Vol. 20, issue 08, article # 6

Romanov N.P. Study of methods and errors in computation of Ricatti-Bessel functions. // Atmospheric and oceanic optics. 2007. V. 20. No. 08. P. 638-646.
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Abstract:

The dependence of the Ricatti-Bessel (RB) function of the 1st (RB1), 2nd (RB2), 3rd (RB3), and 4th (RB4) kinds of the real х and complex z = х + iy arguments on their order l are studied analytically and numerically. The domains of RB function's module increase with increase of l, tolerant to the forward-recurrence errors, and of module decrease, tolerant to the backward-recurrence one, are found out. The domain of stability of the forward recurrence for RB1 function at x >> 1 is determined by the relation 0 ≤ llmax = x - 0,5 - 0,80861x1/3 - 0,1635x-1/3 on condition that |y| ≤ 0,4lgx + 0,5; here the fractional error of the forward recurrence increases with an increase of l proportionally to ll1/2. If l > lmax, the forward recurrence results in generation of the function equal to the sum of RB1 and RB2 functions instead of the function RB1. The fractional error of the backward recurrence for the RB1 function is virtually independent of l in the whole l range, increases by the |z|1/2 law with an increase of z modulus, and is comparable with the forward-recurrence error at l = lmax under the above limitations on y. To obtain the initial RB1 values in the backward recurrence, a simplified calculation procedure for the ratio of these functions of the neighboring orders with the use of a continued fraction is suggested, an additional forward-recurrence computation of the RB2 function at y = 0 or RB3 at y > 0, and the use of the Wronskian of the corresponding functions. A FORTRAN program for computing the above-mentioned ratio of RB1 functions is presented. In the domains of stability, the main computation error of functions of decimal arguments can be determined by the error of their conversion into the binary computer system.

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