[section:bessel_first Bessel Functions of the First and Second Kinds] [h4 Synopsis] `#include ` 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: [expression cyl_bessel_j(v, x) = J[sub v](x)] [expression 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_promotion_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] 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 other 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. Most of the gross errors exhibited by other libraries occur for very large arguments - you will need to drill down into the actual program output if you need more information on this. [table_cyl_bessel_j_integer_orders_] [table_cyl_bessel_j] [table_cyl_neumann_integer_orders_] [table_cyl_neumann] 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. The following error plot are based on an exhaustive search of the functions domain for J0 and Y0, MSVC-15.5 at `double` precision, other compilers and precisions are very similar - the plots simply illustrate the relatively large errors as you approach a zero, and the very low errors elsewhere. [graph j0__double] [graph y0__double] [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: [expression J[sub -n] = (-1)[super n]J[sub n]] [expression 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] and Y[sub 1] 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] and Y[sub 1] 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 is best computed directly from the series: [equation bessel2] In the general case we compute J[sub v] and Y[sub v] simultaneously. To get the initial values, let [mu] = [nu] - floor([nu] + 1/2), then [mu] is the fractional part of [nu] 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] and h[sub k] 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]] 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] [/section:bessel_first Bessel Functions of the First and Second Kinds] [section:bessel_root Finding Zeros of Bessel Functions of the First and Second Kinds] [h4 Synopsis] `#include ` Functions for obtaining both a single zero or root of the Bessel function, and placing multiple zeros into a container like `std::vector` by providing an output iterator. The signature of the single value functions are: template T cyl_bessel_j_zero( T v, // Floating-point value for Jv. int m); // 1-based index of zero. template T cyl_neumann_zero( T v, // Floating-point value for Jv. int m); // 1-based index of zero. and for multiple zeros: template OutputIterator cyl_bessel_j_zero( T v, // Floating-point value for Jv. int start_index, // 1-based index of first zero. unsigned number_of_zeros, // How many zeros to generate. OutputIterator out_it); // Destination for zeros. template OutputIterator cyl_neumann_zero( T v, // Floating-point value for Jv. int start_index, // 1-based index of zero. unsigned number_of_zeros, // How many zeros to generate OutputIterator out_it); // Destination for zeros. There are also versions which allow control of the __policy_section for error handling and precision. template T cyl_bessel_j_zero( T v, // Floating-point value for Jv. int m, // 1-based index of zero. const Policy&); // Policy to use. template T cyl_neumann_zero( T v, // Floating-point value for Jv. int m, // 1-based index of zero. const Policy&); // Policy to use. template OutputIterator cyl_bessel_j_zero( T v, // Floating-point value for Jv. int start_index, // 1-based index of first zero. unsigned number_of_zeros, // How many zeros to generate. OutputIterator out_it, // Destination for zeros. const Policy& pol); // Policy to use. template OutputIterator cyl_neumann_zero( T v, // Floating-point value for Jv. int start_index, // 1-based index of zero. unsigned number_of_zeros, // How many zeros to generate. OutputIterator out_it, // Destination for zeros. const Policy& pol); // Policy to use. [h4 Description] Every real order [nu] cylindrical Bessel and Neumann functions have an infinite number of zeros on the positive real axis. The real zeros on the positive real axis can be found by solving for the roots of [:['J[sub [nu]](j[sub [nu], m]) = 0]] [:['Y[sub [nu]](y[sub [nu], m]) = 0]] Here, ['j[sub [nu], m]] represents the ['m[super th]] root of the cylindrical Bessel function of order ['[nu]], and ['y[sub [nu], m]] represents the ['m[super th]] root of the cylindrical Neumann function of order ['[nu]]. The zeros or roots (values of `x` where the function crosses the horizontal `y = 0` axis) of the Bessel and Neumann functions are computed by two functions, `cyl_bessel_j_zero` and `cyl_neumann_zero`. In each case the index or rank of the zero returned is 1-based, which is to say: [:cyl_bessel_j_zero(v, 1);] returns the first zero of Bessel J. Passing an `start_index <= 0` results in a `std::domain_error` being raised. For certain parameters, however, the zero'th root is defined and it has a value of zero. For example, the zero'th root of `J[sub v](x)` is defined and it has a value of zero for all values of `v > 0` and for negative integer values of `v = -n`. Similar cases are described in the implementation details below. The order `v` of `J` can be positive, negative and zero for the `cyl_bessel_j` and `cyl_neumann` functions, but not infinite nor NaN. [graph bessel_j_zeros] [graph neumann_y_zeros] [h4 Examples of finding Bessel and Neumann zeros] [import ../../example/bessel_zeros_example_1.cpp] [bessel_zeros_example_1] [bessel_zeros_example_2] [import ../../example/bessel_zeros_interator_example.cpp] [bessel_zeros_iterator_example_1] [bessel_zeros_iterator_example_2] [import ../../example/neumann_zeros_example_1.cpp] [neumann_zeros_example_1] [neumann_zeros_example_2] [import ../../example/bessel_errors_example.cpp] [bessel_errors_example_1] [bessel_errors_example_2] The full code (and output) for these examples is at [@../../example/bessel_zeros_example_1.cpp Bessel zeros], [@../../example/bessel_zeros_interator_example.cpp Bessel zeros iterator], [@../../example/neumann_zeros_example_1.cpp Neumann zeros], [@../../example/bessel_errors_example.cpp Bessel error messages]. [h3 Implementation] Various methods are used to compute initial estimates for ['j[sub [nu], m]] and ['y[sub [nu], m]] ; these are described in detail below. After finding the initial estimate of a given root, its precision is subsequently refined to the desired level using Newton-Raphson iteration from Boost.Math's __root_finding_with_derivatives utilities combined with the functions __cyl_bessel_j and __cyl_neumann. Newton iteration requires both ['J[sub [nu]](x)] or ['Y[sub [nu]](x)] as well as its derivative. The derivatives of ['J[sub [nu]](x)] and ['Y[sub [nu]](x)] with respect to ['x] are given by __Abramowitz_Stegun. In particular, [expression d/[sub dx] ['J[sub [nu]](x)] = ['J[sub [nu]-1](x)] - [nu] J[sub [nu]](x) / x] [expression d/[sub dx] ['Y[sub [nu]](x)] = ['Y[sub [nu]-1](x)] - [nu] Y[sub [nu]](x) / x] Enumeration of the rank of a root (in other words the index of a root) begins with one and counts up, in other words ['m,=1,2,3,[ellipsis]] The value of the first root is always greater than zero. For certain special parameters, cylindrical Bessel functions and cylindrical Neumann functions have a root at the origin. For example, ['J[sub [nu]](x)] has a root at the origin for every positive order ['[nu] > 0], and for every negative integer order ['[nu] = -n] with ['n [isin] [negative] [super +]] and ['n [ne] 0]. In addition, ['Y[sub [nu]](x)] has a root at the origin for every negative half-integer order ['[nu] = -n/2], with ['n [isin] [negative] [super +]] and and ['n [ne] 0]. For these special parameter values, the origin with a value of ['x = 0] is provided as the ['0[super th]] root generated by `cyl_bessel_j_zero()` and `cyl_neumann_zero()`. When calculating initial estimates for the roots of Bessel functions, a distinction is made between positive order and negative order, and different methods are used for these. In addition, different algorithms are used for the first root ['m = 1] and for subsequent roots with higher rank ['m [ge] 2]. Furthermore, estimates of the roots for Bessel functions with order above and below a cutoff at ['[nu] = 2.2] are calculated with different methods. Calculations of the estimates of ['j[sub [nu],1]] and ['y[sub [nu],1]] with ['0 [le] [nu] < 2.2] use empirically tabulated values. The coefficients for these have been generated by a computer algebra system. Calculations of the estimates of ['j[sub [nu],1]] and ['y[sub [nu],1]] with ['[nu][ge] 2.2] use Eqs.9.5.14 and 9.5.15 in __Abramowitz_Stegun. In particular, [expression j[sub [nu],1] [cong] [nu] + 1.85575 [nu][super [frac13]] + 1.033150 [nu][super -[frac13]] - 0.00397 [nu][super -1] - 0.0908 [nu][super -5/3] + 0.043 [nu][super -7/3] + [ellipsis]] and [expression y[sub [nu],1] [cong] [nu] + 0.93157 [nu][super [frac13]] + 0.26035 [nu][super -[frac13]] + 0.01198 [nu][super -1] - 0.0060 [nu][super -5/3] - 0.001 [nu][super -7/3] + [ellipsis]] Calculations of the estimates of ['j[sub [nu], m]] and ['y[sub [nu], m]] with rank ['m > 2] and ['0 [le] [nu] < 2.2] use McMahon's approximation, as described in M. Abramowitz and I. A. Stegan, Section 9.5 and 9.5.12. In particular, [:['j[sub [nu],m], y[sub [nu],m] [cong]]] [:[:[beta] - ([mu]-1) / 8[beta]]] [:[:['- 4([mu]-1)(7[mu] - 31) / 3(8[beta])[super 3]]]] [:[:['-32([mu]-1)(83[mu][sup2] - 982[mu] + 3779) / 15(8[beta])[super 5]]]] [:[:['-64([mu]-1)(6949[mu][super 3] - 153855[mu][sup2] + 1585743[mu]- 6277237) / 105(8a)[super 7]]]] [:[:['- [ellipsis]] [emquad] (5)]] where ['[mu] = 4[nu][super 2]] and ['[beta] = (m + [frac12][nu] - [frac14])[pi]] for ['j[sub [nu],m]] and ['[beta] = (m + [frac12][nu] -[frac34])[pi] for ['y[sub [nu],m]]]. Calculations of the estimates of ['j[sub [nu], m]] and ['y[sub [nu], m]] with ['[nu] [ge] 2.2] use one term in the asymptotic expansion given in Eq.9.5.22 and top line of Eq.9.5.26 combined with Eq. 9.3.39, all in __Abramowitz_Stegun explicit and easy-to-understand treatment for asymptotic expansion of zeros. The latter two equations are expressed for argument ['(x)] greater than one. (Olver also gives the series form of the equations in [@http://dlmf.nist.gov/10.21#vi [sect]10.21(vi) McMahon's Asymptotic Expansions for Large Zeros] - using slightly different variable names). In summary, [expression j[sub [nu], m] [sim] [nu]x(-[zeta]) + f[sub 1](-[zeta]/[nu])] where ['-[zeta] = [nu][super -2/3]a[sub m]] and ['a[sub m]] is the absolute value of the ['m[super th]] root of ['Ai(x)] on the negative real axis. Here ['x = x(-[zeta])] is the inverse of the function [expression [frac23](-[zeta])[super 3/2] = [radic](x[sup2] - 1) - cos[supminus][sup1](1/x)] (7) Furthermore, [expression f[sub 1](-[zeta]) = [frac12]x(-[zeta]) {h(-[zeta])}[sup2] [sdot] b[sub 0](-[zeta])] where [expression h(-[zeta]) = {4(-[zeta]) / (x[sup2] - 1)}[super 4]] and [expression b[sub 0](-[zeta]) = -5/(48[zeta][sup2]) + 1/(-[zeta])[super [frac12]] [sdot] { 5/(24(x[super 2]-1)[super 3/2]) + 1/(8(x[super 2]-1)[super [frac12])]}] When solving for ['x(-[zeta])] in Eq. 7 above, the right-hand-side is expanded to order 2 in a Taylor series for large ['x]. This results in [expression [frac23](-[zeta])[super 3/2] [approx] x + 1/2x - [pi]/2] The positive root of the resulting quadratic equation is used to find an initial estimate ['x(-[zeta])]. This initial estimate is subsequently refined with several steps of Newton-Raphson iteration in Eq. 7. Estimates of the roots of cylindrical Bessel functions of negative order on the positive real axis are found using interlacing relations. For example, the ['m[super th]] root of the cylindrical Bessel function ['j[sub -[nu],m]] is bracketed by the ['m[super th]] root and the ['(m+1)[super th]] root of the Bessel function of corresponding positive integer order. In other words, [expression j[sub n[nu], m] < j[sub -[nu], m] < j[sub n[nu], m+1]] where ['m > 1] and ['n[sub [nu]]] represents the integral floor of the absolute value of ['|-[nu]|]. Similar bracketing relations are used to find estimates of the roots of Neumann functions of negative order, whereby a discontinuity at every negative half-integer order needs to be handled. Bracketing relations do not hold for the first root of cylindrical Bessel functions and cylindrical Neumann functions with negative order. Therefore, iterative algorithms combined with root-finding via bisection are used to localize ['j[sub -[nu],1]] and ['y[sub -[nu],1]]. [h3 Testing] The precision of evaluation of zeros was tested at 50 decimal digits using `cpp_dec_float_50` and found identical with spot values computed by __WolframAlpha. [endsect] [/section:bessel Finding Zeros of Bessel Functions of the First and Second Kinds] [/ Copyright 2006, 2013 John Maddock, Paul A. Bristow, Xiaogang Zhang and Christopher Kormanyos. 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). ]