arXiv subjects

Arnie L. Van Buren

Publications and source records attributed to Arnie L. Van Buren.

2 recordsLinked to original sources

arXiv2020

Calculation of oblate spheroidal wave functions with complex argument

A previous article showed that alternative expressions for calculating oblate spheroidal radial functions of both kinds can provide accurate values over very large parameter ranges using double precision arithmetic, even where the traditional expressions fail. The size parameter c was assumed real. This paper considers the case where c = cr + ici is complex with an imaginary part ci often used to represent losses in wave behavior. The methods for c real modified to complex arithmetic work reasonably well as long as ci is very small. This paper describes the substantial changes necessary to obtain useful results for larger values of ci. It shows that accurate eigenvalues can usually be obtained even though the matrix methods used to obtain them for c real provide increasingly inaccurate values, primarily for those with relatively small magnitude, as ci increases. It also shows that some of the eigenvalues can be prolate-like with values that are well approximated using asymptotic estimates for prolate eigenvalues where c is replaced with -ic. A method to order the eigenvalues is presented. The modifications necessary to compute accurately the radial and angular functions for complex c are discussed. A resulting Fortran program coblfcn provides useful function values for a reasonably wide range of c, the order m, and the radial coordinate when using double precision arithmetic. The results can be improved by using quadruple precision for the Bouwkamp procedure to ensure accurate double precision eigenvalues. Further improvement is obtained using full quadruple precision. Coblfcn is freely available at www.mathieuandspheroidalwavefunctions.com

math.NA↗

arXiv2019

Accurate calculation of oblate spheroidal wave functions

Alternative expressions for calculating the oblate spheroidal radial functions of both kinds R1ml and R2ml are shown to provide accurate values over very large parameter ranges using 64 bit arithmetic, even where the traditional expressions fail. First is the expansion of the product of a radial function and the angular function of the first kind in a series of products of the corresponding spherical functions, with the angular coordinate being a free parameter. Setting the angular coordinate equal to zero leads to accurate values for R2ml when the radial coordinate xi is larger than 0.01 and l is somewhat larger than m. Allowing it to vary with increasing l leads to highly accurate values for R1ml over all parameter ranges. Next is the calculation of R2ml as an integral of the product of S1ml and a spherical Neumann function kernel. This is useful for smaller values of xi. Also used is the near equality of pairs of low order eigenvalues when the size parameter c is large that leads to accurate values for R2ml using neighboring accurate values for R1ml. A modified method is described that provides accurate values for the necessary expansion coefficients when c is large and l is near m and traditional methods fail. A resulting Fortran computer program Oblfcn almost always provides radial function values with at least 8 accurate decimal digits using 64 bit arithmetic for m up to at least 1000 with c up to at least 2000 when xi is greater than 0.000001 and c up to at least 5000 when xi is greater than 0.01. Use of 128 bit arithmetic extends the accuracy to 15 or more digits and extends xi to all values other than zero. Oblfcn is freely available.

math.CA↗