On Chebyshev-type quadratures
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- by Walter Gautschi and Hiroki Yanagiwara PDF
- Math. Comp. 28 (1974), 125-134 Request permission
Abstract:
According to a result of S. N. Bernstein, n-point Chebyshev quadrature formulas, with all nodes real, do not exist when $n = 8$ or $n \geqq 10$. Modifications of such quadrature formulas have recently been suggested by R. E. Barnhill, J. E. Dennis, Jr. and G. M. Nielson, and by D. Kahaner. We establish here certain empirical observations made by these authors, notably the presence of multiple nodes. We also show how some of the quadrature rules proposed can be constructed by solving single algebraic equations, and we compute the respective nodes to 25 decimal digits. The same formulas also arise in recent work of P. Rabinowitz and N. Richter as limiting cases of optimal Chebyshev-type quadrature rules in a Hilbert space setting.References
- Tom M. Apostol, Mathematical analysis: a modern approach to advanced calculus, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1957. MR 0087718
- R. E. Barnhill, J. E. Dennis Jr., and G. M. Nielson, A new type of Chebyshev quadrature, Math. Comp. 23 (1969), 437–441. MR 242367, DOI 10.1090/S0025-5718-1969-0242367-4 S. N. Bernšteǐn, "Sur les formules de quadrature de Cotes et Tchebycheff," C. R. Acad. Sci. URSS, v. 14, 1937, pp. 323-326; Reprinted in Collected Works. Vol. II, Izdat. Akad. Nauk SSSR, Moscow, 1954, pp. 200-204. (Russian) MR 16, 433.
- David K. Kahaner, Chebyshev type quadrature formulas, Math. Comp. 24 (1970), 571–574. MR 273818, DOI 10.1090/S0025-5718-1970-0273818-5
- Philip Rabinowitz and Nira Richter-Dyn, Chebyshev-type integration rules of minimum norm, Math. Comp. 24 (1970), 831–845. MR 298947, DOI 10.1090/S0025-5718-1970-0298947-1
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Math. Comp. 28 (1974), 125-134
- MSC: Primary 65D30
- DOI: https://doi.org/10.1090/S0025-5718-1974-0331731-2
- MathSciNet review: 0331731