An application of the finite element approximation method to find the complex zeros of the modified Bessel function 𝐾_{𝑛}(𝑧)

Leung, K. V.; Ghaderpanah, S. S.

doi:10.1090/S0025-5718-1979-0537975-3

An application of the finite element approximation method to find the complex zeros of the modified Bessel function $K_{n}(z)$
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by K. V. Leung and S. S. Ghaderpanah PDF

Math. Comp. 33 (1979), 1299-1306 Request permission

Abstract:

Using a finite element approximation, an iterative optimization scheme is described to find the z zeros of ${K_n}(z)$ for fixed order n. Two computer programs have been implemented to find the complex zeros with a computational accuracy of either 13 or 27 significant digits. The optimization scheme described in the paper may also be readily applied to find real and complex zeros of an arbitrary function with real and complex coefficients. Neither its accuracy nor its efficiency is affected by the number of the roots of the function.

References

James Alan Cochran and Judith N. Hoffspiegel, Numerical techniques for finding $\nu$-zeros of Hankel functions, Math. Comp. 24 (1970), 413–422. MR 272157, DOI 10.1090/S0025-5718-1970-0272157-6
Boro Döring, Complex zeros of cylinder functions, Math. Comp. 20 (1966), 215–222. MR 192632, DOI 10.1090/S0025-5718-1966-0192632-1
Jerry L. Fields, A linear scheme for rational approximations, J. Approximation Theory 6 (1972), 161–175. MR 346383, DOI 10.1016/0021-9045(72)90072-x
Werner L. Frank, Finding zeros of arbitrary functions, J. Assoc. Comput. Mach. 5 (1958), 154–160. MR 144457, DOI 10.1145/320924.320928

Some Applications of Finite Element Method

Proc. Third Internat. Cong. Cybernetics and Systems

Yudell L. Luke, The special functions and their approximations. Vol. II, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR 0249668
Yudell L. Luke, Mathematical functions and their approximations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0501762
Yudell L. Luke, Algorithms for the computation of mathematical functions, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. MR 0494840
F. W. J. Olver, The asymptotic expansion of Bessel functions of large order, Philos. Trans. Roy. Soc. London Ser. A 247 (1954), 328–368. MR 67250, DOI 10.1098/rsta.1954.0021
M. N. Oğuztöreli, D. L. Jeffries, and K. V. Leung, A kinetic study of enzyme systems, Nonlinear Anal. 1 (1976), no. 1, 15–35. MR 432241, DOI 10.1016/0362-546X(76)90005-5
R. Parnes, Complex zeros of the modified Bessel function $K_{n}(Z)$, Math. Comp. 26 (1972), 949–953. MR 314237, DOI 10.1090/S0025-5718-1972-0314237-4
Dura W. Sweeney, On the computation of Euler’s constant, Math. Comp. 17 (1963), 170–178. MR 160308, DOI 10.1090/S0025-5718-1963-0160308-X
James A. Ward, The down-hill method of solving $f(z)=0$, J. Assoc. Comput. Mach. 4 (1957), 148–150. MR 92227, DOI 10.1145/320868.320873
O. C. Zienkiewicz, The finite element method in engineering science, McGraw-Hill, London-New York-Düsseldorf, 1971. The second, expanded and revised, edition of The finite element method in structural and continuum mechanics. MR 0315970
Miloš Zlámal, On the finite element method, Numer. Math. 12 (1968), 394–409. MR 243753, DOI 10.1007/BF02161362