Some polynomials for complex quadrature

Author:
David K. Kahaner

Journal:
Math. Comp. **25** (1971), 827-830

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1971-0298941-1

MathSciNet review:
0298941

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Equal-weight Chebyshev quadrature is not generally used because the nodes become complex for large *n*. However, interest in these schemes remains because of recent work on minimal norm quadrature as well as schemes for doing real integrals of analytic functions by complex methods. This note presents some properties of these Chebyshev quadratures that may be of interest to other researchers in this area. Proofs are sketched to save space.

**[1]**R. E. Barnhill & J. A. Wixom, ``Quadratures with remainders of minimum norm. II,''*Math Comp.*, v. 21, 1967, pp. 382-387. MR**36**#6139. MR**0223090 (36:6139)****[2]**R. E. Barnhill, J. E. Dennis, Jr. & G. M. Nielson, ``A new type of Chebyshev quadrature,''*Math Comp.*, v. 23, 1969, pp. 437-141. MR**39**#3698. MR**0242367 (39:3698)****[3]**N. Richter, ``Properties of minimal integration rules,''*SIAM J. Numer. Anal.*, v. 7, 1970, pp. 67-79. MR**0260176 (41:4804)****[4]**P. J. Davis,*Interpolation and Approximation*, Blaisdell, Waltham, Mass., 1963. MR**28**#393. MR**0157156 (28:393)****[5]**J. N. Lyness, ``Quadrature methods based on complex function values,''*Math. Comp.*, v. 23, 1969, pp. 601-619. MR**40**#1032. MR**0247771 (40:1032)****[6]**R. Kuzmin, ``On the distribution of the roots of polynomials in Chebyshev quadrature,''*Izv. Akad. Nauk SSSR Ser. Mat.*, v. 2, no. 4, 1938, pp. 427-444.**[7]**J. Shohat, ``Definite integrals and Riemann sums,''*Amer. Math. Monthly*, v. 46, 1939, pp. 538-545. MR**0000426 (1:72f)****[8]**D. Kahaner,*Equal and Almost Equal Quadrature Formulas*, Ph.D. Thesis, Stevens Institute of Technology, 1968.**[9]**V. I. Krylov,*Approximate Calculation of Integrals*, Fizmatgiz, Moscow, 1959; English transl., Macmillan, New York, 1962. MR**22**#2002; MR**26**#2008. MR**0144464 (26:2008)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65D30

Retrieve articles in all journals with MSC: 65D30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0298941-1

Keywords:
Numerical quadrature,
Chebyshev quadrature,
complex quadrature

Article copyright:
© Copyright 1971
American Mathematical Society