Showing papers on "Bessel filter published in 1980"

On Temme's Algorithm for the Modified Bessel Function of the Third Kind

Abstract: Some modlficatmns to Temme's algorithm for the evaluation of the modified Bessel functmn of the third kind have been made Temme evaluates K,(x) and K,+dx) for x > 1 and ]~1 --1⁄2 from auxdmry functmns k(,(x) and kdx), which he determines by Miller's backward recurrence algorithm In this paper the starting order for the backward recurrence algorithm for ko(x) and k~(x) is given by a pmcewlse linear functmn of 1/x Also, we have found that ~t is more efficmnt, for small values of x, to calculate only kdx)/ko(x), whmh can be used with the values L(x) and L÷dx) and the Wronskian to obtam K,.(x) and K,+dx)

Further comments on the computation of modified Bessel function ratios

Abstract: A theorem by W. J. Thron is used to enhance and further substantiate empirical results by Gautschi and Slavik on the convergence of continued fractions used to compute ratios of modified Bessel functions. 1. Thron's Theorem Applied to Continued Fractions for Ratios of Modified Bessel Functions. A theorem by Thron* [2] offers analytic confirmation of recent empirical results [1] on the convergence of continued fractions, cfs, for computing ratios of modified Bessel functions. Defining Thron's Theorem 6.1 states: The continued fraction K(an/1) converges to a value u which satisfies R(u) > -Vi provided that J2(an) < R(an) + % and \\an\\ 1. Here M is an arbitrary large positive quantity. As a consequence of Thron's theorem we can obtain a priori bounds on the truncation error, 2Rn, of Gauss's and Perron's cfs for modified Bessel function ratios. In the case of Thron's Theorem 6.1, Rn is the radius of the value region (nested circles within which the value of the cf lies) of the cfs. From Thron's theorem we thus have *7,<1 Üil + Ui4Mk)). I *=i Relatively larger values of M are associated with overall slower convergence. Both Gauss's and Perron's cfs for Iv(x)/Iv_x(x) are in the form required by the theorem [1]. For positive x the assumptions of Thron's theorem are trivially satisfied for Gauss's cf and are easily verified for Perron's cf if v > 1. Thus, to apply the theorem, note that when x » v, the an of Perron's cf will be dominated by l/a0{x) = x/ix + 2v) and the an of Gauss's cf will be dominated by axix) = (l/4)x2/v(v + 1), which we will refer to asMp and MG, respectively. Clearly, as* » v, Mp< 1 and MG » 1. In this case, Rn, or 2Rn (the truncation error), of Gauss's cf is much larger than Perron's, indicating slower convergence of the Gauss cf. These results are given in Table 1.1 which reproduces Table 3.2 of [1] supplemented by values of Mp and MG Received March 20, 1978; revised September 5, 1979. 1980 Mathematics Subject Classification. Primary 33A40; Secondary 40A15, 65D15. * Thron's theorem referred to here is actually a special case of an earlier result due to Scott and Wall (Trans. Amer. Math. Soc, v. 47, 1940, pp. 155-172). However, the results of this paper depend on the proof as given by Thron. © 1980 American Mathematical Society 0025-5718/80/0000-0119/$01.75 937 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 938 MARIETTA J. TRETTER AND G. W. WALSTER (k in Table 1.1 represents the number of terms required in the Gauss and Perron cfs to obtain relative accuracy of tëlO-8, as originally given in Table 3.2 of [1]). One interesting observation in Table 1.1 is the fact that for x » v, Mp is increasing slightly as x increases, even though the table indicates, for 8 decimal digit accuracy, that Perron's cf is converging faster with increasing x. This phenomenon is explained by looking at Figure 3.3 in [1]. The increase in Mp reflects the fact that, although initial convergence is fast, at extreme accuracies the convergence slows down before regaining speed. The phenomenon of slowing down also becomes more pronounced as x increases. Thus, one must be careful to interpret M as an overall measure of convergence. Table 1.1 Number of terms, k, required in the Gauss and Perron cfs to obtain relative accuracy of &10~ 8 (as given in Table 3.2 of [1 ] ), and M as defined above.