Abstract
The numerical construction of the Kronrod extension to then-points Gauss-Gegenbauer integration rule and some extensions of Radau and Lobatto type are considered. A program for their purpose is described and some numerical examples are reported.
Zusammenfassung
Wir behandeln die numerische Konstruktion der von Kronrod stammenden Verallgemeinerung dern-Punkte-Integrationsformeln Gauß-Gegenbauer; daneben werden auch einige Verallgemeinerungen vom Radauschen und vom Lobattoschen Typ betrachtet. Ein entsprechendes Programm wird beschrieben, zusammen mit numerischen Testbeispielen.
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References
Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions. (National Bureau of Standard Applied Mathematical Series, 55.) Washington, D.C.: 1964.
Casaletto, J., Pickett, M., Rice, J.: A comparison of some numerical integration programs. Signum Newsletter4, 30–40 (1969).
Dagnino, C.: Extended product integration rules. BIT (1983, to appear).
Dagnino, C., Fiorentino, C.: A Fortran code for the computation of nodes and weights of extended Gaussian rules. Internal Report, Dipartimento di Matematica, Politecnico di Torino (1983).
Davis, P. J., Rabinowitz, P.: Methods of numerical integration. Academic Press 1975.
Dalquist, G., Bjorck, A.: Numerical methods. Englewood Cliffs, N. J.: Prentice-Hall 1974.
Gautschi, W.: On generating Gaussian quadrature rules. (Numerische Integration, ISNM 45.) Birkhäuser 1979.
Hengels, N.: Numerical quadrature and cubature. Academic Press 1980.
Kronrod, A. S.: Nodes and weights for quadrature formulae. Sixteen places tables. Moscow: Nauka 1964. English transl.: New York: Consultants Bureau 1965.
Monegato, G.: Stieltjes polynomials and related quadrature rules. SIAM Review24, 137–158 (1982).
Monegato, G.: A note on extended Gaussian quadrature rules. Math. Comp.30, 812–817 (1976).
Monegato, G.: Some remarks on the construction of extended Gaussian quadrature rules. Math. Comp.32, 246–252 (1978).
Patterson, T. N. L.: The optimum addition of points to quadrature formulae. Math. Comp.22, 847–856 (1968).
Piessens, R., Branders, M.: A note on the optimal addition of abscissas to quadrature formulas of Gauss and Lobatto type. Math. Comp.28, 135–139 (1974).
Rabinowitz, P.: The exact degree of precision of generalized Gauss-Kronrod integration rules. Math. Comp.35, 1275–1283 (1980).
Ralston, A., Wilf, H.: Mathematical methods for digital computers, J. Wiley 1967.
Stroud, A., Secrest, D.: Gaussian quadrature formulas. Englewood Cliffs, N. J.: Prentice-Hall 1966.
Szegö, G.: Orthogonal polynomials. (Amer. Math. Soc. Publ., 23.) Providence, R. I.: Amer. Math. Soc. 1975.
IBM System/360, FORTRAN IV: Mathematical and service subprograms: SC28-6864.
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Work sponsored by the “Ministero della Pubblica Istruzione” of Italy.
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Dagnino, C., Fiorentino, C. Computation of nodes and weights of extended Gaussian rules. Computing 32, 271–278 (1984). https://doi.org/10.1007/BF02243577
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DOI: https://doi.org/10.1007/BF02243577