New error coefficients for estimating quadrature errors for analytic functions

Authors:
Philip Rabinowitz and Nira Richter

Journal:
Math. Comp. **24** (1970), 561-570

MSC:
Primary 65.55

MathSciNet review:
0275675

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Abstract | References | Similar Articles | Additional Information

Abstract: Since properly normalized Chebyshev polynomials of the first kind satisfy

**[1]**R. E. Barnhill and J. A. Wixom,*Quadratures with remainders of minimum norm. I*, Math. Comp.**21**(1967), 66–75. MR**0223089**, 10.1090/S0025-5718-1967-0223089-0**[2]**M. M. Chawla,*Asymptotic estimates for the error of the Gauss-Legendre quadrature formula*, Comput. J.**11**(1968/1969), 339–340. MR**0237096****[3]**Philip Davis,*Errors of numerical approximation for analytic functions*, J. Rational Mech. Anal.**2**(1953), 303–313. MR**0054348****[4]**Philip J. Davis,*Errors of numerical approximation for analytic functions*, Survey of numerical analysis, McGraw-Hill, New York, 1962, pp. 468–484. MR**0135721****[5]**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****[6]**P. Davis and P. Rabinowitz,*On the estimation of quadrature errors for analytic functions*, Math. Tables and Other Aids to Computation**8**(1954), 193–203. MR**0065256**, 10.1090/S0025-5718-1954-0065256-6**[7]**Günther Hämmerlin,*Zur Abschätzung von Quadraturfehlern für analytische Funktionen*, Numer. Math.**8**(1966), 334–344 (German). MR**0202306****[8]**D. Nicholson, P. Rabinowitz, Nira Richter-Dyn, and D. Zeilberger,*On the error in the numerical integration of Chebyshev polynomials*, Math. Comp.**25**(1971), 79–86. MR**0300443**, 10.1090/S0025-5718-1971-0300443-0**[9]**Philip Rabinowitz,*Practical error coefficients for estimating quadrature errors for analytic functions*, Comm. ACM**11**(1968), 45–46. MR**0240979****[10]**Philip Rabinowitz and Nira Richter-Dyn,*Asymptotic properties of minimal integration rules*, Math. Comp.**24**(1970), 593–609. MR**0298946**, 10.1090/S0025-5718-1970-0298946-X**[11]**A. H. Stroud and Don Secrest,*Gaussian quadrature formulas*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR**0202312**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0275675-X

Keywords:
Error coefficients,
error in numerical integration,
analytic functions,
Chebyshev polynomials,
complete orthonormal set,
error estimates,
trapezoidal rule,
Simpson rule,
norm of error functional,
interpolatory quadrature

Article copyright:
© Copyright 1970
American Mathematical Society