Numerical quadrature rules for some infinite range integrals

Author:
Avram Sidi

Journal:
Math. Comp. **38** (1982), 127-142

MSC:
Primary 65D32

DOI:
https://doi.org/10.1090/S0025-5718-1982-0637291-5

MathSciNet review:
637291

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 and , where 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 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 too are considered.

**[1]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[2]**George A. Baker Jr.,*Essentials of Padé approximants*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR**0454459****[3]**S. Chandrasekhar,*The Transfer of Radiant Energy*, Clarendon Press, Oxford, 1953.**[4]**Bernard Danloy,*Numerical construction of Gaussian quadrature formulas for ∫₀¹(-𝐿𝑜𝑔𝑥)⋅𝑥^{𝛼}⋅𝑓(𝑥)⋅𝑑𝑥 and ∫₀^{∞}𝐸_{𝑚}(𝑥)⋅𝑓(𝑥)⋅𝑑𝑥*, Math. Comp.**27**(1973), 861–869. MR**0331730**, https://doi.org/10.1090/S0025-5718-1973-0331730-X**[5]**W. Gautschi, ``Algorithm 331, Gaussian quadrature formulas,''*Comm. ACM*, v. 11, 1968, pp. 432-436.**[6]**David Levin,*Development of non-linear transformations of improving convergence of sequences*, Internat. J. Comput. Math.**3**(1973), 371–388. MR**0359261**, https://doi.org/10.1080/00207167308803075**[7]**Avram Sidi,*Convergence properties of some nonlinear sequence transformations*, Math. Comp.**33**(1979), no. 145, 315–326. MR**514827**, https://doi.org/10.1090/S0025-5718-1979-0514827-6**[8]**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. 151, 851–874. MR**572861**, https://doi.org/10.1090/S0025-5718-1980-0572861-2**[9]**Avram Sidi,*Analysis of convergence of the 𝑇-transformation for power series*, Math. Comp.**35**(1980), no. 151, 833–850. MR**572860**, https://doi.org/10.1090/S0025-5718-1980-0572860-0**[10]**A. Sidi,*Converging Factors for Some Asymptotic Moment Series That Arise in Numerical Quadrature*, TR # 165, Computer Science Dept., Technion, Haifa.

Retrieve articles in *Mathematics of Computation*
with MSC:
65D32

Retrieve articles in all journals with MSC: 65D32

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0637291-5

Article copyright:
© Copyright 1982
American Mathematical Society