity (Russian). Prikl. Mat. i Meh. 17, 639–644.

    Google Scholar 

  • Bezikovi, J.S. [1937]: On formulas of mechanical quadrature with n ordinates, exact for polynomials of degree not higher than 2n-2 and 2n-3 (Russian). Trudy Leningr. Indust. Inst. 4, 1–18.

    Google Scholar 

  • Bezikovi, J.S. [1939 ): Process of mechanical quadratures for improper integrals (Russian) Leningrad. Gos. Univ. U6. Zap. Ser. Mat. Nauk 6, 36–42.

    Google Scholar 

  • Bilharz, H. [1951]: Über die Gaußsche Methode zur angenäherten Berechnung bestimmter Integrale. Math. Nachr. 6, 171–192.

    Google Scholar 

  • Billauer, A. [1974]: On Gaussian quadrature by divided differences of a modified function. BIT 14, 359–361.

    Google Scholar 

  • Blue, J.L. [1979]: A Legendre polynomial integral. Math. Comp. 33, 739–741.

    Google Scholar 

  • Boland, W.R. [1972]: The convergence of product-type quadrature formulas. SIAM J. Numer. Anal. 9, 6–13.

    Google Scholar 

  • Boland, W.R. [1973]: Properties of product-type quadrature formulas. BIT 13, 287–291.

    Google Scholar 

  • Boland, W.R., Duris, C.S. [1971 ): Product-type quadrature formulas. BIT 11, 139–158.

    Google Scholar 

  • Boujot, J.-P. and Maroni, P. [1968]: Algorithme général de construction de tables de Gauss pour les problèmes de quadratures. Institut Blaise Pascal, Publ. n° NMX/8.1.8/AI, Paris.

    Google Scholar 

  • Bouzitat, J. [1952]: Intégration numérique approchée par la méthode de Gauss généralisée et extension de cette méthode. In: Mineur, H., Techniques de Calcul Numérique, pp. 557–605. Beranger, Paris.

    Google Scholar 

  • Branders, M. [1976]: Application of Chebyshev polynomials in numerical integration (Flemish). Dissertation, Catholic University of Leuven, Leuven.

    Google Scholar 

  • Branders, M., Piessens, R. [1971]: Gaussian quadrature formulas for integrals with logarithmic singularity. Report TW8, Appl. Math. Progr. Div., Catholic University of Leuven, Leuven.

    Google Scholar 

  • Branders, M., Piessens, R. [1975]: An extension of Clenshaw-Curtis quadrature. J. Comput. Appl. Math. 1, 55–65.

    Google Scholar 

  • Brass, H. [1977]: Quadraturverfahren. Vandenhoeck and Ruprecht, Göttingen.

    Google Scholar 

  • Brass, H. [1978]: Monotonie bei den Quadraturverfahren von Gauss und Newton-Cotes. Numer. Math. 30, 349–354.

    Google Scholar 

  • Bruin, M.G., Saff, E.B. and Varga, R.S. [to appear]: On the zeros of generalized Bessel polynomials. Indag. Math.

    Google Scholar 

  • Butzer, P.L. [1979/80]: The Banach-Steinhaus theorem with rates, and applications to various branches of analysis. In: General Inequalities II (Ed. by Beckenbach, E.F.), Birkhäuser, Basel.

    Google Scholar 

  • Butzer, P.L., Scherer, K. and Westphal, U. [1973]: On the Banach-Steinhaus theorem and application in locally convex spaces. Acta Sci. Math. (Szeged) 34, 25–34.

    Google Scholar 

  • Byrd, P.F. and Galant, D.C. [1970]: Gauss quadrature rules involving some nonclassical weight functions. NASA Techn. Note D-5785, Ames Research Center, Moffett Field, California.

    Google Scholar 

  • Capovani, M., Ghelardoni, G. and Lombardi, G. [1976a]: Utilizzazione di proprietà delle matrici per lo studio dei polinomi ultrasferici. Rend. Mat. 9, 57–69.

    Google Scholar 

  • Capovani, M., Ghelardoni, G. and Lombardi, G. [1976b]: Un metodo per il calcolo dei nodi e dei pesi delle formule di quadratura di Gauss-Hermite. Calcolo 13, 441–452.

    Google Scholar 

  • Cassity, C.R. [1965]: Abscissas, coefficients, and error term for the generalized Gauss-Laguerre quadrature formula using the zero ordinate. Math. Comp. 19, 287–296.

    Google Scholar 

  • Cassity, C.R., Hopper, H.R. [1964]: Abscissas and coefficients for the generalized Gauss-Laguene quadrature formula using the zero ordinate. NASA Report TMX-53099, George C. Marshall Space Flight Center, Huntsville, Alabama.

    Google Scholar 

  • Cecchi, M.M. [1967]: L’integrazione numerica di una classe di integrali utili nei calcoli quantomeccanici. Calcolo 4, 363–368.

    Google Scholar 

  • Cecchi, M.M. [no date]: Tavole di nodi e pesi per ff(x)dx Af(x,). Ist. Elaborazione della Informazione, Felici, Pisa.

    Google Scholar 

  • Chakalov, L. [1930/31]: Sur les formules de quadrature à nombre minimum de termes. Bull. Math. Phys. Bucarest Z 160–163.

    Google Scholar 

  • Chakalov, L. [1954]: Geleral quadrature formulae of Gaussian type (Bulgarian). Bulgar. Akad. Nauk Izv. Mat. Inst. 1, 67–84.

    Google Scholar 

  • Chakalov, L. [1957]: Formules générales de quadrature mécanique du type de Gauss. Colloq. Math. 5, 69–73.

    Google Scholar 

  • Chawla, M.M. [1967]: On the Chebyshev polynomials of the second kind. SIAM Rev. 9, 729–733.

    Google Scholar 

  • Chawla, M.M. [1968]: Error bounds for the Gauss -Chebyshev quadrature formula of the closed type. Math. Comp. 22, 889–891.

    Google Scholar 

  • Chawla, M.M. [1969]: Asymptotic estimates for the error of the Gauss-Legendre quadrature formula. Comput. J. 11, 339–340.

    Google Scholar 

  • Chawla, M.M. [1969]: On Davis’ method for the estimation of errors of Gauss-Chebyshev quadratures. SIAM J. Numer. Anal. 6, 108–117.

    Google Scholar 

  • Chawla, M.M. [1970a]: Estimation of errors of Gauss-Chebyshev quadratures. Comput. J. 13, 107–109.

    Google Scholar 

  • Chawla, M.M. [1970b]: Hilbert spaces for estimating errors of quadratures for analytic functions. BIT 10, 145–155.

    Google Scholar 

  • Chawla, M.M. [1971a]: Asymptotic Gauss quadrature errors as Fourier coefficients of the integrand. J. Austral. Math. Soc. 12, 315–322.

    Google Scholar 

  • Chawla, M.M. [1971b]: Estimating errors of numerical approximation for analytic functions. Numer. Math. 16, 370–374.

    Google Scholar 

  • Chawla, M.M., Jain, M.K. [1968a]: Error estimates for Gauss quadrature formulas for analytic functions. Math. Comp. 22, 82–90.

    Google Scholar 

  • Chawla, M.M. [1968b]: Asymptotic error estimates for the Gauss quadrature formula. Math. Comp. 22, 91–97.

    Google Scholar 

  • Chawla, M.M., Jayarajan, N. [1975]: Quadrature formulas for Cauchy principal value integrals. Computing 15, 347–355.

    Google Scholar 

  • Chawla, M.M., Kumar, S. [1978]: Convergence of Fejér type quadrature formulas for Cauchy principal value integrals. J. Math. Phys. Sci. 12, 49–59.

    Google Scholar 

  • Chawla, M.M., Kumar, S. [1979]: Convergence of quadratures for Cauchy principal value integrals. Computing 23, 67–72.

    Google Scholar 

  • Chawla, M.M., Kumar, S., Ramakrishnan, T.R. [1974]: Modified Gauss-Jacobi quadrature formulas for the numerical evaluation of Cauchy-type singular integrals. BIT 14, 14–21.

    Google Scholar 

  • Chebyshev, P.L. [1855]: On continued fractions (Russian). Men. Zap. Imp. Akad. Nauk 3, 636–664.

    Google Scholar 

  • Chebyshev, P.L. [1855]: French translation: J. Math. Pures Appt. (2) 3 (1858), 289–322

    Google Scholar 

  • Chebyshev, P.L. [1859a]: Sur l’interpolation par la méthode des moindres carrés. Mém. Acad. Impér. Sci. St. Pétersbourg (7) 1, no. 15, 1–24.

    Google Scholar 

  • Chawla, M.M., Kumar, S. [1859b]: Sur le développement des fonctions à une seule variable. Bull. Phys.-Math. Acad. Imp. Sci. St. Pétersbourg 1, 193–200.

    Google Scholar 

  • Chisholm, J.S.R., Genz, A. and Rowlands, G.E. [1972]: Accelerated convergence of sequences of quadrature approximations. J. Computational Phys. 10, 284–307.

    Google Scholar 

  • Christoffel, E.B. [1858]: Ober die Gaußische Quadratur und eine Verallgemeinerung derselben. J. Reine Angew. Math. 55, 61–82.

    Google Scholar 

  • Christoffel, E.B. [1877]: Sur une classe particulière de fonctions entières et de fractions continues. Ann. Mat. Pura Appl. (2) 8, 1–10

    Google Scholar 

  • Chui, C.K. [1972]: Concerning Gaussian-Chebyshev quadrature errors. SIAM J. Numer. Anal. 9, 237–240.

    Google Scholar 

  • Cohen, H. [1978]: Accurate numerical solutions of integral equations with kernels containing poles. J. Computational Phys. 26, 257–276.

    Google Scholar 

  • Concus, P. [1964]: Additional tables for the evaluations of fox°e ‘f(x)dx by Gauss-Laguerre quadrature. Math. Comp. 18, Review 81, 523.

    Google Scholar 

  • Concus, P., Cassatt, D., Jaehnig, G., Melby, E. [1963]: Tables for the evaluation of xge f(x)dx by Gauss -Laguerre quadrature. Math. Comp. 17, 245–256.

    Google Scholar 

  • Cosma Cagnazzi, L. [1970]: Su un nuovo tipo di formule di maggiorazione del resto di una formula di quadratura. Rend. Mat. (6) 3, 203–210.

    Google Scholar 

  • Cranley, R. and Patterson, T.N.L. [1971]: On the automatic numerical evaluation of definite integrals. Comput. J. 14, 189–198.

    Google Scholar 

  • Crout, P.D. [1930]: The approximation of functions and integrals by a linear combination of functions J. Math. Phys. (M.I.T.) 9, 278–314.

    Google Scholar 

  • Curtis, A.R. and Rabinowitz, P. [1972]: On the Gaussian integration of Chebyshev polynomials. Math. Comp. 26, 207–211.

    Google Scholar 

  • Danloy, B. [1973]: Numerical construction of Gaussian quadrature formulas for Log x)x°f(x)dx and E,,(x)f(x)dx. Math. Comp. 27, 861–869.

    Google Scholar 

  • Darboux, G. [1878]: Mémoire sur l’approximation des fonctions de très-grands nombres et sur une classe étendue de développements en série. J. Math. Pures Appl. 4 (3), 5–56.

    Google Scholar 

  • Davis, P.J. [1953]: Errors of numerical approximation for analytic functions. J. Rational Mech. Anal. 2, 303–313.

    Google Scholar 

  • Davis, P.J. [1962]: Errors of numerical approximation for analytic functions. In: Survey of Numerical Analysis (Ed. by Todd, J. ), 468–484. McGraw-Hill, New York.

    Google Scholar 

  • Davis, P.J., Rabinowitz, P. [1954]: On the estimation of quadrature errors for analytic functions. Math. Tables Aids Comput. 8, 193–203.

    Google Scholar 

  • Davis, P.J., Rabinowitz, P. [1975]: Methods of Numerical Integration. Academic Press, New York.

    Google Scholar 

  • Dekanosidze, E.N. [1966]: Tables of Roots and Weight-factors of Generalized Laguerre Polynomials (Russian). Vy6isl. Centr. Akad. Nauk SSSR, Moscow.

    Google Scholar 

  • Delves, L.M. [1967/68]: The numerical evaluation of principal value integrals. Comput. J. 10, 389–391.

    Google Scholar 

  • Deruyts, J. [1886]: Sur le calcul approché de certaines intégrales définies. Bull. Acad. Roy. Belgique 11 (3), 307–311.

    Google Scholar 

  • Donaldson, J.D. [1973]: Estimates of upper bounds for quadrature errors. SIAM J. Numer. Anal. 10, 13–22.

    Google Scholar 

  • Donaldson, J.D., Elliott, D. [1972]: A unified approach to quadrature rules with asymptotic estimates of their remainders. SIAM J. Numer. Anal. 9, 573–602.

    Google Scholar 

  • Dyn, N. [1979]: On the existence of Hermite-Birkhoff quadrature formulas of Gaussian type. MRC Tech. Summary Rep. # 1978, Mathematics Research Center, University of Wisconsin, Madison, Wisconsin.

    Google Scholar 

  • Elliott, D. [1979]: On the convergence of Hunter’s quadrature rule for Cauchy principal value integrals. BIT 19, 457–462.

    Google Scholar 

  • Elliott, D., Paget, D.F. [1975]: On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals. Numer. Math. 23, 311–319.

    Google Scholar 

  • Elliott, D., Paget, D.F. [1976a]: On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals: an addendum. Numer. Math. 25, 287–289.

    Google Scholar 

  • Elliott, D., Paget, D.F. [1976b]: Product-integration rules and their convergence. BIT 16, 32–40.

    Google Scholar 

  • Elliott, D., Paget, D.F. [1978]: The convergence of product integration rules. BIT 18, 137–141.

    Google Scholar 

  • Elliott, D., Paget, D.F. [1979]: Gauss type quadrature rules for Cauchy principal value integrals. Math Comp. 33, 301–309.

    Google Scholar 

  • Engels, H. [1972]: Ober allgemeine Gaußsche Quadraturen. Computing 10, 83–95.

    Google Scholar 

  • Engels, H. [1973]: Spezielle Interpolationsquadraturen vom Gauss’schen Typ. In: Numerische, insbesondere approximationstheoretische Behandlung von Funktionalgleichungen (Ed. By Ansorge, R. and Törnig, W.), 54–68. Lecture Notes in Math. 333, Springer, Berlin.

    Google Scholar 

  • Engels, H. [1974]: Positive Interpolationsquadraturen mit teilweise vorgeschriebenen Stützstellen. Computing 13, 121–141.

    Google Scholar 

  • Engels, H. [1977]: Eine Familie interpolatorischer Quadraturformeln mit ableitungsfreien Fehlerschranken. Numer. Math. 28, 49–58.

    Google Scholar 

  • Erdogan, F. and Gupta, G.D. [1972]: On the numerical solution of singular integral equations. Quart. Appl. Math. 29, 525–534.

    Google Scholar 

  • Esser, H. [1971a]: Konvergenz von Quadraturverfahren vom Radau-Typ. Computing 7, 254–263.

    Google Scholar 

  • Esser, H. [1971b]: Bemerkungen zu einem Satz von G. Freud über Quadraturkonvergenz. Computing 8, 216–220.

    Google Scholar 

  • Esser, H. [1972]: Konvergenz von Quadraturverfahren vom Gaußschen Typ. Studia Sci. Math. Hungar. 7, 375–378.

    Google Scholar 

  • Feldstein, A. and Miller, R.K. [1971]: Error bounds for compound quadrature of weakly singular integrals. Math. Comp. 25, 505–520.

    Google Scholar 

  • Filippi, S. and Esser, H. [1970]: Darstellungs-und Konvergenzsätze für Quadraturverfahren auf C und. Forschungsber. des Landes Nordrhein-Westfalen No. 2137. Westdeutscher Verlag, Köln.

    Google Scholar 

  • Fishman, H. [1957]: Numerical integration constants. Math. Tables Aids Comput. 11, 1–9.

    Google Scholar 

  • Fock, V. [1932]: On the remainder term of certain quadrature formulae (Russian). Bull. Acad. Sci. Leningrad (7), 419–448.

    Google Scholar 

  • Freud, G. [1971]: Orthogonal Polynomials. Pergamon Press, New York.

    Google Scholar 

  • Freud, G. [1973]: Error estimates for Gauss-Jacobi quadrature formulae. In: Topics in Numerical Analysis (Ed. by Miller, J.J.H. ), 113–121. Academic Press, London.

    Google Scholar 

  • Freud, G. [1975a]: Numerical estimates for the error of Gauss-Jacobi quadrature formulae. In: Topics in Numerical Analysis II (Ed. by Miller, J.J.H. ), 43–50. Academic Press, London.

    Google Scholar 

  • Freud, G. [1975b]: Error estimates for Gauss-Jacobi quadrature formulae and some applications of them (Hungarian). Alkalmaz. Mat. Lapok 1, 23–36.

    Google Scholar 

  • Gabdulhaev, B.G. [1975]: Cubature formulae for multidimensional singular integrals II (Russian). Izv. Vysg. U6ebn. Zaved. Matematika, no. 4 (155), 3–13.

    Google Scholar 

  • Gabdulhaev, B.G. [1976]: Quadrature formulae with multiple nodes for singular integrals (Russian). Dokl. Akad. Nauk SSSR 227, 531–534.

    Google Scholar 

  • Gabdulhaev, B.G., Onegov, L.A. [1976]: Cubature formulae for singular integrals (Russian). Izv. Vyss. U6ebn. Zaved. Matematika, no. 7 (170), 100–105.

    Google Scholar 

  • Gahov, R.D. [1958]: Boundary Problems (Russian). Izdat. Fiz. -Mat. Lit., Moscow. [2nd ed., ibid., 1963. English translation: Pergamon Press, Oxford, 1966.]

    Google Scholar 

  • Gaier, D. [1964]: Konstruktive Methoden der konformen Abbildung. Springer, Berlin.

    Google Scholar 

  • Galant, D. [1969]: Gauss quadrature rules for the evaluation of 27r-rn fo exp(-xz)f(x)dx. Math. Comp. 23, Review 42, 676–677. Loose microfiche suppl. E.

    Google Scholar 

  • Galant, D.[1971]: An implementation of Christoffel’s theorem in the theory of orthogonal polynomials. Math. Comp. 25, 111–113.

    Google Scholar 

  • Gatteschi, L. [1964]: Su una formula di quadratura “quasi gaussiana”. Tabulazione delle ascisse d’integrazione e delle relative costanti di Christoffel. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 98, 641–661.

    Google Scholar 

  • Gatteschi, L. [1979]: On the construction of some Gaussian quadrature rules. In: Numerische Integration. ISNM 45 (Ed. by Hämmerlin, G. ), 138–146. Birkhäuser, Basel.

    Google Scholar 

  • Gatteschi, L. [1980]: On some orthogonal polynomial integrals. Math. Comp. 35, 1291–1298.

    Google Scholar 

  • Gatteschi, L., Monegato, G. and Vinardi, G. [1976]: Alcuni problemi relativi alle formule di quadratura del tipo di Tchebycheff. Calcolo 13, 79–104.

    Google Scholar 

  • Gauss, C.F. [1812]: Disquisitiones generales circa seriem infinitam 1 + ((a ß)/(1 + y))x +etc. Commentationes Societatis Regiae Scientarium Gottingensis Recentiores 2. [Werke III, 123–162.]

    Google Scholar 

  • Gauss, C.F. [1814]: Methodus nova integralium valores per approximationem inveniendi. Commentationes Societatis Regiae Scientarium Gottingensis Recentiores 3. [Werke III, 163–196.]

    Google Scholar 

  • Gautschi, W. [1967]: Numerical quadrature in the presence of a singularity. SIAM J. Numer. Anal. 4, 357–362.

    Google Scholar 

  • Gautschi, W. [1968a]: Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, 251–270.

    Google Scholar 

  • Gautschi, W. [1968b]: Algorithm 331 — Gaussian quadrature formulas. Comm. ACM 11, 432–436.

    Google Scholar 

  • Gautschi, W. [1970a]: On the construction of Gaussian quadrature rules from modified moments. Math. Comp. 24, 245–260.

    Google Scholar 

  • Gautschi, W. [1970b]: Tables of Gaussian quadrature rules for the calculation of Fourier coefficients. Math. Comp. 24, no. 110, loose microfiche suppl. A-D.

    Google Scholar 

  • Gautschi, W. [1978]: Questions of numerical condition related to polynomials. In: Recent Advances in Numerical Analysis (Ed. by de Boor, C. and Golub, G.H. ), 45–72. Academic Press, New York.

    Google Scholar 

  • Gautschi, W. [1979a]: On the preceding paper “A Legendre polynomial integral” by James L. Blue. Math. Comp. 33, 742–743.

    Google Scholar 

  • Gautschi, W. [1979b]: On generating Gaussian quadrature rules. In: Numerische Integration. ISNM 45 (Ed. by Hämmerlin, G. ), 147–154. Birkhäuser, Basel.

    Google Scholar 

  • Ghizzetti, A. [1954/55]: Sulle formule di quadratura. Rend. Sem. Mat. Fis. Milano 26, 1–16.

    Google Scholar 

  • Ossicini, A. [1967]: Su un nuovo tipo di sviluppo di una funzione in serie di polinomi. Rend. Accad. Naz. Lincei 43 (8), 21–29.

    Google Scholar 

  • Ossicini, A. [1970]: Quadrature Formulae. Academic Press, New York.

    Google Scholar 

  • Ossicini, A. [1974]: Polinomi s-ortogonali e sviluppi in serie ad essi collegati. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (4), no. 18, 1–16.

    Google Scholar 

  • Ossicini, A. [1975]: Generalizzazione dei polinomi s-ortogonali e dei corrispondenti sviluppi in serie. Atti Accad. Sci. Torino 109, 371–379.

    Google Scholar 

  • Ossicini, A. [1975]: Sull’ esistenza e unicità delle formule di quadratura gaussiane. Rend. Mat. 8 (6), 1–15.

    Google Scholar 

  • Glonti, E.N. [1971]: Tables of Roots and Quadrature Coefficients of Jacobi Polynomials (Russian). Vyéisl. Centr Akad. Nauk SSSR, Moscow.

    Google Scholar 

  • Golub, G.H. [1973]: Some modified matrix eigenvalue problems. SIAM Rev. 15, 318–334.

    Google Scholar 

  • Welsch, J. H. [1969]: Calculation of Gauss quadrature rules. Math. Comp. 23, 221–230.

    Google Scholar 

  • Gordon, R.G. [1968]: Error bounds in equilibrium statistical mechanics. J. Mathematical Phys. 9, 655–663.

    Google Scholar 

  • Gourier, G. [1883]: Sur une méthode capable de fournir une valeur approchée de l’intégrale F(x)dx. C.R. Acad. Sci. Paris 97, 79–82.

    Google Scholar 

  • Gribble, J.D. [1977]: Further properties of inner product quadrature formulas. BIT 17, 392–408.

    Google Scholar 

  • Grinpun, Z.S. [1966]: On a class of orthogonal polynomials (Russian). Vestnik Leningrad. Univ. 21, no. 19, 147–149.

    Google Scholar 

  • Grosswald, E. [1978]: Bessel Polynomials. Lecture Notes in Math. 698, Springer, Berlin.

    Google Scholar 

  • Haber, S. [1964]: A note on some quadrature formulas for the interval (- ce, so). Math. Comp. 18, 313–314.

    Google Scholar 

  • Haber, S. [1971]: On certain optimal quadrature formulas. J. Res. Nat. Bur. Standards 75B, 85–88.

    Google Scholar 

  • Haber, S. [1972]: The error in numerical integration of analytic functions. Quart. Appl. Math. 29, 411–420.

    Google Scholar 

  • Hammer, P.C. and Wicke, H.H. [1960]: Quadrature formulas involving derivatives of the integrand. Math. Comp. 14, 3–7.

    Google Scholar 

  • Hämmerlin, G. [1972]: Fehlerabschätzung bei numerischer Integration nach Gauss. Methoden und Verfahren der mathematischen Physik 6, 153–163. B.I.-Hochschultaschenbücher, No. 725, Bibliographisches Institut, Mannheim.

    Google Scholar 

  • Harper, W.M. [1962]: Quadrature formulas for infinite intervals. Math. Comp. 16, 170–175.

    Google Scholar 

  • Harris, C.G. and Evans, W.A.B. [1977/78]: Extension of numerical quadrature formulae to cater for end point singular behaviours over finite intervals. Internat. J. Comput. Math. 6, 219–227.

    Google Scholar 

  • Heine, E. [1881]: Anwendungen der Kugelfunctionen und der verwandten Functionen. 2nd ed., Reimer, Berlin. [I. Theil: Mechanische Quadratur, 1–31.]

    Google Scholar 

  • Hermite, C. [1864]: Sur un nouveau développement en série des fonctions. C.R. Acad. Paris 58, 93–100, 266–273. [Oeuvres II, 293–308.]

    Google Scholar 

  • Heun, K. [1900]: Neue Methode zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen. Z. Math. Phys. 45, 23–38.

    Google Scholar 

  • Hunter, D.B. [1972]: Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals. Numer. Math. 19, 419–424.

    Google Scholar 

  • Ioakimidis, N.I. and Theocaris, P.S. [1977a]: On the numerical solution of a class of singular integral equations. J. Math. Phys. Sci. 11, 219–235.

    Google Scholar 

  • Ioakimidis, N.I. and Theocaris, P.S. [1977b]: On the numerical evaluation of Cauchy principal value integrals. Rev. Roumaine Sci. Techn. Sér. Méc. Appl. 22, 803–818.

    Google Scholar 

  • Ioakimidis, N.I. and Theocaris, P.S. [1978a]: Numerical solution of Cauchy type singular integral equations by use of the Lobatto-Jacobi numerical integration rule. Apl. Mat. 23, 439–452.

    Google Scholar 

  • Ioakimidis, N.I. and Theocaris, P.S. [1978b]: The numerical evaluation of a class of generalized stress intensity factors by use of the Lobatto-Jacobi numerical integration rule. Internat. J. Fracture 14, 469–484.

    Google Scholar 

  • Ioakimidis, N.I. and Theocaris, P.S. [1979]: On the numerical solution of singular integro-differential equations. Quart. Appl. Math. 37, 325–331.

    Google Scholar 

  • Ioakimidis, N.I. and Theocaris, P.S. [1980]: On convergence of two direct methods for solution of Cauchy type singular integral equations of the first kind. BIT 20, 83–87.

    Google Scholar 

  • Ionescu, D.V. [1951]: Some formulas of mechanical quadrature (Romanian). Acad. R.P. Rot/line Fil. Cluj. Stud. Cerc. Mat., 16–37.

    Google Scholar 

  • Ionescu, D.V. [1967]: Restes des formules de quadrature de Gauss et de Turän. Acta Math. Acad. Sci. Hungar. 18, 283–295.

    Google Scholar 

  • Ivanova, A.N. [1955]: On convergence of sequences of quadrature formulas of Gauss type on an infinite interval (Russian). Dokl. Akad. Nauk SSSR 104, 169–172.

    Google Scholar 

  • Jacobi, C.G.J. [1826]: Ueber Gauß neue Methode, die Werthe der Integrale näherungsweise ru finden. J. Reine Angew. Math. 1, 301–308

    Google Scholar 

  • Jacobi, C.G.J. [1859]: Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe. J. Reine Angew. Math. 56, 149–165.

    Google Scholar 

  • Jayarajan, N. [1974]: Error estimates for Gauss-Chebyshev and Clenshaw-Curtis quadrature formulas. Calcolo 11, 289–296.

    Google Scholar 

  • Johnson, L.W. and Riess, R.D. [1979]: Gauss harmonic interpolation formulas for circular regions. J. Inst. Math. Appl. 23, 215–222.

    Google Scholar 

  • Johnson, W.W. [1915]: On Cotesian numbers: Their history, computation, and values to n = 20. Quart. J. Pure Appl. Math. 46, 52–65.

    Google Scholar 

  • Jouraysky, A. [1928]: Sur la convergence des formules des quadratures mécaniques dans un intervalle infini. J. Soc. Phys.-Math. Leningrad 2, 31–52.

    Google Scholar 

  • Kahaner, D.K. and Monegato, G. [1978]: Nonexistence of extended Gauss-Laguerre and Gauss -Hermite quadrature rules with positive weights. Z. Angew. Math. Phys. 29, 983–986.

    Google Scholar 

  • Kambo, N.S. [1970]: Error of the Newton-Cotes and Gauss-Legendre quadrature formulas. Math. Comp. 24, 261–269.

    Google Scholar 

  • Kambo, N.S. [1970/71]: Error bounds for the Lobatto and Radau quadrature formulas. Numer. Math. 16, 383–388.

    Google Scholar 

  • Kambo, N.S. [1971]: Error of certain Gauss quadrature formulas. J. Inst. Math. Appl. 7, 303–307.

    Google Scholar 

  • Karlin, S. and Pinkus, A. [1976a]: Gaussian quadrature formulae with multiple nodes. In: Studies in Spline Functions and Approximation Theory (Ed. by Karlin, S., Micchelli, C.A., Pinkus, A. and Schoenberg, I.J. ), 113–141. Academic Press, New York.

    Google Scholar 

  • Karlin, S. and Pinkus, A. [1976b]: An extremal property of multiple Gaussian nodes. In: Studies in Spline Functions and Approximation Theory (Ed. by Karlin, S., Micchelli, C.A., Pinkus, A. and Schoenberg, I.J. ), 143–162. Academic Press, New York.

    Google Scholar 

  • Studden, W.J. [1966]: Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience, New York.

    Google Scholar 

  • Kasjanov, V.I. [1977]: The finite-dimensional approximation of singular integrals over the real line (Russian). Izv. VysI. ULebn. Zaved. Matematika, 11 (186), 27–33.

    Google Scholar 

  • Kastlunger, K. and Wanner, G. [1972]: On Turin type implicit Runge -Kutta methods. Computing 9, 317–325.

    Google Scholar 

  • Keda, N.P. [1961a]: On the theory of quadrature for periodic functions (Russian). Dokl. Akad. Nauk BSSR 5, 375–379.

    Google Scholar 

  • Keda, N.P. [1961b]: Quadrature formulae with derivatives for periodic functions (Russian). Vesci Akad. Navuk BSSR Ser. Fiz: Tehn. Navuk, no. 4, 38–44.

    Google Scholar 

  • Kegel, W.H. [1962]: Zur numerischen Berechnung der Integrale fix (x)dx. Z. Astrophys. 54, 34–40.

    Google Scholar 

  • King, H.F. and Dupuis, M. [1976]: Numerical integration using Rys polynomials. J. Computational Phys. 21, 144–165.

    Google Scholar 

  • Kis, O. [1957]: Remark on mechanical quadrature (Russian). Acta Math. Acad. Sci. Hungar. 8, 473–476.

    Google Scholar 

  • Knauff, W. [1977]: Fehlernonnen zur Quadratur analytischer Funktionen. Computing 17, 309–322.

    Google Scholar 

  • Knight, C.J. and Newbery, A.C.R. [1970]: Trigonometric and Gaussian quadrature. Math. Comp. 24, 575–581.

    Google Scholar 

  • Kofroh, J. [1972]: Die ableitungsfreien Fehlerabschätzungen von Quadraturformeln I. Apl. Mat. 17, 39–52.

    Google Scholar 

  • KomeiZuk, A.A. [1964]: Quadrature formulae for singular integrals (Russian). 2. Vycisl. Mat. i Mat. Fiz. 4, no. 4, suppl., 64–74.

    Google Scholar 

  • Kowalewski, A. [1917]: Newton, Cotes, Gauss, Jacobi: Vier grundlegende Abhandlungen über Interpolation und genäherte Quadratur. von Veit, Leipzig.

    Google Scholar 

  • Krall, H.L. and Frink, O. [1949]: A new class of orthogonal polynomials: The Bessel polynomials. Trans. Amer. Math. Soc. 65, 100–115.

    Google Scholar 

  • Krenk, S. [1976]: On quadrature formulas for singular integral equations of the first and second kind. Quart. Appl. Math. 33, 225–232.

    Google Scholar 

  • Krenk, S. [1978]: Quadrature formulae of closed type for solution of singular integral equations. J. Inst. Math. Appl. 22, 99–107.

    Google Scholar 

  • Kronecker, L. [1894]: Vorlesungen über die Theorie der einfachen und vielfachen Integrale. Leipzig.

    Google Scholar 

  • Kronrod, A.S. [1964a]: Integration with control of accuracy (Russian). Doki. Akad. Nauk SSSR 154, 283–286.

    Google Scholar 

  • Kronecker, L. [1964b]: Nodes and Weights for Quadrature Formulae. Sixteen-place Tables (Russian).

    Google Scholar 

  • Izdat. “Nauka”, Moscow. [English translation: Consultants Bureau, New York, 1965.]

    Google Scholar 

  • Kruglikova, L.G. and Krylov, V.I. [1961]: A numerical Fourier transform (Russian) Dokl. Akad. Nauk BSSR 5, 279–283.

    Google Scholar 

  • Krylov, V.I. [1959]: Approximate Calculation of Integrals (Russian). Izdat. Fiz: Mat. Lit., Moscow. [2nd ed., Izdat. “Nauka”, Moscow, 1967. English translation of 1st ed.: McMillan, New York, 1962.]

    Google Scholar 

  • Krylov, V.I., Fedenko, N.P. [1962]: Approximate representation of the integral fó x’e-f(x)dx by mechanical quadrature containing the value f(0) (Russian). Vesci Akad. Navuk BSSR Ser. Fiz: Tehn. Navuk, no. 2, 5–9.

    Google Scholar 

  • Krylov, V.I., Kruglikova, L.G. [1968]: A Handbook on Numerical Harmonic Analysis (Russian). Izdat. “Nauka i Tehnika”, Minsk. [English translation: Israel Progr. Sci. Transi., Jerusalem, 1969.]

    Google Scholar 

  • Krylov, V.I., Pal’cev, A.A. [1967]: Tables for the Numerical Integration of Functions with Logarithmic and Exponential Singularities (Russian). Izdat. “Nauka i Tehnika”, Minsk. [English translation: Israel Progr. Sci. Transi., Jerusalem, 1971.]

    Google Scholar 

  • Krylov, V.I., Skoblja, N.S. [1961]: On the numerical inversion of the Laplace transform (Russian). InJ. Fiz. Z. 4, 85–101.

    Google Scholar 

  • Krylov, V.I., Skoblja, N.S. [1968]: Handbook on the Numerical Inversion of the Laplace Transform (Russian). Izdat. “Nauka i Tehnika”, Minsk. [English translation: Israel Progr. Sci. Transi., Jerusalem, 1969.]

    Google Scholar 

  • Krylov, V.I., Skoblja, N.S. [1974]: Approximate Methods for the Fourier Transformation and Inversion of the Laplace Transform (Russian). Izdat. “Nauka”, Moscow.

    Google Scholar 

  • Krylov, V.I., Skoblja, N.S., Vorob’eva, A.A. [1971]: Tables for the Computation of the Integrals of Functions with Power Singularities: f ó x ° f (x)dx, j“,; x ’ (1-x)°f (x)dx (Russian). Izdat. ”Nauka i Tehnika“, Minsk.

    Google Scholar 

  • Krylov, V.I., Skoblja, N.S., Korolev, N.I. and Skobija, N.S. [1959]: A remark on computing the integral f xe-’f (x)dx (Russian). Dokl. Akad. Nauk BSSR 3, 3–7.

    Google Scholar 

  • Krylov, V.I., Skoblja, N.S., Lugin, V.V. and Janovi, L.A. [1963]: Tables for the Numerical Integration of Functions with Power Singularities: f xi (1- x)°f (x)dx (Russian). Izdat. Akad. Nauk Belorussk. SSR, Minsk.

    Google Scholar 

  • Kumar, R. [1974a]: A class of quadrature formulas. Math. Comp. 28, 769–778.

    Google Scholar 

  • Kumar, R. [1974b]: Certain Gaussian quadratures. J. Inst. Math. Appl. 14, 175–182.

    Google Scholar 

  • Kumar, R., Jain, M.K. [1974]: Quadrature formulas for semi-infinite integrals. Math. Comp. 28, 499–503.

    Google Scholar 

  • Kutt, H.R. [1976]: Gaussian quadrature formulae for improper integrals involving a logarithmic singularity. CSIR Special Report WISK 232, CSIR, Pretoria, South Africa.

    Google Scholar 

  • Kutta, W. [1901]: Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Z. Math. Phys. 46, 435–453.

    Google Scholar 

  • Lagrange, J.-L. [1762–1765]: Solution de différents problèmes de calcul intégral. Miscellanea Taurinensia 3. [Oeuvres 1, 471–668.]

    Google Scholar 

  • Laguerre, E.N. [1879]: Sur l’intégrale f, e-dx/x. Bull. Soc. Math. France 7, 72–81.

    Google Scholar 

  • Lambin, P. and Vigneron, J.P. [1979]: Tables for the Gaussian computation of fó xe-’f(x)dx for values of a varying continuously between -1 and + 1. Math. Comp. 33, 805–811.

    Google Scholar 

  • Laplace, P.S. [1810/11): Mémoire sur les intégrales définies et leur application aux probabilités. Mém. Acad. Sci. Paris (1) 11, 279–347. [Oeuvres 12, 357–412.]

    Google Scholar 

  • Laurie, D.P. [1977]: Gaussian quadrature rules using Chebyshev expansions. CSIR Special Report WISK 255, Nat. Res. Inst. Math. Sciences, Pretoria, South Africa.

    Google Scholar 

  • Laurie, D.P., Rolfes, L. [1977]: A FORTRAN subroutine for computing Gaussian quadrature rules. NRIMS Note No. 31, Nat. Res. Inst. Math. Sciences, Pretoria, South Africa.

    Google Scholar 

  • Laurie, D.P., Rolfes, L. [1979]: Computation of Gaussian quadrature rules from modified moments — Algorithm 015. J. Comput. Appl. Math. 5, 235–243.

    Google Scholar 

  • Lebedev, V.I. and Baburin, O.V. [1965]: On the computation of principal value integrals, the weights and nodes of Gaussian quadrature formulae (Russian). Vy6isl. Mat. i Mat. Fiz. 5, 454–462.

    Google Scholar 

  • Lether, F.G. [1977]: Modified quadrature formulas for functions with nearby poles. J. Comput. Appl. Math. 3, 3–9.

    Google Scholar 

  • Lether, F.G. [1978]: On the construction of Gauss-Legendre quadrature rules. J. Comput. Appl. Math. 4, 47–52.

    Google Scholar 

  • Lether, F.G. [1980]: Error estimates for Gaussian quadrature. Appl. Math. Comput. 7, 237–246.

    Google Scholar 

  • Levin, M. [1974]: A remark on a quadrature formula (Russian). Tallin. Polütehn. Inst. Toimetised Seer. A, no. 312, 71–74.

    Google Scholar 

  • Lewanowicz, S. [1979]: Construction of a recurrence relation for modified moments. J. Comput. Appl. Math. 5, 193–206.

    Google Scholar 

  • Lipòw, P.R. and Stenger, F. [1972]: How slowly can quadrature formulas converge ? Math. Comp. 26, 917–922.

    Google Scholar 

  • Ljasenko, N.J. and Oleinik, A.G. [1974 ): Two-sided quadrature formulae of closed type for the computation of the approximate values of iterated integrals with a preassigned number of correct decimal places (Russian). Vycisl. Prikl. Mat. (Kiev) Vyp. 23, 136–148.

    Google Scholar 

  • Ljasenko, N.J. and Oleinik, A.G.[1975]: Two-sided quadrature formulae of semiclosed type for the computation of the approximate values of iterated integrals (Russian). Vy6isl. Prikl. Mat. (Kiev) Vyp. 27, 19–27.

    Google Scholar 

  • Lo, Y.T., Lee, S.W. and Sun, B. [1965 ): On Davis’ method of estimating quadrature errors. Math. Comp. 19, 133–138.

    Google Scholar 

  • Lobatto, R. [1852]: Lessen over de Differentiaal-en Integraal-Rekening. Part I I. IntegraalRekening. Van Cleef, The Hague.

    Google Scholar 

  • Lo Cascio, M.L. [1973]: Alcuni risultati numerici sugli s-polinomi di Legendre. Pubbl. dell’Ist. Mat. Appl. No. 111, Quaderno No. 2,29–48. Ist. Mat. Appl., Fac. Ingegn., Univ. degli Studi Roma, Rome.

    Google Scholar 

  • Locher, F. [1974]: Normschranken für Interpolations-und Quadraturverfahren. In: Numerische Methoden bei Differentialgleichungen und mit funktionalanalytischen Hilfsmitteln. ISNM 19 (Ed. by Albrecht, J. and Collatz, L. ), 159–167. Birkhäuser, Basel.

    Google Scholar 

  • Locher, F. [1980]: Dividierte Differenzen und Monotonie von Quadraturformeln. Numer. Math. 34, 99–109.

    Google Scholar 

  • Locher, F., Zeller, K. [1968]: Approximationsgüte und numerische Integration. Math. Z. 104, 249–251.

    Google Scholar 

  • Lorentz, G.G. and Riemenschneider, S.D. [1978]: Birkhoff quadrature matrices. In: Linear Spaces and Approximation. ISNM 40 (Ed. by Butzer, P.L. and Sz. Nagy, B. ), 359–374. Birkhäuser, Basel.

    Google Scholar 

  • Luke, Y.L. [1969]: The Special Functions and their Approximations I I. Academic Press, New York.

    Google Scholar 

  • Luke, Y.L. [1975 ): On the error in a certain interpolation formula and in the Gaussian integration formula. J. Austral. Math. Soc. 19, 196–209.

    Google Scholar 

  • Luke, Y.L. [1977]: Algorithms for the Computation of Mathematical Functions. Academic Press, New York.

    Google Scholar 

  • Luke, Y.L., Ting, B.Y. and Kemp, M.J. [1975]: On generalized Gaussian quadrature. Math. Comp. 29, 1083–1093.

    Google Scholar 

  • Luvison, A. [1974]: On the construction of Gaussian quadrature rules for inverting the Laplace transform. Proc. IEEE 62, 637–638.

    Google Scholar 

  • Markov, A. [1885]: Sur la méthode de Gauss pour le calcul approché des intégrales. Math. Ann. 25, 427–432.

    Google Scholar 

  • Martinez, J.R. [1977]: Transfer functions of generalized Besse! polynomials. IEEE CAS 24, 325–328.

    Google Scholar 

  • Maskell, S.J. and Sack, R.A. [1974]: Generalised Lobatto quadrature formulas for contour integrals. In: Studies in Numerical Analysis (Ed. by Scaife, B.K.P. ), 295–310. Academic Press, London.

    Google Scholar 

  • Mehler, F.G. [1864]: Bemerkungen zur Theorie der mechanischen Quadraturen. J. Reine Angew. Math. 63, 152–157.

    Google Scholar 

  • Micchelli, C.A. and Pinkus, A. [1977]: Moment theory for weak Chebyshev systems with applications to monosplines, quadrature formulae and best one-sided L’-approximation by spline functions with fixed knots. SIAM J. Math. Anal. 8, 206–230.

    Google Scholar 

  • Micchelli, C.A. and Pinkus, A., Rivlin, T.J. [1972]: Turítn formulae and highest precision quadrature rules for Chebyshev coefficients. IBM J. Res. Develop. 16, 372–379.

    Google Scholar 

  • Micchelli, C.A. and Pinkus, A., Rivlin, T.J. [1973a]: Numerical integration rules near Gaussian quadrature. Israel J. Math. 16, 287–299.

    Google Scholar 

  • Micchelli, C.A. and Pinkus, A., Rivlin, T.J. [1973b]: Quadrature formulae and Hermite-Birkhoff interpolation. Advances in Math. 11, 93–112.

    Google Scholar 

  • Michels, H.S. [1963]: Abscissas and weight coefficients for Lobatto quadrature. Math. Comp. 17, 237–244.

    Google Scholar 

  • Miklosko, J. [1970a]: Numerical integration with highly oscillating weight functions. Apl. Mat. 15, 133–145.

    Google Scholar 

  • Miklosko, J. [1970b]: Asymptotic properties and the convergence of numerical quadratures. Numer. Math. 15, 234–249.

    Google Scholar 

  • Miller, R.K. [1971]: On ignoring the singularity in numerical quadrature. Math. Comp. 25, 521–532.

    Google Scholar 

  • Milne, W.E. [1949]: The remainder in linear methods of approximation. J. Res. Nat. Bur. Standards 43, 501–511.

    Google Scholar 

  • Mises, R. [1933]: Zur mechanischen Quadratur. Z. Angew. Math. Mech. 13, 53–56.

    Google Scholar 

  • Mises, R. [1936]: Über allgemeine Quadraturformeln. J. Reine Angew. Math. 174, 56–67.

    Google Scholar 

  • Monegato, G. [1976]: A note on extended Gaussian quadrature rules. Math. Comp. 30, 812–817.

    Google Scholar 

  • Monegato, G. [1978a]: Positivity of the weights of extended Gauss-Legendre quadrature rules. Math. Comp. 32, 243–245.

    Google Scholar 

  • Monegato, G. [1978b]: Some remarks on the construction of extended Gaussian quadrature rules. Math. Comp. 32, 247–252.

    Google Scholar 

  • Monegato, G. [1979]: An overview of results and questions related to Kronrod schemes. In: Numerische Integration. ISNM 45 (Ed. by Hämmerlin, G. ), 231–240. Birkhäuser, Basel.

    Google Scholar 

  • Monegato, G. [1980]: On polynomials orthogonal with respect to particular variable-signed weight functions. Z. Angew. Math. Phys. 31, 549–555.

    Google Scholar 

  • Morelli, A. [1968]: Formula di quadratura con valori della funzione e delle sue derivate anche in punti fuori dell’intervallo di integrazione. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 102, 569–579.

    Google Scholar 

  • Morelli, A., Verna, I. [1969]: Formula di quadratura in cui compaiono i valori della funzione e delle derivate con ordine massimo variabile da nodo a nodo. Rend. Circ. Mat. Palermo 18 (2), 91–98.

    Google Scholar 

  • Murphy, R. [1835]: Second memoir on the inverse method of definite integrals. Trans. Cambridge Philos. Soc. 5, 113–148.

    Google Scholar 

  • Mushelisvili, N.I. [1946]: Singular Integral Equations (Russian). OGIZ, Moscow-Leningrad. [3rd ed., Izdat. “Nauka”, Moscow, 1968. English translation of the 1st ed.: Noordhoff, Groningen, 1953. German translation of the 2nd ed.: Akademie-Verlag, Berlin, 1965.]

    Google Scholar 

  • Mysovskih, I.P. [1964]: A special case of quadrature formulae containing preassigned nodes (Russian). Vesci Akad. Navuk BSSR Ser. Fiz.-Tehn. Navuk, 4, 125–127.

    Google Scholar 

  • McNamee, J. [1964 ): Error-bounds for the evaluation of integrals by the Euler-Maclaurin formula and by Gauss-type formulae. Math. Comp. 18, 368–381.

    Google Scholar 

  • Newbery, A.C.R. [1969]: Some extensions of Legendre quadrature. Math. Comp. 23, 173–176.

    Google Scholar 

  • Nicholson, D., Rabinowitz, P., Richter, N. and Zeilberger, D. [1971]: On the error in the numerical integration of Chebyshev polynomials. Math. Comp. 25, 79–86.

    Google Scholar 

  • Nuttal, J. and Wherry, C.J. [1978 ): Gaussian integration for complex weight functions. J. Inst. Math. Appl. 21, 165–170.

    Google Scholar 

  • Olson, A.P. [1969]: Gaussian quadratures for f; exp(x)f(x)dxx’“ and f; g(x)dxx”. Math. Comp. 23, 447.

    Google Scholar 

  • Omladi, M., Pahor, S. and Suhadolc, A. [1975/76]: On a new type of quadrature formulas. Numer. Math. 25, 421–426.

    Google Scholar 

  • Ossicini, A. [1966]: Costruzione di formule di quadratura di tipo Gaussiano. Ann. Mat. Pura Appl. 72 (4), 213–237.

    Google Scholar 

  • Ossicini, A.[1968]: Le funzioni di influenza nel problema di Gauss sulle formule di quadratura. Matematiche (Catania) 23, 7–30.

    Google Scholar 

  • Ossicini, A., Rosati, F. [1975]: Funzioni caratteristiche nelle formule di quadratura gaussiane con nodi multipli. Boll. Un. Mat. Ital. 11 (4), 224–237.

    Google Scholar 

  • Ossicini, A., Rosati, F. [1978]: Sulla convergenza dei funzionali ipergaussiani. Rend. Mat. 11 (6), 97–108.

    Google Scholar 

  • Paget, D.F. and Elliott, D. [1972]: An algorithm for the numerical evaluation of certain Cauchy principal value integrals. Numer. Math. 19, 373–385.

    Google Scholar 

  • Pal’cev, A.A. and Skoblja, N.S. [1965]: The integration of bounded functions with a Laguerre weight (Russian). Vesci Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk, no. 3, 15–23.

    Google Scholar 

  • Patterson, T.N.L. [1968]: The optimum addition of points to quadrature formulae. Math. Comp. 22, 847–856. Loose microfiche suppl. Cl-Cll. [Errata: ibid. 23 (1969), 892.]

    Google Scholar 

  • Patterson, T.N.L. [1969]: Integration formulae involving derivatives. Math. Comp. 23, 411–412.

    Google Scholar 

  • Patterson, T.N.L. [1973]: Algorithm 468 — Algorithm for automatic numerical integration over a finite interval. Comm. ACM 16, 694–699.

    Google Scholar 

  • Patterson, T.N.L. [1977]: On high precision methods for the evaluation of Fourier integrals with finite and infinite limits. Numer. Math. 27, 41–52.

    Google Scholar 

  • Pavel, P. [1967 ): On the remainder of some Gaussian formulae. Studia Univ. Babes-Bolyai Ser. Math.-Phys. 12, no. 2, 65–70.

    Google Scholar 

  • Patterson, T.N.L. [1968a]: On some quadrature formulae of Gaussian type (Romanian). Studia Upiv. Babes-Bolyai Ser. Math.-Phys. 13, no. 1, 51–58.

    Google Scholar 

  • Pavel, P. [1968b]: On the remainder of certain quadrature formulae of Gauss-Christoffel type (Romanian). Studia Univ. Babes-Bolyai Ser. Math.-Phys. 13, no. 2, 67–72.

    Google Scholar 

  • Peano, G. [1913]: Resto nelle formule di quadratura, espresso con un integrale definito. Atti R. Accad. Lincei (5) Rend. Cl. Sci. Fis. Mat. Nat. 22, 562–569.

    Google Scholar 

  • Peano, G. [1914]: Residuo in formulas de quadratura. Mathesis 4 (34), 5–10.

    Google Scholar 

  • Perron, O. [1957]: Die Lehre von den Kettenbrüchen IL 3rd ed., Teubner, Stuttgart.

    Google Scholar 

  • Pexton, R.L. [1976]: Application of Salzer’s inverse Laplace transform algorithm. J. Computational Phys. 20, 492–494.

    Google Scholar 

  • Phillips. J.L. and Hanson, R.J. [1974]: Gauss quadrature rules with B-spline weight functions. Math. Comp. 28, Review 18, 666. Loose microfiche suppl. Al-C4.

    Google Scholar 

  • Piessens, R. [1969a]: Gaussian quadrature formulas for the numerical integration of Bromwich’s integral and the inversion of the Laplace transform, Rep. TW1, Inst. Appl. Math., University of Leuven, Leuven.

    Google Scholar 

  • Piessens, R. [1969b]: New quadrature formulas for the numerical inversion of the Laplace transform. BIT 9, 351–361.

    Google Scholar 

  • Piessens, R. [1970a]: Gaussian quadrature formulas for the integration of oscillating functions. Z. Angew. Math. Mech. 50, 698–700.

    Google Scholar 

  • Piessens, R. [1970b]: Gaussian quadrature formulas for the integration of oscillating functions. Math. Comp. 24, Review 24, 478–479.

    Google Scholar 

  • Piessens, R. [1970e]: Numerical evaluation of Cauchy principal values of integrals. BIT 10, 476–480.

    Google Scholar 

  • Piessens, R. [1971a]: Gaussian quadrature formulas for the numerical integration of Bromwich’s integral and the inversion of the Laplace transform. J. Engrg. Math. 5, 1–9.

    Google Scholar 

  • Piessens, R. [1971b]: Some aspects of Gaussian quadrature formulae for the numerical inversion of the Laplace transform. Comput. J. 14, 433–436.

    Google Scholar 

  • Piessens, R. [1971c]: On a numerical method for the calculation of transient responses. J. Franklin Inst. 292, 57–64.

    Google Scholar 

  • Piessens, R. [1972a]: Gaussian quadrature formulas for the evaluation of Fourier-cosine coefficients. Z. Angew. Math. Mech. 52, 56–58.

    Google Scholar 

  • Piessens, R. [1972b]: Gaussian quadrature formulae for integrals involving Besse! functions. Math. Comp. 26, Review 44, 1016–1017. Loose microfiche suppl. Dl-D14, El-E14, F1–F5.

    Google Scholar 

  • Piessens, R. [1973a]: An algorithm for automatic integration. Angew. Informatik, Heft 9, 399–401.

    Google Scholar 

  • Piessens, R. [1973b]: Algorithm 453 — Gaussian quadrature formulas for Bromwich’s integral Comm. ACM 16, 486–487.

    Google Scholar 

  • Piessens, R. [1975]: Comments on: “On the construction of Gaussian quadrature rules for inverting the Laplace transform” by A. Luvison. Proc. IEEE 63, 817–818.

    Google Scholar 

  • Piessens, R., Branders, M. [1973]: The evaluation and application of some modified moments. BIT 13, 443–450.

    Google Scholar 

  • Piessens, R., Branders, M. [1974]: A note on the optimal addition of abscissas to quadrature formulas of Gauss and Lobatto type. Math. Comp. 28, 135–139.

    Google Scholar 

  • Piessens, R., Branders, M. [1975]: Tables of Gaussian quadrature formulas. Appl. Math. Progr. Div., University of Leuven, Leuven.

    Google Scholar 

  • Piessens, R., Branders, M., Poleunis, F. [1971]: A numerical method for the integration of oscillatory functions. BIT 11, 317–327.

    Google Scholar 

  • Piessens, R., Branders, M., Chawla, M.M. and Jayarajan, N. [1976]: Gaussian quadrature formulas for the numerical calculation of integrals with logarithmic singularity. J. Computational Phys. 21, 356–360.

    Google Scholar 

  • Piessens, R., Branders, M., Van Roy-Branders, M. and Mertens, I. [1976]: The automatic evaluation of Cauchy principal value integrals. Angew. Informatik, Heft 1, 31–35.

    Google Scholar 

  • Pittnauer, F. and Reimer, M. [1976]: Interpolation mit Intervallfunktionalen. Math. Z. 146, 7–15.

    Google Scholar 

  • Pittnauer, F. and Reimer, M. [1979a]: Intervallfunktionale vom Gauss -Legendre Typ. Math. Nachr. 87, 239–248.

    Google Scholar 

  • Pittnauer, F. and Reimer, M. [1979b]: Zur Fehler-und Konvergenztheorie der Integration mit Intervallfunktionalen. Rev. Roumaine Math. Pures Appl. 24, 1105–1115.

    Google Scholar 

  • Pólya, G. [1933]: Über die Konvergenz von Quadraturverfahren. Math. Z. 37, 264–286.

    Google Scholar 

  • Popoviciu, T. [1955]: Sur une généralisation de la formule d’intégration numérique de Gauss. Acad. R.P. Romîne Fil. lali Stud. Cerc. Sti. 6, 29–57.

    Google Scholar 

  • Porath, G. and Wenzlaff, G. [1976]: Ober eine verallgemeinerte Lobattosche Quadraturformel. Beiträge Numer. Math. 5, 147–156.

    Google Scholar 

  • Posse, C. [1875]: Sur les quadratures. Nouv. Ann. Math. 14 (2), 49–62.

    Google Scholar 

  • Price, J.F. [1960]: Discussion of quadrature formulas for use on digital computers. Math. Note No. 217, Math. Res. Lab., Boeing Scientific Res. Laboratories.

    Google Scholar 

  • Price, T.E., [1979]: Orthogonal polynomials for nonclassical weight functions. SIAM J. Numer. Anal. 16, 999–1006.

    Google Scholar 

  • Rabinowitz, P. [1960]: Abscissas and weights for Lobatto quadrature of high order. Math. Comp. 14, 47–52.

    Google Scholar 

  • Rabinowitz, P. [1967]: Gaussian integration in the presence of a singularity. SIAM J. Numer. Anal. 4, 191–201.

    Google Scholar 

  • Rabinowitz, P. [1968]: Error bounds in Gaussian integration of functions of low-order continuity. Math. Comp. 22, 431–434.

    Google Scholar 

  • Rabinowitz, P. [1969]: Rough and ready error estimates in Gaussian integration of analytic functions. Comm. ACM 12, 268–270.

    Google Scholar 

  • Rabinowitz, P. [1970]: Gaussian integration of functions with branch point singularities. Internat. J. Comput. Math. 2, 297–306.

    Google Scholar 

  • Rabinowitz, P. [1977]: Ignoring the singularity in numerical integration. In: Topics in Numerical Analysis III (Ed. by Miller, J.J.H. ), 361–368. Academic Press, London.

    Google Scholar 

  • Rabinowitz, P. [1978]: The numerical evaluation of Cauchy principal value integrals. Proc. Sympos. Numerical Analysis, Durban, South Africa, 1–29.

    Google Scholar 

  • Rabinowitz, P. [1979]: On avoiding the singularity in the numerical integration of proper integrals. BIT 19, 104–110.

    Google Scholar 

  • Rabinowitz, P. [1980]: The exact degree of precision of generalized Gauss -Kronrod integration rules. Math. Comp. 35, 1275–1283.

    Google Scholar 

  • Rabinowitz, P., Richter, N. [1970]: New error coefficients for estimating quadrature errors for analytic functions. Math. Comp. 24, 561–570.

    Google Scholar 

  • Rabinowitz, P., Weiss, G. [1959]: Tables of abscissas and weights for numerical evaluation of integrals of the form xf(x)dx. Math. Tables Aids Comput. 13, 285–294. [Errata by Shao, T.S. and Chen, T.C., ibid. 18 (1964), 177.]

    Google Scholar 

  • Radau, R. [1880]: Etude sur les formules d’approximation qui servent à calculer la valeur numérique d’une intégrale définie. J. Math. Pures Appl. 6 (3), 283–336.

    Google Scholar 

  • Radau, R. [1883]: Remarque sur le calcul d’une intégrale définie. C.R. Acad. Sci. Paris 97, 157–158.

    Google Scholar 

  • Radon, J. [1935]: Restausdrücke bei Interpolations und Quadraturformeln durch bestimmte Integrale. Monatsh. Math. Phys. 42, 389–396.

    Google Scholar 

  • Ramakrishnan, T.R. [1973]: Asymptotic estimates for the error of Gauss-Jacobi quadrature formulas. Calcolo 10, 233–244.

    Google Scholar 

  • Ramskiï, J.S. [1968]: Quadrature formulae of A.A. Markov’s type for functions of class C,’’[a, b] (Russian). Vyéisl. Prikl. Mat. (Kiev) Vyp. 6, 73–81.

    Google Scholar 

  • Ramskiï, J.S. [1974]: The improvement of a certain quadrature formula of Gauss type (Russian). Vyèisl. Prikl. Mat. (Kiev) Vyp. 22, 143–146.

    Google Scholar 

  • Rebolia, L. [1973]: Formule di quadratura “chiuse” di tipo Gaussiano: Tabulazione dei nodi e dei coefficienti. alcolo 10, 245–256.

    Google Scholar 

  • Reiz, A. [1950a]: Quadrature formulae for the numerical calculation of mean intensities and fluxes in a stellar atmosphere. Ark. Astr. 1, 147–153.

    Google Scholar 

  • Reiz, A. [1950b]: On a special case of a quadrature formula of Christoffel. Math. Tables Aids Comput. 4, 181–185.

    Google Scholar 

  • Rémès, E.J. [1940]: Sur les termes complémentaires de certaines formules d’analyse approximative. C.R. (Doklady) Acad. Sci. URSS 26, 129–133.

    Google Scholar 

  • Richert, W.R. [1979]: Gauss-Quadraturformeln mit mehrfachen Knoten. In: Numerische Integration. ISNM 45 (Ed. by Hämmerlin, G. ), 241–244. Birkhäuser, Basel.

    Google Scholar 

  • Riess, R.D. [1971]: A note on error bounds for Gauss-Chebyshev quadrature. SIAM J. Numer. Anal. 8, 509–511.

    Google Scholar 

  • Riess, R.D. [1975]: Gauss-Turân quadratures of Chebyshev type and error formulae. Computing 15, 173–179.

    Google Scholar 

  • Riess, R.D. [1976]: A Gaussian approach to equally spaced quadratures. J. Inst. Math. Appl. 17, 261–265.

    Google Scholar 

  • Riess, R.D., Johnson, L.W. [1969]: Estimating Gauss-Chebyshev quadrature errors. SIAM J. Numer. Anal. 6, 557–559.

    Google Scholar 

  • Riess, R.D., Johnson, L.W.[1974]: On the determination of quadrature formulae of highest degree of precision for approximating Fourier coefficients. J. Inst. Math. Appl. 13, 345–351.

    Google Scholar 

  • Riesz, M. [1922/23]: Sur le problème des moments III. Ark. Mat. Astr. Fys. 17, no. 16, 1–52.

    Google Scholar 

  • Roghi, G. [1967]: Sul resto delle formule di quadratura di tipo gaussiano. Matematiche (Catania) 22, 143–159.

    Google Scholar 

  • Roghi, G. [1978]: Alcune osservazioni sulla convergenza di formule di quadratura ipergaussiane.

    Google Scholar 

  • Roghi, G. [1978]: Pubbl. Ist. Mat. Appi. Fac. Ingr. Univ. degli Studi Roma, Quaderno No. 12, 15–20.

    Google Scholar 

  • Rosati, F. [1968]: Problemi di Gauss e Tchebychef relativi a formule di quadratura esatte per polinomi trigonometrici. Matematiche (Catania) 23, 31–49.

    Google Scholar 

  • Rothmann, H.A. [1961]: Gaussian quadrature with weight function x“ on the interval (-1,1). Math. Comp. 15, 163–168.

    Google Scholar 

  • Runge, C. and Willers, F.A. [1915]: Numerische und graphische Integration. Encyklopädie der Math. Wiss., vol. 2, sec. 3, 47–176, Leipzig.

    Google Scholar 

  • Rutishauser, H. [1962a]: On a modification of the OD-algorithm with Graeffe-type convergence. Z. Angew. Math. Phys. 13, 493–496.

    Google Scholar 

  • Rutishauser, H. [1962b]: Algorithm 125 — Weightcoeff. Comm. ACM 5, 510–511.

    Google Scholar 

  • Sack, R.A. and Donovan, A.F. [1972]: An algorithm for Gaussian quadrature given modified moments. Numer. Math. 18, 465–478.

    Google Scholar 

  • Salzer, H.E. [1955]: Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9, 164–177.

    Google Scholar 

  • Salzer, H.E. [1961]: Additional formulas and tables for orthogonal polynomials originating from inversion integrals. J. Math. Phys. 40, 72–82.

    Google Scholar 

  • Salzer, H.E. [1976]: Inverse Laplace transforms of osculatory and hyperosculatory interpolation polynomials. J. Computational Phys. 20, 480–491.

    Google Scholar 

  • Sanikidze, D.G. [1968]: The convergence of certain quadrature processes (Russian). Sakharth. SSR Mecn. Akad. Moambe 52, 577–581.

    Google Scholar 

  • Sanikidze, D.G. [1970a]: Quadrature formulae, for singular integrals, that have the highest algebraic degree of accuracy (Russian). Sakharth. SSR Mecn. Akad. Moambe 57, 281–284.

    Google Scholar 

  • Sanikidze, D.G. [1970b]: The approximate computation of singular integrals with summable density by methods of mechanical quadrature (Russian). Ukrain. Mat. Z. 22, 106–144.

    Google Scholar 

  • Sanikidze, D.G. [1970c]: The convergence of a quadrature process for certain singular integrals (Russian). Z. Vyisl. Mat. i Mat. Fiz. 10, 189–999.

    Google Scholar 

  • Sanikidze, D.G. [1970d]: The order of approximation of certain singular operators by quadrature sums (Russian). Izv. Akad. Nauk Armjan. SSR Ser. Mat. 5, no. 4, 371–384.

    Google Scholar 

  • Sanikidze, D.G. [1972]: Quadrature processes for Cauchy type integrals (Russian). Mat. Zametki 11, 517–526.

    Google Scholar 

  • Sard, A. [1948]: Integral representations of remainders. Duke Math. J. 15, 333–345.

    Google Scholar 

  • Schmidt, E. [1907]: Zur Theorie der linearen und nichtlinearen Integralgleichungen, I. Theil: Entwicklung willkürlicher Funktionen nach System vorgeschriebener. Math. Ann. 63, 433–476.

    Google Scholar 

  • Schmidt, R. [1947]: Mechanische Quadratur nach Gauss für periodische Funktionen Bayer. Akad. Wiss. Math.-Natur. Kl. S.-B., 155–173.

    Google Scholar 

  • Schoenberg, I.J. [1958]: Spline functions, convex curves and mechanical quadrature. Bull. Amer. Math. Soc. 64, 352–357.

    Google Scholar 

  • Schulten, Z., Anderson, D.G.M. and Gordon, R.G. [1979]: An algorithm for the evaluation of the complex Airy functions. J. Computational Phys. 31, 60–75.

    Google Scholar 

  • Esko, M.A. [1976]: Convergence of quadrature processes for singular integrals (Russian). Izv. Vys3. U6ebn. Zaved. Matematika, no. 12 (175), 108–118.

    Google Scholar 

  • Esko, M.A. [1979]: On the convergence of cubature processes for a two-dimensional singular integral (Russian). Dokl. Akad. Nauk BSSR 23, 293–296.

    Google Scholar 

  • Esko, M.A., Jakimenko, T.S. [1980]: On the convergence of a quadrature process for a singular integral with power-logarithmic singularity (Russian). Izv. Vyss. U6ebn. Zaved. Matematika, no. 1 (212), 82–84.

    Google Scholar 

  • Shao, T.S., Chen, T.C. and Frank, R.M. [1964a]: Tables of zeros and Gaussian weights of certain associated Laguene polynomials and the related generalized Hermite polynomials. Math. Comp. 18, 598–616. [Errata by Danloy, B., ibid. 26 (1972), 813.]

    Google Scholar 

  • Shao, T.S., Chen, T.C. and Frank, R.M. [1964b]: Tables of zeros and Gaussian weights of certain associated Laguerre polynomials and the related generalized Hermite polynomials. Tech. Rep. 00. 1100, Development Laboratory, Data Systems Div., IBM Corp., Poughkeepsie, New York.

    Google Scholar 

  • Shohat, J.A. [1927]: Sur les quadratures mécaniques et sur les zeros des polynomes de Tchebycheff dans un intervalle infini. C.R. Acad. Sci. Paris 185, 597–598.

    Google Scholar 

  • Shohat, J.A. [1928]: On the asymptotic properties of a certain class of Tchebycheff polynomials. Proc. Intern. Math. Congress Toronto 1924, 611–618. The University of Toronto Press, Toronto.

    Google Scholar 

  • Shohat, J.A. [1929]: On a certain formula of mechanical quadratures with non-equidistant ordinates. Trans. Amer. Math. Soc. 31, 448–463.

    Google Scholar 

  • Shohat, J.A., Tamarkin, J.D. [1943]: The Problem of Moments. Mathem. Surveys, No. 1, Amer. Math. Soc., New York.

    Google Scholar 

  • Skoblja, N.S. [1961]: On the calculation of Mellin’s integral (Russian). Dokl. Akad. Nauk BSSR 5, 142–145.

    Google Scholar 

  • Sloan, I.H. [1978]: On the numerical evaluation of singular integrals. BIT 18, 91–102.

    Google Scholar 

  • Smith, H.V. [1977]: Error estimates for Gauss-Legendre quadrature of integrands possessing Dirichlet series expansions. BIT 17, 108–112.

    Google Scholar 

  • Smith, W.E. and Sloan, I.H. [1980]: Product-integration rules based on the zeros of Jacobi polynomials. SIAM J. Numer. Anal. 17, 1–13.

    Google Scholar 

  • Sohockii, Ju. [1873]: On definite integrals and functions used in series expansions (Russian). St. Pétersbourg.

    Google Scholar 

  • Song, C.C.S. [1969]: Numerical integration of a double integral with Cauchy-type singularity. AIAA J. 7, 1389–1390.

    Google Scholar 

  • Sonin, N.J. [1880]: Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries. Math. Ann. 16, 1–80.

    Google Scholar 

  • Sprung, D.W.L. and Hughes, D.J. [1965]: Gauss weights and ordinates for f(x)xdx. Math. Comp. 19, 139–142.

    Google Scholar 

  • Stancu, D.D. [1957a]: Generalization of the quadrature formula of Gauss-Christoffel (Romanian). Acad. R.P. Romfne Fil. la§i Stud. Cerc. Sti. Mat. 8, no. 1, 1–18.

    Google Scholar 

  • Stancu, D.D. [19576]: On a class of orthogonal polynomials and on some general quadrature formulae with minimum number of terms (Romanian). Bull. Math. Soc. Sci. Math. Phys. R.P. Roumaine (N.S.) 1, no. 49, 479–498.

    Google Scholar 

  • Stancu, D.D. [1959]: Sur quelques formules générales de quadrature du type Gauss-Christoffel. Mathematica (Cluj) 1 (24), 167–182.

    Google Scholar 

  • Stancu, D.D., Stroud, A.H. [1963]: Quadrature formulas with simple Gaussian nodes and multiple fixed nodes. Math. Comp. 17, 384–394.

    Google Scholar 

  • Stark, V.J.E. [1971]: A generalized quadrature formula for Cauchy integrals. AIAA J. 9, 1854–1855.

    Google Scholar 

  • Steen, N.M., Byrne, G.D. and Gelbard, E.M. [1969]: Gaussian quadratures for the integrals fexp(-x)f(x)dx and foexp(x2)f(x)dx. Math. Comp. 23, 661–671.

    Google Scholar 

  • Steklov, V.A. [1916]: On the approximate calculation of definite integrals by means of formulae of mechanical quadrature (Russian). Izv. Akad. Nauk SSSR 10 (6), 169–186.

    Google Scholar 

  • Stenger, F. [1966]: Bounds on the error of Gauss-type quadratures. Numer. Math. 8, 150–160.

    Google Scholar 

  • Stetter, F. [1968]: Error bounds for Gauss-Chebyshev quadrature. Math. Comp. 22, 657–659.

    Google Scholar 

  • Stieltjes, T.J. [1883]: Sur l’évaluation approchée des intégrales. C.R. Acad. Sci. Paris 97, 740–742, 798–799. [Oeuvres I, 314–316, 317–318.]

    Google Scholar 

  • Stieltjes, T.J. [1884a]: Quelques recherches sur la théorie des quadratures dites mécaniques. Ann. Sci. Éc. Norm. Paris, Sér. 3, 1, 409–426. [Oeuvres I, 377–396.]

    Google Scholar 

  • Stieltjes, T.J. [1884b]: Note sur quelques formules pour l’évaluation de certaines intégrales. Bul. Astr. Paris 1, 568. [Oeuvres I, 426–4271

    Google Scholar 

  • Stieltjes, T.J. [1884e]: Sur une généralisation de la théorie des quadratures mécaniques. C.R. Acad. Sci. Paris 99, 850–851. [Oeuvres I, 428–429.]

    Google Scholar 

  • Stieltjes, T.J. [1894]: Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8, 1–122; ibid. 9 (1895), 1–47. [Oeuvres II, 402–566.]

    Google Scholar 

  • Stroud, A.H. [1963]: Coefficients in quadrature formulas. Math. Comp. 17, 289–291.

    Google Scholar 

  • Stroud, A.H. [1965]: Error estimates for Romberg quadrature. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2, 480–488.

    Google Scholar 

  • Stroud, A.H. [1966]: Estimating quadrature errors for functions with low continuity. SIAM J. Numer. Anal. 3, 420–424.

    Google Scholar 

  • Stroud, A.H. [1974]: Gauss harmonic interpolation formulas. Comm. ACM 17, 471–475.

    Google Scholar 

  • Stroud, A.H., Chen, Kwan-wei [1972]: Peano error estimates for Gauss-Laguerre quadrature formulas. SIAM J. Numer. Anal. 9, 333–340.

    Google Scholar 

  • Stroud, A.H., Secrest, D. [1966]: Gaussian Quadrature Formulas. Prentice-Hall, Englewood Cliffs, N.J.

    Google Scholar 

  • Stroud, A.H., Stancu, D.D. [1965]: Quadrature formulas with multiple Gaussian nodes. J. Soc. Indust. Appt. Math. Ser. B Numer. Anal. 2, 129–143.

    Google Scholar 

  • Struble, G. [1960]: Tables for use in quadrature formulas involving derivatives of the integrand. Math. Comp. 14, 8–12.

    Google Scholar 

  • Struble, G. [1963]: Orthogonal polynomials: Variable-signed weight functions. Numer. Math. 5, 88–94.

    Google Scholar 

  • Sydow, B. [1977/78]: Error estimates for Gaussian quadrature formulae. Numer. Math. 29, 59–64.

    Google Scholar 

  • Szegö, G. [1918]: Ein Beitrag zur Theorie der Polynome von Laguerre und Jacobi. Math. Z. 1, 341–356.

    Google Scholar 

  • Szegö, G. [1919]: Über Orthogonalsysteme von Polynomen. Math. Z. 4, 139–151.

    Google Scholar 

  • Szegö, G. [1921]: Über die Entwickelung einer analytischen Funktion nach den Polynomen eines Orthogonalsystems. Math. Ann. 82, 188–212.

    Google Scholar 

  • Szegö, G. [1922]: Über die Entwicklung einer willkürlichen Funktion nach den Polynomen eines Orthogonalsystems. Math. Z. 12, 61–94.

    Google Scholar 

  • Szegö, G. [1935]: Über gewisse orthogonale Polynome die zu einer oszillierenden Belegungsfunktion gehören. Math. Ann. 110, 501–513.

    Google Scholar 

  • Takahasi, H. and Mori, M. [1970]: Error estimation in the numerical integration of analytic functions. Rep. Comput. Centre Univ. Tokyo 3, 41–108.

    Google Scholar 

  • Takahasi, H. and Mori, M. [1971]: Estimation of errors in the numerical quadrature of analytic functions. Applicable Anal. 1, 201–229.

    Google Scholar 

  • Theocaris, P.S. [1976]: On the numerical solution of Cauchy-type singular integral equations. Serdica 2, 252–275.

    Google Scholar 

  • Theocaris, P.S., loakimidis, N.I. [1977]: On the numerical solution of Cauchy type singular integral equations and the determination of stress intensity factors in case of complex singularities. Z. Angew. Math. Phys. 28, 1085–1098.

    Google Scholar 

  • Theocaris, P.S., loakimidis, N.I. [1978]: Numerical integration methods for the solution of singular integral equations. Quart. Appl. Math. 35, 173–187.

    Google Scholar 

  • Theocaris, P.S., loakimidis, N.I. [1978a]: On the Gauss-Jacobi numerical integration method applied to the solution of singular integral equations. Bull. Calcutta Math. Soc. 71, 29–43.

    Google Scholar 

  • Theocaris, P.S., loakimidis, N.I. [1978b]: Application of the Gauss, Radau and Lobatto numerical integration rules to the solution of singular integral equations. Z. Angew. Math. Mech. 58, 520–522.

    Google Scholar 

  • Theocaris, P.S., loakimidis, N.I. [1979a]: A method of numerical solution of Cauchy-type singular equations with generalized kernels and arbitrary complex singularities. J. Computational Phys. 30, 309–323.

    Google Scholar 

  • Theocaris, P.S., loakimidis, N.I. [1979b]: A remark on the numerical solution of singular integral equations and the determination of stress-intensity factors. J. Engrg. Math. 13, 213–222.

    Google Scholar 

  • Theocaris, P.S., loakimidis, N.I., Kazantzakis, J.G. [1980]: On the numerical evaluation of two-dimensional principal value integrals. Internat. J. Numer. Methods Engrg. 15, 629–634.

    Google Scholar 

  • Theocaris, P.S., loakimidis, N.I., Tsamasphyros, G. [1979]: Numerical solution of systems of singular integral equations with variable coefficients. Applicable Anal. 9, 37–52.

    Google Scholar 

  • Theocaris, P.S., loakimidis, N.I., Chrysakis, A.C. and Ioakimidis, N.I. [1979]: Cauchy-type integrals and integral equations with logarithmic singularities. J. Engrg. Math. 13, 63–74.

    Google Scholar 

  • Tom, M.E.A. [1971]: On ignoring the singularity in approximate integration. SIAM J. Numer. Anal. 8, 412–424.

    Google Scholar 

  • Tsamasphyros, G.J. and Theocaris, P.S. [1977]: On the convergence of a Gauss quadrature rule for evaluation of Cauchy type singular integrals. BIT 17, 458–464.

    Google Scholar 

  • Tsamasphyros, G.J. and Theocaris, P.S. [1979]: Cubature formulas for the evaluation of surface singular integrals. BIT 19, 368–377.

    Google Scholar 

  • Turin, P. [1950]: On the theory of the mechanical quadrature. Acta Sci. Math. Szeged 12, 30–37.

    Google Scholar 

  • Tureckii, A.H. [1959]: On quadrature formulas exact for trigonometric polynomials (Russian). Uten. Zap. Belorussk. Univ., no. 1 (49), 31–54.

    Google Scholar 

  • Tureckii, A.H. [1960]: Quadrature formulae with an even number of interpolation points accurate for trigonometric polynomials (Russian). Dokl. Akad. Nauk BSSR 4, 365–368.

    Google Scholar 

  • Uspensky, J.V. [1916]: On the convergence of mechanical quadratures between infinite limits (Russian). Izv. Akad. Nauk SSSR 10, 851–866.

    Google Scholar 

  • Uspensky, J.V. [1928]: On the convergence of quadrature formulas related to an infinite interval. Trans. Amer. Math. Soc. 30, 542–559.

    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). Z. Vytisl. Mat. i Mat. Fiz. 9, 1253–1262.

    Google Scholar 

  • Velev, G.D., Semenov, I.P. and Soliev, Ju. [1977]: Interpolatory quadrature and cubature formulae with multiple nodes for some singular integrals (Russian). Izv. Vyss. Utebn. Zaved. Matematika, no. 2 (177), 10–20.

    Google Scholar 

  • Verna, I. [1969]: Formule di quadratura gaussiana su intervalli infiniti. Rend. Mat. (6) 409–424.

    Google Scholar 

  • Vigneron, J.P. and Lambin, P. [1980]: Gaussian quadrature of integrands involving the error function. Math. Comp. 35, 1299–1307.

    Google Scholar 

  • Wall, H.S. [1948]: Analytic Theory of Continued Fractions. Van Nostrand, New York.

    Google Scholar 

  • Wellekens, C.J. Wall, H.S. [1970]: Generalisation of Vlach’s method for the numerical inversion of the Laplace transform. Electron. Lett. 6, 741–743.

    Google Scholar 

  • Wheeler, J.C. [1974]: Modified moments and Gaussian quadratures. Rocky Mountain J. Math. 4, 287–296.

    Google Scholar 

  • Wheeler, J.C., Blumstein, C. [1972]: Modified moments for harmonic solids, Phys. Rev. B6, 4380–4382.

    Google Scholar 

  • Wilf, H.S. [1962]: Mathematics for the Physical Sciences. Wiley, New York.

    Google Scholar 

  • Wilf, H.S. [1964]: Exactness conditions in numerical quadrature. Numer. Math. 6, 315–319.

    Google Scholar 

  • Wilf, H.S. [1980]: personal communication.

    Google Scholar 

  • Wilkinson, J.H. and Reinsch, C. [1971]: Linear Algebra. Handbook for Automatic Computation, Vol. II, Springer, New York.

    Google Scholar 

  • Wimp, J. [1965]: On the zeros of a confluent hypergeometric function. Proc. Amer. Math. Soc. 16, 281–283.

    Google Scholar 

  • Zamfirescu, I. [1963]: An extension of Gauss’ method for the calculation of improper integrals (Romanian). Acad. R.P. Romfne Stud. Cerc. Mat. 14, 615–631.

    Google Scholar 

  • Železnova, K.M., Korneĭčuk, A.A. and Markov, A.S. [1965]: Approximate calculation of singular integrals (Russian). In: Proc. Conf. Math. Methods of Solution of Problems in Nuclear Phys., 38–40. Ob“edin. Inst. Jadernyh Issled., Dubna.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Sciences, Purdue University, West Lafayette, Ind., USA

    Walter Gautschi

Authors
  1. Walter Gautschi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Aachen, Germany

    P. L. Butzer  & F. Fehér  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1981 Springer Basel AG

About this chapter

Cite this chapter

Gautschi, W. (1981). A Survey of Gauss-Christoffel Quadrature Formulae. In: Butzer, P.L., Fehér, F. (eds) E. B. Christoffel. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5452-8_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-5452-8_6

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-5453-5

  • Online ISBN: 978-3-0348-5452-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation