Evaluation and Inversion of the Ratios of Modified Bessel Functions, I1(x) /I0 (x) and I 1.5(x)/ I0.5(x)

Abstract

Adaptation to specific processor precision is illustrated for procedures for evaluating the ratios A(k) = I1(k)/Io(k) and B(k) = 115(K)/I05(K), which provide the expected value R of the mean modulus of random umt vectors, distributed with concentration parameter k in (a) von Mises’ distribution of directions m two dimensmns (on a circle) or (b) Fisher's chstribution in three dnnensions (on a sphere), respectively. For large k, the exponential equivalent of B(k) = coth k - 1/K is used. Three continued fractmn expansmns are shown to be statable for recurswe evaluatmn from a depth of recursion n(k | S), approximated to ensure at least S significant decimal chgits m the result. Improved approximatmns for the inverse functions A-1(R) and B-1(R) provide maxunum likelihood estimates of K for gaven R, directly achmvmg at least 3S and 8S, respectively, which may be extended near processor preclsmn by few Newton-Raphson improvements. © 1981, ACM. All rights reserved.