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Automatic integration of a function with a parameter

Author:

Philip RabinowitzAuthors Info & Claims

Communications of the ACM, Volume 9, Issue 11

Pages 804 - 806

https://doi.org/10.1145/365876.365911

Published: 01 November 1966 Publication History

Abstract

Two efficient methods for automatic numerical integration are Romberg integration and adaptive Simpson integration. For integrands of the form ƒ(x)g(x, α) where α is a parameter, it is shown that Romberg's method is more efficient. A FORTRAN program shows how to achieve this greater efficiency.

References

[1]

BAUER, F. L. Algorithm 60, Romberg integration. Comm. ACM 4 (1961), 255; see also Comm. ACM 5 (1962), 168, 281.

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[2]

BAUER, F. L. La m6thode d'int6gration num4rique de Romberg. Colloque Bur l'analyse num4rique, 22-24 mars 1961 a Mons, 119-129.

[3]

BAUER, F. L., RUTISHAUSER, H., AND STIEFEL, E. New aspects in numerical quadrature. Proc. Symposia Appl. Math. Vol. 15, 1962, 199-218.

[4]

BAUMANN, R., FELICIANO, M., BAUER, F. L., AND SAMELSEN, K. Introduction to ALGOL. Prentice-Hall, Englewood Cliffs, N. J., 1964, 73.

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BULIRSCH, R. Bemerkungen zur Romberg-Integration. Numer. Math. 6 (1964), 6-16.

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BULIRSCH, R., AND STOER, J. Fehlerabschatzungen und Extrapolation mit rationalen Funktionen bei Verfahren yon Richardson-Typus. Numer. Math. 6 (1964), 413-427.

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BULIRSCH, R., AND STOER, J. Asymptotic upper and lower bounds for results of extrapolation methods. Numer. Math. 8 (1966), 93-104.

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DUNKL, C .F . Romberg quadrature to prescribed accuracy. SHARE File No. 7090-1481 TYQUAD.

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ELLIOTT, J., AND PRAGER, W. Quadrature routine to evaluate fpQ F(u) du. Brown U. Computing Lab., Providence, R. I.

[10]

FILIPPI, S. Des Verfahren yon Romberg-Stiefel-Bauer als Spezialfall des allgemeinen Prinzips von Richardson. Mathematik-Technik-Wirtschaft 11 (1964), 49-54, 98-100.

[11]

GRAM, C. Definite integral by Romberg's method. ALGOL procedure. BIT $ (1964), 54-60; see also BIT 4 (1964), 118-120.

[12]

HAVIE, T. One modification of Romberg's algorithm. BIT 6 (1966), 24--30.

[13]

KRASUN, A. M., AND PRAGER, W. Remark on Romberg quadrature. Comm. ACM 8 (1965), 236-237.

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KUBIK, R.N. Algorithm 257, Havie integrator. Comm. ACM 8 (1965), 381.

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[15]

KUNCIR, G. F. Algorithm 103, Simpson's rule integration. Comm. ACM 5 (1962), 347.

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McKEEMAN, W. M. Algorithm 145, adaptive numerical integration by Simpson's rule. Comm. ACM 5 (1962), 604; see also Comm. ACM 8 (1965), 171.

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[17]

MCKEEMAN, W.M. Certification of algorithm 145, adaptive numerical integration by Simpson's rule. Comm. ACM 6 (1963), 167-168.

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[18]

MCKEEMAN, W. M., AND TESLER, LARR:f. Algorithm 182, nonrecursive adaptive integration. Comm. ACM 6 (1963), 315; see also Comm. ACM 7 (1964), 244.

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[19]

MCKEEMAN, W.M. Algorithm 198, adaptive integration and multiple integration. Comm. ACM 6 (1963), 443-444.

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[20]

MEINGUET, J. Methods for estimating the remainder in linear rules of approximation: application to the Romberg algorithm. S6minaire de Math6matique Appliqu6e et Mecanique, Universit6 Catholique de Louvain, Rept. no. 5, 1966.

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PRAGER, W. Introduction to Basic FORTRAN Programming and Numerical Methods. Blaisdell Publishers, New York, 1965, 123-124.

[22]

ROMBERG, W. Vereinfachte numerische Integration. Det. Kong. Norske Videnskab. Selskab Forhandlinger, 28, 7, 1955.

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RUTISHAUSER, H. Ausdehnung des Rombergscher Prinzips. Numer. Math. g (1963), 48-54.

[24]

STIEFEL, E. Altes und Neues fiber numerische Quadratur. Z. Angew. Math. Mech. 41 (1961), 408-413.

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STIEFEL, E., AND RUTISHAUSER, H. Remarques concernant l'intdgration numerique. Compt. Rend. Acad. Sci. Paris 252 (1961), 1899-1900.

[26]

STROUD, A.H. Error estimates for Romberg quadrature. J. SIAM Numer. Anal., Ser. B, P (1965), 480-488.

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THACHER, JR., H. C. Remark on algorithm 60, Romberg integration. Comm. ACM 7 (1964), 420-421.

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Cited By

Joyce D(1971)Survey of Extrapolation Processes in Numerical AnalysisSIAM Review10.1137/101309213:4(435-490)Online publication date: 1-Oct-1971
https://dl.acm.org/doi/10.1137/1013092
Wallick G(1970)Remark on algorithm 351 [D1]: modified Romberg quadratureCommunications of the ACM10.1145/362384.36251113:6(374-376)Online publication date: 1-Jun-1970
https://dl.acm.org/doi/10.1145/362384.362511
Gentleman W(1969)Off-the-Shelf Black Boxes for ProgrammingIEEE Transactions on Education10.1109/TE.1969.432043812:1(43-50)Online publication date: 1-Mar-1969
https://dl.acm.org/doi/10.1109/TE.1969.4320438

Automatic integration of a function with a parameter
1. Theory of computation

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Published In

cover image Communications of the ACM

Communications of the ACM Volume 9, Issue 11

Nov. 1966

44 pages

ISSN:0001-0782

EISSN:1557-7317

DOI:10.1145/365876

Editor:
Gerard Salton
Cornell Univ., Ithaca, NY

Issue’s Table of Contents

Copyright © 1966 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 01 November 1966

Published in CACM Volume 9, Issue 11

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Cited By

Joyce D(1971)Survey of Extrapolation Processes in Numerical AnalysisSIAM Review10.1137/101309213:4(435-490)Online publication date: 1-Oct-1971
https://dl.acm.org/doi/10.1137/1013092
Wallick G(1970)Remark on algorithm 351 [D1]: modified Romberg quadratureCommunications of the ACM10.1145/362384.36251113:6(374-376)Online publication date: 1-Jun-1970
https://dl.acm.org/doi/10.1145/362384.362511
Gentleman W(1969)Off-the-Shelf Black Boxes for ProgrammingIEEE Transactions on Education10.1109/TE.1969.432043812:1(43-50)Online publication date: 1-Mar-1969
https://dl.acm.org/doi/10.1109/TE.1969.4320438

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