By Malcolm M. Bibby, Andrew F. Peterson
This lecture offers a latest technique for the computation of Mathieu capabilities. those services locate software in boundary worth research equivalent to electromagnetic scattering from elliptic cylinders and flat strips, in addition to the analogous acoustic and optical difficulties, and lots of different functions in technology and engineering. The authors evaluation the conventional method used for those features, convey its barriers, and supply another "tuned" strategy allowing more advantageous accuracy and convergence. The functionality of this technique is investigated for quite a lot of parameters and desktop precision. Examples from electromagnetic scattering are supplied for representation and to teach the convergence of the common sequence that hire Mathieu features for boundary price analysis.
Table of Contents: advent / Mathieu features / saw Accuracy utilizing conventional and Tuned equipment / instructed set of rules for Mathieu functionality Computation / Electromagnetic Scattering from engaging in Elliptic Cylinders / Electromagnetic Scattering from an enormous carrying out Strip / References / Authors' Biographies
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Extra resources for Accurate Computation of Mathieu Functions
5, it is observed that a strong relationship exists between the magnitude of log10 jce2r j and the associated subtraction error which is independent of both q and v , although it is not known at this time how to exploit this relationship. 6 illustrates the eﬀect of varying N on the value of log10 jce2r j. For the parameters used, a value of N D 100 suﬃces. As this is for a value of v D 0, smaller values of N may be suitable. To date no deterministic method for a priori deciding on the necessary minimum value of N is known.
Is problem was suggested by Erricolo , who observed serious convergence problems in its evaluation. We will see how this convergence problem is avoided by using the tuned method for evaluating the radial functions. kR/, where k D 2 deﬁned by the expansion in Mathieu functions [6, p. 4) is summation was studied using the traditional formulation by Erricolo , who found that the real part of the calculated value quickly converged. 1. His remedy to this undesirable outcome was to apply an adaptation of the Shanks transformation to the imaginary component to accelerate the convergence.
1: Plot of the Subtraction Error for cem as a function of m. Although the matrix size is only 100, the even and odd functions, when combined, produce a total of 200 values. Only the ﬁrst 100 are shown. 2: Plot of the magnitude of cem as a function of m for v D 0. Although the matrix size is only 100, the even and odd functions, when combined, produce a total of 200 values. 1. 3: Plots of the Subtraction Error encountered in the ﬁrst two ce and se functions versus the value of the angular function for N D 100.