[section:bessel Bessel Functions of the First and Second Kinds] [h4 Synopsis] template ``__sf_result`` cyl_bessel_j(T1 v, T2 x); template ``__sf_result`` cyl_bessel_j(T1 v, T2 x, const ``__Policy``&); template ``__sf_result`` cyl_neumann(T1 v, T2 x); template ``__sf_result`` cyl_neumann(T1 v, T2 x, const ``__Policy``&); [h4 Description] The functions __cyl_bessel_j and __cyl_neumann return the result of the Bessel functions of the first and second kinds respectively: cyl_bessel_j(v, x) = J[sub v](x) cyl_neumann(v, x) = Y[sub v](x) = N[sub v](x) where: [equation bessel2] [equation bessel3] The return type of these functions is computed using the __arg_pomotion_rules when T1 and T2 are different types. The functions are also optimised for the relatively common case that T1 is an integer. [optional_policy] The functions return the result of __domain_error whenever the result is undefined or complex. For __cyl_bessel_j this occurs when `x < 0` and v is not an integer, or when `x == 0` and `v != 0`. For __cyl_neumann this occurs when `x <= 0`. The following graph illustrates the cyclic nature of J[sub v]: [graph cyl_bessel_j] The following graph shows the behaviour of Y[sub v]: this is also cyclic for large /x/, but tends to -[infin][space] for small /x/: [graph cyl_neumann] [h4 Testing] There are two sets of test values: spot values calculated using [@http://functions.wolfram.com functions.wolfram.com], and a much larger set of tests computed using a simplified version of this implementation (with all the special case handling removed). [h4 Accuracy] The following tables show how the accuracy of these functions varies on various platforms, along with comparisons to the __gsl and __cephes libraries. Note that the cyclic nature of these functions means that they have an infinite number of irrational roots: in general these functions have arbitrarily large /relative/ errors when the arguments are sufficiently close to a root. Of course the absolute error in such cases is always small. Note that only results for the widest floating-point type on the system are given as narrower types have __zero_error. All values are relative errors in units of epsilon. [table Errors Rates in cyl_bessel_j [[Significand Size] [Platform and Compiler] [J[sub 0][space] and J[sub 1]] [J[sub v]] [J[sub v][space] (large values of x > 1000)] ] [[53] [Win32 / Visual C++ 8.0] [Peak=2.5 Mean=1.1 GSL Peak=6.6 __cephes Peak=2.5 Mean=1.1] [Peak=11 Mean=2.2 GSL Peak=11 __cephes Peak=17 Mean=2.5] [Peak=413 Mean=110 GSL Peak=6x10[super 11] __cephes Peak=2x10[super 5] ] ] [[64] [Red Hat Linux IA64 / G++ 3.4] [Peak=7 Mean=3] [Peak=117 Mean=10] [Peak=2x10[super 4][space] Mean=6x10[super 3]] ] [[64] [SUSE Linux AMD64 / G++ 4.1] [Peak=7 Mean=3] [Peak=400 Mean=40] [Peak=2x10[super 4][space] Mean=1x10[super 4]] ] [[113] [HP-UX / HP aCC 6] [Peak=14 Mean=6] [Peak=29 Mean=3] [Peak=2700 Mean=450] ] ] [table Errors Rates in cyl_neumann [[Significand Size] [Platform and Compiler] [J[sub 0][space] and J[sub 1]] [J[sub n] (integer orders)] [J[sub v] (fractional orders)] ] [[53] [Win32 / Visual C++ 8.0] [Peak=330 Mean=54 GSL Peak=34 Mean=9 __cephes Peak=330 Mean=54] [Peak=923 Mean=83 GSL Peak=500 Mean=54 __cephes Peak=923 Mean=83] [Peak=561 Mean=36 GSL Peak=1.4x10[super 6][space] Mean\=7x10[super 4][space] __cephes Peak=+INF]] [[64] [Red Hat Linux IA64 / G++ 3.4] [Peak=470 Mean=56] [Peak=843 Mean=51] [Peak=741 Mean=51] ] [[64] [SUSE Linux AMD64 / G++ 4.1] [Peak=1300 Mean=424] [Peak=2x10[super 4][space] Mean=8x10[super 3]] [Peak=1x10[super 5][space] Mean=6x10[super 3]] ] [[113] [HP-UX / HP aCC 6] [Peak=180 Mean=63] [Peak=340 Mean=150] [Peak=2x10[super 4][space] Mean=1200] ] ] Note that for large /x/ these functions are largely dependent on the accuracy of the `std::sin` and `std::cos` functions. Comparison to GSL and __cephes is interesting: both __cephes and this library optimise the integer order case - leading to identical results - simply using the general case is for the most part slightly more accurate though, as noted by the better accuracy of GSL in the integer argument cases. This implementation tends to perform much better when the arguments become large, __cephes in particular produces some remarkably inaccurate results with some of the test data (no significant figures correct), and even GSL performs badly with some inputs to J[sub v]. Note that by way of double-checking these results, the worst performing __cephes and GSL cases were recomputed using [@http://functions.wolfram.com functions.wolfram.com], and the result checked against our test data: no errors in the test data were found. [h4 Implementation] The implementation is mostly about filtering off various special cases: When /x/ is negative, then the order /v/ must be an integer or the result is a domain error. If the order is an integer then the function is odd for odd orders and even for even orders, so we reflect to /x > 0/. When the order /v/ is negative then the reflection formulae can be used to move to /v > 0/: [equation bessel9] [equation bessel10] Note that if the order is an integer, then these formulae reduce to: J[sub -n] = (-1)[super n]J[sub n] Y[sub -n] = (-1)[super n]Y[sub n] However, in general, a negative order implies that we will need to compute both J and Y. When /x/ is large compared to the order /v/ then the asymptotic expansions for large /x/ in M. Abramowitz and I.A. Stegun, ['Handbook of Mathematical Functions] 9.2.19 are used (these were found to be more reliable than those in A&S 9.2.5). When the order /v/ is an integer the method first relates the result to J[sub 0], J[sub 1], Y[sub 0][space] and Y[sub 1][space] using either forwards or backwards recurrence (Miller's algorithm) depending upon which is stable. The values for J[sub 0], J[sub 1], Y[sub 0][space] and Y[sub 1][space] are calculated using the rational minimax approximations on root-bracketing intervals for small ['|x|] and Hankel asymptotic expansion for large ['|x|]. The coefficients are from: W.J. Cody, ['ALGORITHM 715: SPECFUN - A Portable FORTRAN Package of Special Function Routines and Test Drivers], ACM Transactions on Mathematical Software, vol 19, 22 (1993). and J.F. Hart et al, ['Computer Approximations], John Wiley & Sons, New York, 1968. These approximations are accurate to around 19 decimal digits: therefore these methods are not used when type T has more than 64 binary digits. When /x/ is smaller than machine epsilon then the following approximations for Y[sub 0](x), Y[sub 1](x), Y[sub 2](x) and Y[sub n](x) can be used (see: [@http://functions.wolfram.com/03.03.06.0037.01 http://functions.wolfram.com/03.03.06.0037.01], [@http://functions.wolfram.com/03.03.06.0038.01 http://functions.wolfram.com/03.03.06.0038.01], [@http://functions.wolfram.com/03.03.06.0039.01 http://functions.wolfram.com/03.03.06.0039.01] and [@http://functions.wolfram.com/03.03.06.0040.01 http://functions.wolfram.com/03.03.06.0040.01]): [equation bessel_y0_small_z] [equation bessel_y1_small_z] [equation bessel_y2_small_z] [equation bessel_yn_small_z] When /x/ is small compared to /v/ and /v/ is not an integer, then the following series approximation can be used for Y[sub v](x), this is also an area where other approximations are often too slow to converge to be used (see [@http://functions.wolfram.com/03.03.06.0034.01 http://functions.wolfram.com/03.03.06.0034.01]): [equation bessel_yv_small_z] When /x/ is small compared to /v/, J[sub v]x[space] is best computed directly from the series: [equation bessel2] In the general case we compute J[sub v][space] and Y[sub v][space] simultaneously. To get the initial values, let [mu][space] = [nu] - floor([nu] + 1/2), then [mu][space] is the fractional part of [nu][space] such that |[mu]| <= 1/2 (we need this for convergence later). The idea is to calculate J[sub [mu]](x), J[sub [mu]+1](x), Y[sub [mu]](x), Y[sub [mu]+1](x) and use them to obtain J[sub [nu]](x), Y[sub [nu]](x). The algorithm is called Steed's method, which needs two continued fractions as well as the Wronskian: [equation bessel8] [equation bessel11] [equation bessel12] See: F.S. Acton, ['Numerical Methods that Work], The Mathematical Association of America, Washington, 1997. The continued fractions are computed using the modified Lentz's method (W.J. Lentz, ['Generating Bessel functions in Mie scattering calculations using continued fractions], Applied Optics, vol 15, 668 (1976)). Their convergence rates depend on ['x], therefore we need different strategies for large ['x] and small ['x]. ['x > v], CF1 needs O(['x]) iterations to converge, CF2 converges rapidly ['x <= v], CF1 converges rapidly, CF2 fails to converge when ['x] [^->] 0 When ['x] is large (['x] > 2), both continued fractions converge (CF1 may be slow for really large ['x]). J[sub [mu]], J[sub [mu]+1], Y[sub [mu]], Y[sub [mu]+1] can be calculated by [equation bessel13] where [equation bessel14] J[sub [nu]] and Y[sub [mu]] are then calculated using backward (Miller's algorithm) and forward recurrence respectively. When ['x] is small (['x] <= 2), CF2 convergence may fail (but CF1 works very well). The solution here is Temme's series: [equation bessel15] where [equation bessel16] g[sub k][space] and h[sub k][space] are also computed by recursions (involving gamma functions), but the formulas are a little complicated, readers are refered to N.M. Temme, ['On the numerical evaluation of the ordinary Bessel function of the second kind], Journal of Computational Physics, vol 21, 343 (1976). Note Temme's series converge only for |[mu]| <= 1/2. As the previous case, Y[sub [nu]][space] is calculated from the forward recurrence, so is Y[sub [nu]+1]. With these two values and f[sub [nu]], the Wronskian yields J[sub [nu]](x) directly without backward recurrence. [endsect] [/ Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang. Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt). ]