Numerical quadrature rules for some infinite range integrals
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Abstract:
Recently the present author has given a new approach to numerical quadrature and derived new numerical quadrature formulas for finite range integrals with algebraic and/or logarithmic endpoint singularities. In the present work this approach is used to derive new numerical quadrature formulas for integrals of the form $\smallint _0^\infty {x^\alpha }{e^{ - x}}f(x) dx$ and $\smallint _0^\infty {x^\alpha }{E_p}(x)f(x) dx$, where ${E_p}(x)$ is the exponential integral. It turns out the new rules are of interpolatory type, their abscissas are distinct and lie in the interval of integration and their weights, at least numerically, are positive. For fixed $\alpha$ the new integration rules have the same set of abscissas for all p. Finally, the new rules seem to be at least as efficient as the corresponding Gaussian quadrature formulas. As an extension of the above, numerical quadrature formulas for integrals of the form $\smallint _{ - \infty }^{ + \infty }|x{|^\beta }{e^{ - {x^2}}}f(x) dx$ too are considered.References
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
- George A. Baker Jr., Essentials of PadΓ© approximants, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0454459 S. Chandrasekhar, The Transfer of Radiant Energy, Clarendon Press, Oxford, 1953.
- Bernard Danloy, Numerical construction of Gaussian quadrature formulas for $\int _{0}^{1}(-\textrm {Log}\ x)\cdot x^{\alpha }\cdot f(x)\cdot dx$ and $\int _{0}^{\infty } E_{m}(x)\cdot f(x)\cdot dx$, Math. Comp. 27 (1973), 861β869. MR 331730, DOI 10.1090/S0025-5718-1973-0331730-X W. Gautschi, βAlgorithm 331, Gaussian quadrature formulas,β Comm. ACM, v. 11, 1968, pp. 432-436.
- David Levin, Development of non-linear transformations of improving convergence of sequences, Internat. J. Comput. Math. 3 (1973), 371β388. MR 359261, DOI 10.1080/00207167308803075
- Avram Sidi, Convergence properties of some nonlinear sequence transformations, Math. Comp. 33 (1979), no.Β 145, 315β326. MR 514827, DOI 10.1090/S0025-5718-1979-0514827-6
- Avram Sidi, Numerical quadrature and nonlinear sequence transformations; unified rules for efficient computation of integrals with algebraic and logarithmic endpoint singularities, Math. Comp. 35 (1980), no.Β 151, 851β874. MR 572861, DOI 10.1090/S0025-5718-1980-0572861-2
- Avram Sidi, Analysis of convergence of the $T$-transformation for power series, Math. Comp. 35 (1980), no.Β 151, 833β850. MR 572860, DOI 10.1090/S0025-5718-1980-0572860-0 A. Sidi, Converging Factors for Some Asymptotic Moment Series That Arise in Numerical Quadrature, TR # 165, Computer Science Dept., Technion, Haifa.
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Math. Comp. 38 (1982), 127-142
- MSC: Primary 65D32
- DOI: https://doi.org/10.1090/S0025-5718-1982-0637291-5
- MathSciNet review: 637291