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

Abstract

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

Abstract

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.

Keywords

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

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