On Chebyshev-type quadratures
Walter Gautschi and Hiroki Yanagiwara
Math. Comp. 28 (1974), 125-134
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Abstract: According to a result of S. N. Bernstein, n-point Chebyshev quadrature formulas, with all nodes real, do not exist when or . 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.
M. Apostol, Mathematical analysis: a modern approach to advanced
calculus, Addison-Wesley Publishing Company, Inc., Reading, Mass.,
E. Barnhill, J.
E. Dennis Jr., and G.
M. Nielson, A new type of Chebyshev
quadrature, Math. Comp. 23 (1969), 437–441. MR 0242367
(39 #3698), http://dx.doi.org/10.1090/S0025-5718-1969-0242367-4
S. N. Bernštein, "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.
K. Kahaner, Chebyshev type quadrature
formulas, Math. Comp. 24 (1970), 571–574. MR 0273818
(42 #8694), http://dx.doi.org/10.1090/S0025-5718-1970-0273818-5
Rabinowitz and Nira
Richter-Dyn, Chebyshev-type integration rules of
minimum norm, Math. Comp. 24 (1970), 831–845. MR 0298947
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- T. M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley, Reading, Mass., 1957. MR 19, 398. MR 0087718 (19:398e)
- R. E. Barnhill, J. E. Dennis, Jr. & G. M. Nielson, "A new type of Chebyshev quadrature," Math. Comp., v. 23, 1969, pp. 437-441. MR 39 #3698. MR 0242367 (39:3698)
- S. N. Bernštein, "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.
- D. Kahaner, "Chebyshev type quadrature formulas," Math. Comp., v. 24, 1970, pp. 571-574. MR 42 #8694. MR 0273818 (42:8694)
- P. Rabinowitz & N. Richter, "Chebyshev-type integration rules of minimum norm," Math. Comp., v. 24, 1970, pp. 831-845. MR 45 #7996. MR 0298947 (45:7996)
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