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An error analysis for numerical multiple integration. I


Author: Robert E. Barnhill
Journal: Math. Comp. 22 (1968), 98-109
MSC: Primary 65.55
DOI: https://doi.org/10.1090/S0025-5718-1968-0226852-6
MathSciNet review: 0226852
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  • [1] R. E. Barnhill, Numerical Contour Integration U. S. Army Mathematics Research Center Report No. 519, Madison, Wis., 1964.
  • [2] R. E. Barnhill, "Complex quadratures with remainders of minimum norm," Numer. Math., v. 7, 1965, pp. 384-390. MR 32 #8497. MR 0191088 (32:8497)
  • [3] R. E. Barnhill & J. A. Wixom, "Quadratures with remainders of minimum norm. I," Math. Comp., v. 21, 1967, pp. 66-75. MR 0223089 (36:6138)
  • [4] R. E. Barnhill & J. A. Wixom, "Quadratures with remainders of minimum norm. II," Math. Comp., v. 21, 1967, pp. 382-387. MR 0223090 (36:6139)
  • [5] R. E. Barnhill, "Optimal quadratures in $ {L^2}({E_p})$. I," SI AM J. Numer. Anal., v. 4, 1967, pp. 390-397.
  • [6] R. E. Barnhill, "Optimal quadratures in $ {L^2}({E_p})$. II." SIAM J. Numer. Anal. (To appear.)
  • [7] R. E. Barnhill, "Asymptotic properties of minimum norm and optimal quadratures." (Submitted for publication.)
  • [8] S. Bergman, The Kernet Function and Conformal Mapping, Math. Surveys, No. 5, Amer. Math. Soc, Providence, R. I., 1950. MR 12, 402. MR 0038439 (12:402a)
  • [9] G. Birkhoff & C. R. deBoor, "Piecewise polynomial interpolation and approximation," Approximation of Functions, edited by H. L. Garabedian, Elsevier, Amsterdam, 1965. MR 32 #2789.
  • [10] S. Bochner & W. T. Martin, Several Complex Variables, Princeton Math. Series, Vol. 10, Princeton Univ. Press, Princeton, N. J., 1948. MR 10, 366. MR 0027863 (10:366a)
  • [11] R. Courant & D. Hilbert, Methods of Mathematical Physics, Vol. 1, Interscience, New York, 1953. MR 16, 426. MR 0065391 (16:426a)
  • [12] P. J. Davis, "Errors of numerical approximation for analytic functions," J. Rational Mech. Anal., v. 2, 1953, pp. 303-313. MR 14, 907. MR 0054348 (14:907f)
  • [13] P. J. Davis, "Errors of numerical approximation for analytic functions," Survey of Numerical Analysis, McGraw-Hill, New York, 1962. MR 24 B1766. MR 0135721 (24:B1766)
  • [14] P. J. Davis, Interpolation and Approximation, Blaisdell, New York, 1963. MR 28 #393. MR 0157156 (28:393)
  • [15] P. J. Davis & P. Rabinowitz, Numerical Integration, Blaisdell, New York, 1967. MR 0211604 (35:2482)
  • [16] P. J. Davis & P. Rabinowitz, "On the estimation of quadrature errors for analytic functions," MTAC, v. 8, 1954, pp. 193-203. MR 16, 404. MR 0065256 (16:404e)
  • [17] M. Golomb & H. F. Weinberger, "Optimal approximation and error bounds," On Numerical Approximation, edited by R. E. Langer, Proceedings of a Symposium, Univ. of Wisconsin Press, Madison, Wis., 1959. MR 22 #12697. MR 0121970 (22:12697)
  • [18] M. Golomb, Lectures on Theory of Approximation, Argonne National Laboratory, Argonne, 111., 1962.
  • [19] P. C. Hammer & A. H. Stroud, "Numerical evaluation of multiple integrals. II," MTAC, v. 12, 1958, pp. 272-280. MR 21 #970. MR 0102176 (21:970)
  • [20] P. C. Hammer, "Numerical evaluation of multiple integrals," On Numerical Approximation, edited by R. E. Langer, Univ. of Wisconsin Press, Madison, Wis., 1959. MR 0100355 (20:6788)
  • [21] V. I. Krtlov, Approximate Calculation of Integrals, translated from Russian, Macmillan, New York, 1962. MR 26 #2008. MR 0144464 (26:2008)
  • [22] J. N. Lyness, "Symmetric integration rules for hypercubes. I: Error coefficients," Math. Comp., v. 19, 1965, pp. 260-276. MR 34 #952. MR 0201067 (34:952)
  • [23] J. Meinguet, "Methods for estimating the remainder in linear rules of approximation. Application to the Romberg algorithm," Numer. Math., v. 8, 1966, pp. 345-366. MR 33 #8102. MR 0199962 (33:8102)
  • [24] A. Sard, Linear Approximation, Math. Surveys, No. 9, Amer. Math. Soc., Providence, R. I., 1963. MR 28 #1429. MR 0158203 (28:1429)
  • [25] D. D. Stancu, "The remainder of certain linear approximation formulas in two variables," SI AM J. Numer. Anal., v. 1, 1964, pp. 137-163. MR 31 #1503. MR 0177240 (31:1503)
  • [26] F. Stenger, "Bounds on the error of Gauss-type quadratures," Numer. Math., v. 8, 1966, pp. 150-160. MR 33 #5120. MR 0196936 (33:5120)
  • [27] F. Stenger, "Error bounds for the evaluation of integrals by repeated Gauss-type formulae," Numer. Math., v. 9, 1966, pp. 200-213. MR 0205462 (34:5289)
  • [28] A. H. Stroud, "Quadrature methods for functions of more than one variable," Ann. New York Acad. Sci., v. 86, 1960, pp. 776-791. MR 22 #10179. MR 0119417 (22:10179)
  • [29] A. H. Stroud & D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, N. J., 1966. MR 0202312 (34:2185)
  • [30] R. A. Valentin, Applications of Functional Analysis to Optimal Numerical Approximation for Analytic Functions, Ph.D. Thesis, Brown University, Providence, R. I., 1965.
  • [31] H. Yanagihara, "A new method of numerical integration of Gaussian type," Bull. Fukuoka Gakugei Univ. III. v. 6, 1956, pp. 17-24. (Japanese) MR 26 #5729. MR 0148221 (26:5729)
  • [32] G. Hämmerlin, "Über ableitungsfreie Schranken für Quadraturfehler," Numer. Math., v. 5, 1963, pp. 226-233. MR 28 #1756.
  • [33] G. Hämmerlin, "Über ableitungsfreie Schranken für Quadraturfehler. II. Ergänzungen und Möglichkeiten zur Verbesserung," Numer. Math., v. 7, 1965, pp. 232-237. MR. 32 #1899. MR 0184427 (32:1899)
  • [34] G. Hämmerlin, "Zur Abschätzung von Quadraturfehlern für analytische Funktionen," Numer. Math., v. 8, 1966, pp. 334-344. MR 34 #2179. MR 0202306 (34:2179)

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DOI: https://doi.org/10.1090/S0025-5718-1968-0226852-6
Article copyright: © Copyright 1968 American Mathematical Society

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