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Journal articles on the topic 'Integration numerical'

Author: Grafiati

Published: 30 May 2022

Last updated: 31 May 2022

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1

Matušů, Josef, Gejza Dohnal, and Martin Matušů. "On one method of numerical integration." Applications of Mathematics 36, no. 4 (1991): 241–63. http://dx.doi.org/10.21136/am.1991.104464.

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2

Faou, Erwan, Ernst Hairer, Marlis Hochbruck, and Christian Lubich. "Geometric Numerical Integration." Oberwolfach Reports 13, no. 1 (2016): 869–948. http://dx.doi.org/10.4171/owr/2016/18.

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3

Elliott, David, H. Brass, and G. Hammerlin. "Numerical Integration IV." Mathematics of Computation 64, no. 210 (April 1995): 901. http://dx.doi.org/10.2307/2153467.

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4

G., W., H. Brass, and G. H. Hammerlin. "Numerical Integration III." Mathematics of Computation 53, no. 187 (July 1989): 451. http://dx.doi.org/10.2307/2008381.

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5

Dyer, Stephen, and Justin Dyer. "bythenumbers - Numerical integration." IEEE Instrumentation & Measurement Magazine 11, no. 2 (April 2008): 47–49. http://dx.doi.org/10.1109/mim.2008.4483733.

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6

Clegg, D. B., and A. N. Richmond. "Perfect numerical integration." International Journal of Mathematical Education in Science and Technology 18, no. 4 (July 1987): 519–25. http://dx.doi.org/10.1080/0020739870180403.

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7

Zadiraka, V. K., L. V. Luts, and I. V. Shvidchenko. "Optimal Numerical Integration." Cybernetics and Computer Technologies, no. 4 (December 31, 2020): 47–64. http://dx.doi.org/10.34229/2707-451x.20.4.4.

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Introduction. In many applied problems, such as statistical data processing, digital filtering, computed tomography, pattern recognition, and many others, there is a need for numerical integration, moreover, with a given (often quite high) accuracy. Classical quadrature formulas cannot always provide the required accuracy, since, as a rule, they do not take into account the oscillation of the integrand. In this regard, the development of methods for constructing optimal in accuracy (and close to them) quadrature formulas for the integration of rapidly oscillating functions is rather important and topical problem of computational mathematics. The purpose of the article is to use the example of constructing optimal in accuracy (and close to them) quadrature formulas for calculating integrals for integrands of various degrees of smoothness and for oscillating factors of different types and constructing a priori estimates of their total error, as well as applying to them of the theory of testing the quality of algorithms-programs to create a theory of optimal numerical integration. Results. The optimal in accuracy (and close to them) quadrature formulas for calculating the Fourier transform, wavelet transforms, and Bessel transform were constructed both in the classical formulation of the problem and for interpolation classes of functions corresponding to the case when the information operator about the integrand is given by a fixed table of its values. The paper considers a passive pure minimax strategy for solving the problem. Within the framework of this strategy, we used the method of “caps” by N. S. Bakhvalov and the method of boundary functions developed at the V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine. Great attention is paid to the quality of the error estimates and the methods to obtain them. The article describes some aspects of the theory of algorithms-programs testing and presents the results of testing the constructed quadrature formulas for calculating integrals of rapidly oscillating functions and estimates of their characteristics. The problem of determining the ranges of admissible values of control parameters of programs for calculating integrals with the required accuracy, as well as their best values for integration with the minimum possible error, is considered for programs calculating a priori estimates of characteristics. Conclusions. The results obtained make it possible to create a theory of optimal integration, which makes it possible to reasonably choose and efficiently use computational resources to find the value of the integral with a given accuracy or with the minimum possible error. Keywords: quadrature formula, optimal algorithm, interpolation class, rapidly oscillating function, quality testing.

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8

Burk, Frank. "Numerical Integration via Integration by Parts." College Mathematics Journal 17, no. 5 (November 1986): 418. http://dx.doi.org/10.2307/2686254.

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9

Burk, Frank. "Numerical Integration via Integration by Parts." College Mathematics Journal 17, no. 5 (November 1986): 418–22. http://dx.doi.org/10.1080/07468342.1986.11972993.

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10

Arthur, D. W., Philip J. Davis, and Philip Rabinowitz. "Methods of Numerical Integration." Mathematical Gazette 70, no. 451 (March 1986): 70. http://dx.doi.org/10.2307/3615859.

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11

Bland, J. A., and H. V. Smith. "Numerical Methods of Integration." Mathematical Gazette 79, no. 484 (March 1995): 244. http://dx.doi.org/10.2307/3620126.

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12

S., F., and H. V. Smith. "Numerical Methods of Integration." Mathematics of Computation 64, no. 210 (April 1995): 900. http://dx.doi.org/10.2307/2153466.

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13

Chen, Chuanmiao, Michal Křížek, and Liping Liu. "Numerical Integration over Pyramids." Advances in Applied Mathematics and Mechanics 5, no. 03 (June 2013): 309–20. http://dx.doi.org/10.4208/aamm.12-m12110.

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AbstractPyramidal elements are often used to connect tetrahedral and hexahedral elements in the finite element method. In this paper we derive three new higher order numerical cubature formulae for pyramidal elements.

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14

Kalaida, A. F. "Matrix numerical integration algorithm." Journal of Mathematical Sciences 69, no. 6 (May 1994): 1369–78. http://dx.doi.org/10.1007/bf01250578.

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15

Jacka, K., and David Lewin. "Numerical measures of integration." Journal of Advanced Nursing 11, no. 6 (November 1986): 679–85. http://dx.doi.org/10.1111/j.1365-2648.1986.tb03385.x.

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16

Schmee, Josef. "Tables for Numerical Integration." Technometrics 27, no. 1 (February 1985): 90–91. http://dx.doi.org/10.1080/00401706.1985.10488025.

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17

Laue, Hans. "Elementary numerical integration methods." American Journal of Physics 56, no. 9 (September 1988): 849–50. http://dx.doi.org/10.1119/1.15441.

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18

Voronin, S. M., and V. I. Skalyga. "On numerical integration algorithms." Izvestiya: Mathematics 60, no. 5 (October 31, 1996): 887–91. http://dx.doi.org/10.1070/im1996v060n05abeh000084.

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19

Huang, Xiaowei, Chuansheng Wu, and Jun Zhou. "Numerical differentiation by integration." Mathematics of Computation 83, no. 286 (June 4, 2013): 789–807. http://dx.doi.org/10.1090/s0025-5718-2013-02722-6.

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20

S., F., Philip J. Davis, and Philip Rabinowitz. "Methods of Numerical Integration." Mathematics of Computation 46, no. 174 (April 1986): 760. http://dx.doi.org/10.2307/2008014.

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21

Monahan, John F., Philip J. Davis, and Philip Rabinowitz. "Methods of Numerical Integration." Journal of the American Statistical Association 80, no. 392 (December 1985): 1081. http://dx.doi.org/10.2307/2288607.

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22

Hashish, H., S. H. Behiry, and N. A. El-Shamy. "Numerical integration using wavelets." Applied Mathematics and Computation 211, no. 2 (May 2009): 480–87. http://dx.doi.org/10.1016/j.amc.2009.01.084.

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23

Valliappan, S., and K. K. Ang. "? method of numerical integration." Computational Mechanics 5, no. 5 (1989): 321–36. http://dx.doi.org/10.1007/bf01047049.

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24

Xin, Wu, and Huang Tian-yi. "Constraints and numerical integration." Chinese Astronomy and Astrophysics 29, no. 1 (January 2005): 81–91. http://dx.doi.org/10.1016/j.chinastron.2005.01.008.

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25

Obradovic, Dragan, Lakshmi Narayan Mishra, and Vishnu Narayan Mishra. "Numerical Differentiation and Integration." JOURNAL OF ADVANCES IN PHYSICS 19 (January 25, 2021): 1–5. http://dx.doi.org/10.24297/jap.v19i.8938.

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There are several reasons why numerical differentiation and integration are used. The function that integrates f (x) can be known only in certain places, which is done by taking a sample. Some supercomputers and other computer applications sometimes need numerical integration for this very reason. The formula for the function to be integrated may be known, but it may be difficult or impossible to find the antiderivation that is an elementary function. One example is the function f (x) = exp (−x2), an antiderivation that cannot be written in elementary form. It is possible to find antiderivation symbolically, but it is much easier to find a numerical approximation than to calculate antiderivation (anti-derivative). This can be used if antiderivation is given as an unlimited array of products, or if the budget would require special features that are not available to computers.

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26

Murphy, Robin. "103.45 Improving elementary numerical integration using numerical differentiation." Mathematical Gazette 103, no. 558 (October 21, 2019): 548–56. http://dx.doi.org/10.1017/mag.2019.127.

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27

Abdulle, Assyr. "The role of numerical integration in numerical homogenization." ESAIM: Proceedings and Surveys 50 (March 2015): 1–20. http://dx.doi.org/10.1051/proc/201550001.

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28

Malmquist, Jens, and Robert Strichartz. "Numerical integration for fractal measures." Journal of Fractal Geometry 5, no. 2 (June 4, 2018): 165–226. http://dx.doi.org/10.4171/jfg/60.

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29

Bakirov, N. K., and I. R. Gallyamov. "Comparison of numerical integration formulas." Russian Mathematics 54, no. 12 (November 27, 2010): 1–16. http://dx.doi.org/10.3103/s1066369x10120017.

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30

Assouline, F., and P. Lailly. "Numerical Integration for Kirchhoff Migration." Oil & Gas Science and Technology 58, no. 3 (May 2003): 385–412. http://dx.doi.org/10.2516/ogst:2003024.

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31

de Doncker, Elise. "Methods for enhancing numerical integration." Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 502, no. 2-3 (April 2003): 358–63. http://dx.doi.org/10.1016/s0168-9002(03)00443-1.

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32

Soper, Davison E. "QCD calculations by numerical integration." Nuclear Physics B - Proceedings Supplements 79, no. 1-3 (October 1999): 444–46. http://dx.doi.org/10.1016/s0920-5632(99)00748-3.

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33

Caligaris, Marta, Georgina Rodríguez, and Lorena Laugero. "Designing Tools for Numerical Integration." Procedia - Social and Behavioral Sciences 176 (February 2015): 270–75. http://dx.doi.org/10.1016/j.sbspro.2015.01.471.

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34

Holloway, Damien Scott. "Numerical stabilisation of motion integration." ANZIAM Journal 49 (July 16, 2007): 249. http://dx.doi.org/10.21914/anziamj.v48i0.51.

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35

Smith, Harry V. "Numerical integration — a different approach." Mathematical Gazette 90, no. 517 (March 2006): 21–24. http://dx.doi.org/10.1017/s0025557200178994.

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In common with, I suspect, many people the author does not have access to the NAG library and so, when I was asked recently to calculate the value of the integralcorrect to 10 decimal places my first reaction was to try several different calculators as well as several mathematical software packages. On doing so it was disappointing to find they either gave widely differing values such as 7.9065200767, 4.1317217452 or 0.9174196842 or an error message indicating that the method had not converged.

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36

Treutler, Oliver, and Reinhart Ahlrichs. "Efficient molecular numerical integration schemes." Journal of Chemical Physics 102, no. 1 (January 1995): 346–54. http://dx.doi.org/10.1063/1.469408.

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37

Balkir, S., M. Yanilmaz, and M. Plonus. "Numerical integration using Bezier splines." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 13, no. 6 (June 1994): 737–45. http://dx.doi.org/10.1109/43.285248.

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38

de Doncker, Elise, Ajay Gupta, and Rodger R. Zanny. "Large-scale parallel numerical integration." Journal of Computational and Applied Mathematics 112, no. 1-2 (November 1999): 29–44. http://dx.doi.org/10.1016/s0377-0427(99)00210-1.

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39

Ginestar, D., G. Verdú, and J. March-Leuba. "Thermohydraulics Oscillations and Numerical Integration." Nuclear Science and Engineering 140, no. 2 (February 2002): 172–80. http://dx.doi.org/10.13182/nse02-a2253.

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40

Rees, W. G. "Numerical integration of orbital motion." European Journal of Physics 6, no. 4 (October 1, 1985): 302–6. http://dx.doi.org/10.1088/0143-0807/6/4/017.

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41

McLachlan, Robert I. "Perspectives on geometric numerical integration." Journal of the Royal Society of New Zealand 49, no. 2 (February 4, 2019): 114–25. http://dx.doi.org/10.1080/03036758.2018.1564676.

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42

Cools, Ronald, Daan Huybrechs, and Dirk Nuyens. "Recent topics in numerical integration." International Journal of Quantum Chemistry 109, no. 8 (2009): 1748–55. http://dx.doi.org/10.1002/qua.22101.

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43

Novak, Erich, and Ingo Roschmann. "Numerical Integration of Peak Functions." Journal of Complexity 12, no. 4 (December 1996): 358–79. http://dx.doi.org/10.1006/jcom.1996.0023.

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44

Mathé, Peter. "Asymptotically Optimal Weighted Numerical Integration." Journal of Complexity 14, no. 1 (March 1998): 34–48. http://dx.doi.org/10.1006/jcom.1997.0467.

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45

Scherf, O. "Numerical Integration of Viscoplastic Problems." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 80, S2 (2000): 543–44. http://dx.doi.org/10.1002/zamm.200008014141.

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46

Griffiths, D. V. "Generalized numerical integration of moments." International Journal for Numerical Methods in Engineering 32, no. 1 (July 1991): 129–47. http://dx.doi.org/10.1002/nme.1620320108.

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47

Sidi, Avram. "Variable transformations in numerical integration." PAMM 7, no. 1 (December 2007): 2020019–20. http://dx.doi.org/10.1002/pamm.200700104.

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48

Soper, Davison E. "QCD Calculations by Numerical Integration." Physical Review Letters 81, no. 13 (September 28, 1998): 2638–41. http://dx.doi.org/10.1103/physrevlett.81.2638.

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49

OCHIAI, Yoshihiro. "Numerical Integration for Meshless BEM." Transactions of the Japan Society of Mechanical Engineers Series B 69, no. 677 (2003): 82–87. http://dx.doi.org/10.1299/kikaib.69.82.

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50

Rabinowitz, Philip. "Extrapolation methods in numerical integration." Numerical Algorithms 3, no. 1 (December 1992): 17–28. http://dx.doi.org/10.1007/bf02141912.

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