A Fast Newton-Raphson Based Roots Finding Algorithm Design and its Applications to Circular Waveguides

Yıl 2017, Cilt: 4 Sayı: 1, 0 - 0, 31.01.2017

Coşkun Deniz

https://doi.org/10.31202/ecjse.289635

Öz

Determination of zeros of first two kinds of Bessel functions and their derivatives by fast and reliable accurate calculations is essential to determine the necessary TE and TM modes supported by the circular waveguides. Here, a fast computational algorithm design based on the numerical Newton-Raphson method to determine the first n zeros of these special functions is being presented. Our suggestion involves: scanning the given function in the given domain with the given iteration step and finding their zeros. Repeated roots and roots out of the domain is rejected and the remaining desired roots are ordered by the bubble sorting simultaneously. Consequently, TE and TM modes of the circular waveguides is obtained successfully. Our design running under the free “Wolfram CDF player” software has been open to the users for free in the web page of our institution as being presented here.

Newton-Raphson Temelli Kökleri Bulma Algoritması Tasarımı ve Dairesel Dalgakılavuzlarına Uygulamaları

Öz

Dairesel dalgakılavuzlarının (DDK) desteklediği Transvers Elektrik (TE) ve Transvers Manyetik (TM)
modların belirlenmesinde, ilk iki türden Bessel fonksiyonlarının ve türevlerinin sıfırlarının hızlı ve güvenilir
doğruluklarda hesaplanarak tespiti elzemdir. Burada, bu özel fonksiyonlarınistenen aralıkta ilk n sıfırını
bulan, Newton-Raphson (N-R) temelli, hızlı hesaplama yapabilen bir algoritma tasarımı sunulmaktadır.
Önerimiz, fonksiyonun seçilen iterasyon adım sayısına göre (veya domain bölme sayısına göre), seçilen tanım
aralığında taranmasınıve her adımda köklerin N-R yöntemiyle bulunmasını içermektedir. Bulunan tekrarlı
köklerden biri ve girilen tanım aralığı dışında bulunan kökler atılmakta ve geriye kalan kökler, opsiyonel
olarak konulan köpük (bubble) sıralama algoritmasına göre tekrardan sıralanmaktadır. Netice itibariyle,
elektomanyetik dalga davranışlarını belirleyen, dairesel dalgakılavuzlarının TE ve TM modları başarıyla elde
edilmektedir. Ücretsiz “Wolfram CDF player” altında çalışan tasarımımız, kurumumuzun ilgili internet
adresinden kullanıcıların hizmetine ücretsiz olarak sunulmuştur. Tasarımımızın özel bir DDK’na örnek
uygulamasının modlara, kesme frekansına ve ilerleyen dalga frekansına ilişkin sonuçları sunulmaktadır.

Anahtar Kelimeler

Bessel fonksiyonları, Dairesel dalga klavuzları, Kayıpsız ortam, Silindirik dalga klavuzları, TE modlar, TM modlar, Numerik kök bulma, Newton-Raphson metodu

Kaynakça

• Korenev, B.G., “Bessel functions and their applications”, Taylor and Francis, 2002, NY.
• Werner, A. and Eliezer C.J., “The Lengthening Pendulum”, Journal of The Australian Mathematical Society (J. Aust. Math. Soc.), 1969, 9(3-4), pp. 331—336.
• McMillan, M., Blasing, D., and Whitney, H.M., “Radial forcing and Edgar Allan Poe’s lengthening pendulum,” American Journal of Physics, 2013, 81(9), pp. 682–687.
• Asadi-Zeydabadi, M., “Bessel function and damped simple harmonic motion”, Journal of Applied Mathematics and Physics (JAMP), 2014, 2, pp. 26—34.
• Bell, W.W., “Special functions for scientists and engineers”, D. Van Nostrand Compant Ltd., 1968, London, pp. 92—110.
• Boas, L.M., “Mathematical methods in the physical sciences”, Wiley, 2006, 3rd ed., NY, pp. 587—606.
• Vallée, O. and Soares, M., “Airy functions and applications to physics”, Imperial College Press (distributed by World Scientific), 2004, NJ, pp. 115—176.
• Watson, G.N., “A treatise on the theory of Bessel functions”, Cambridge University Press, 1995, 2nd ed., NY.
• Arfken H.J. and Weber, G.B., “Mathematical methods for physicists”, Elsevier Academic Press, 2005, 6th ed., pp. 675—686.
• Sekeljic, N., “Asymptotic expansion of Bessel functions; applications to electromagnetics”, Dynamics at the Horsetooth, 2010, Focussed Issue: Asymptotics and Perturbations 2A, pp. 1—11.
• Niedziela, J., “Bessel function and their applications”, Knoxville: University of Tennessee, 2008.
• Balanis, C., “Advanced engineering electromagnetics”, Wiley, 1989, 2nd ed., NY, pp. 483—500.
• Cheng, D. K., “Field and wave electromagnetics”, Addison-Wesley, 1989, 2nd ed., London, pp., 562—572.
• Beattie, C.L., “Table of first 700 zeros of Bessel functions—Jl(x) and Jl’(x)”, Bell System Technical Journal, 1958, 37, pp. 689—697.
• Millane R.P. and Eads, J.L., “Polynomial approximations to Bessel functions”, IEEE Transactions On Antennas And Propagation, 2003, 51(6), pp. 1398—1400.
• Abuelma’atti, M.T., “Trigonometric approximations for some Bessel functions”, Active and Passive Elec. Comp., 1999, 22, pp. 75-85.
• Blachman, N. M. and Mousavinezhad, S. H. “Trigonometric approximations for Bessel functions”, EEE Transactions on Aerospace and Electronic Systems, 1986, AES-22(1): 2—7.
• Waldron, R. A. “Formulas for computation of approximate values of some Bessel functions”, Proceedings of the IEEE, 69, pp. 1686-1588, (1981).
• Luke, Y.L., “Approximation of special Functions”, Academic Press, 1975, NY.
• Newman, J.N., “Approximations for the Bessel and Struve functions”, Mathematics of Computation, 1984, 43(168), pp. 551—556.
• Harrison, J., “Fast and accurate Bessel function computation”, Computer Arithmetic-proc. of 19th EEE symp. on Computer Arithmetic, Portland-Oregon, pp. 104—113, (2009).
• Vrahatism, M. N., Ragos, O., Zafiropoulos, F. A., Grapsa, T. N., “Locating and computing zeros of Airy functions”, Z. Angew. Math. Mech., 1996, 76 (7), pp. 419—422.
• Abramowitz A. and Stegun I.A., “Handbook of mathematical functions, with formulas, Graphs, and Mathematical Tables”, 3rd printing with corrections, Vol. 55 of NBS Applied mathematics series, superintendent of documents, US Government Printing Office, 1965, Washington DC, pp. 355—479.
• Wolfram, S., “The mathematica book”, Wolfram Media Inc., 2003, 5th edition, USA, pp. 29—35&pp. 102—110.
• http://reference.wolfram.com/mathematica/ref/BesselJ.html
• Richards, D., “Advanced mathematical methods with maple”, Cambridge University Press, 2002, UK, pp. 325—331.
• Chapra, S. and Canale, R., “Numerical methods for engineers”, WCB/McGraw-Hill, 2014, 7th Edition, NY, pp. 148—154.
• Hamming, R.W., “Numerical methods for scientists and engineers (Dover Books on Mathematics)”, Dover Publications, 1987, 2nd Revised ed., NY, pp. 68—72.
• Hoffman, J.D., “Numerical methods for engineers and scientists”, Marcel Dekkel, 2001, 2nd edition, NY-Basel, pp. 141—154.
• https://www.wolfram.com/cdf-player/
• https://en.wikipedia.org/wiki/Computable_Document_Format
• http://www.wolfram.com/events/siam-2016/files/CDF-4.pdf
• http://akademik.adu.edu.tr/fakulte/muhendislik/personel/uploads/cdeniz/nr-based-root-finder-for-cylindrical-waveguides-1471264126.rar (after downloaded and unzipped, it should be opened by the free wolfam cdf player, which is downloadable via [30])
• https://en.wikipedia.org/wiki/Bubble_sort
• Cormen, T.H., Leiserson, C.E., Rivest, R.L., and Stein, C., “Introduction to algorithms”, MIT Press, 2009, 3rd ed., USA, p. 40.
• Astrachan, O., “Bubble Sort: An archaeological algorithmic analysis”, SIGCSE '03 Proceedings of the 34th SIGCSE Technical Symposium on Computer Science Education, NY, pp. 1—5, (2003).
• Khairullah, Md., “Enhancing worst sorting algorithms”, International Journal of Advanced Science and Technology, 2013, 56, pp. 13—26.
• Rohil, H. and Manisha, “Run time Bubble sort–An enhancement of Bubble sort”, International Journal of Computer Trends and Technology (IJCTT), 2014, 14(1), pp. 36—38.
• Arora, N., Kumar, S., and Tamta, V.K., “A novel sorting algorithm and comparison with Bubble sort and Insertion sort”, International Journal of Computer Applications, 2012, 45(1): pp. 31—32.