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-rw-r--r--boost/math/special_functions/detail/bessel_i1.hpp612
1 files changed, 528 insertions, 84 deletions
diff --git a/boost/math/special_functions/detail/bessel_i1.hpp b/boost/math/special_functions/detail/bessel_i1.hpp
index b85bc67..0f9ba96 100644
--- a/boost/math/special_functions/detail/bessel_i1.hpp
+++ b/boost/math/special_functions/detail/bessel_i1.hpp
@@ -1,7 +1,9 @@
-// Copyright (c) 2006 Xiaogang Zhang
-// Use, modification and distribution are subject to 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)
+// Modified Bessel function of the first kind of order zero
+// we use the approximating forms derived in:
+// "Rational Approximations for the Modified Bessel Function of the First Kind I1(x) for Computations with Double Precision"
+// by Pavel Holoborodko,
+// see http://www.advanpix.com/2015/11/12/rational-approximations-for-the-modified-bessel-function-of-the-first-kind-i1-for-computations-with-double-precision/
+// The actual coefficients used are our own, and extend Pavel's work to precision's other than double.
#ifndef BOOST_MATH_BESSEL_I1_HPP
#define BOOST_MATH_BESSEL_I1_HPP
@@ -21,21 +23,36 @@
namespace boost { namespace math { namespace detail{
template <typename T>
-T bessel_i1(T x);
+T bessel_i1(const T& x);
-template <class T>
+template <class T, class tag>
struct bessel_i1_initializer
{
struct init
{
init()
{
- do_init();
+ do_init(tag());
}
- static void do_init()
+ static void do_init(const mpl::int_<64>&)
{
bessel_i1(T(1));
+ bessel_i1(T(15));
+ bessel_i1(T(80));
+ bessel_i1(T(101));
}
+ static void do_init(const mpl::int_<113>&)
+ {
+ bessel_i1(T(1));
+ bessel_i1(T(10));
+ bessel_i1(T(14));
+ bessel_i1(T(19));
+ bessel_i1(T(34));
+ bessel_i1(T(99));
+ bessel_i1(T(101));
+ }
+ template <class U>
+ static void do_init(const U&) {}
void force_instantiate()const{}
};
static const init initializer;
@@ -45,86 +62,513 @@ struct bessel_i1_initializer
}
};
-template <class T>
-const typename bessel_i1_initializer<T>::init bessel_i1_initializer<T>::initializer;
+template <class T, class tag>
+const typename bessel_i1_initializer<T, tag>::init bessel_i1_initializer<T, tag>::initializer;
+
+template <typename T, int N>
+T bessel_i1_imp(const T&, const mpl::int_<N>&)
+{
+ BOOST_ASSERT(0);
+ return 0;
+}
+
+template <typename T>
+T bessel_i1_imp(const T& x, const mpl::int_<24>&)
+{
+ BOOST_MATH_STD_USING
+ if(x < 7.75)
+ {
+ //Max error in interpolated form : 1.348e-08
+ // Max Error found at float precision = Poly : 1.469121e-07
+ static const float P[] = {
+ 8.333333221e-02f,
+ 6.944453712e-03f,
+ 3.472097211e-04f,
+ 1.158047174e-05f,
+ 2.739745142e-07f,
+ 5.135884609e-09f,
+ 5.262251502e-11f,
+ 1.331933703e-12f
+ };
+ T a = x * x / 4;
+ T Q[3] = { 1, 0.5f, boost::math::tools::evaluate_polynomial(P, a) };
+ return x * boost::math::tools::evaluate_polynomial(Q, a) / 2;
+ }
+ else
+ {
+ // Max error in interpolated form: 9.000e-08
+ // Max Error found at float precision = Poly: 1.044345e-07
+
+ static const float P[] = {
+ 3.98942115977513013e-01f,
+ -1.49581264836620262e-01f,
+ -4.76475741878486795e-02f,
+ -2.65157315524784407e-02f,
+ -1.47148600683672014e-01f
+ };
+ T ex = exp(x / 2);
+ T result = ex * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
+ result *= ex;
+ return result;
+ }
+}
+
+template <typename T>
+T bessel_i1_imp(const T& x, const mpl::int_<53>&)
+{
+ BOOST_MATH_STD_USING
+ if(x < 7.75)
+ {
+ // Bessel I0 over[10 ^ -16, 7.75]
+ // Max error in interpolated form: 5.639e-17
+ // Max Error found at double precision = Poly: 1.795559e-16
+
+ static const double P[] = {
+ 8.333333333333333803e-02,
+ 6.944444444444341983e-03,
+ 3.472222222225921045e-04,
+ 1.157407407354987232e-05,
+ 2.755731926254790268e-07,
+ 4.920949692800671435e-09,
+ 6.834657311305621830e-11,
+ 7.593969849687574339e-13,
+ 6.904822652741917551e-15,
+ 5.220157095351373194e-17,
+ 3.410720494727771276e-19,
+ 1.625212890947171108e-21,
+ 1.332898928162290861e-23
+ };
+ T a = x * x / 4;
+ T Q[3] = { 1, 0.5f, boost::math::tools::evaluate_polynomial(P, a) };
+ return x * boost::math::tools::evaluate_polynomial(Q, a) / 2;
+ }
+ else if(x < 500)
+ {
+ // Max error in interpolated form: 1.796e-16
+ // Max Error found at double precision = Poly: 2.898731e-16
+
+ static const double P[] = {
+ 3.989422804014406054e-01,
+ -1.496033551613111533e-01,
+ -4.675104253598537322e-02,
+ -4.090895951581637791e-02,
+ -5.719036414430205390e-02,
+ -1.528189554374492735e-01,
+ 3.458284470977172076e+00,
+ -2.426181371595021021e+02,
+ 1.178785865993440669e+04,
+ -4.404655582443487334e+05,
+ 1.277677779341446497e+07,
+ -2.903390398236656519e+08,
+ 5.192386898222206474e+09,
+ -7.313784438967834057e+10,
+ 8.087824484994859552e+11,
+ -6.967602516005787001e+12,
+ 4.614040809616582764e+13,
+ -2.298849639457172489e+14,
+ 8.325554073334618015e+14,
+ -2.067285045778906105e+15,
+ 3.146401654361325073e+15,
+ -2.213318202179221945e+15
+ };
+ return exp(x) * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
+ }
+ else
+ {
+ // Max error in interpolated form: 1.320e-19
+ // Max Error found at double precision = Poly: 7.065357e-17
+ static const double P[] = {
+ 3.989422804014314820e-01,
+ -1.496033551467584157e-01,
+ -4.675105322571775911e-02,
+ -4.090421597376992892e-02,
+ -5.843630344778927582e-02
+ };
+ T ex = exp(x / 2);
+ T result = ex * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
+ result *= ex;
+ return result;
+ }
+}
+
+template <typename T>
+T bessel_i1_imp(const T& x, const mpl::int_<64>&)
+{
+ BOOST_MATH_STD_USING
+ if(x < 7.75)
+ {
+ // Bessel I0 over[10 ^ -16, 7.75]
+ // Max error in interpolated form: 8.086e-21
+ // Max Error found at float80 precision = Poly: 7.225090e-20
+ static const T P[] = {
+ BOOST_MATH_BIG_CONSTANT(T, 64, 8.33333333333333333340071817e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 6.94444444444444442462728070e-03),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 3.47222222222222318886683883e-04),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.15740740740738880709555060e-05),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.75573192240046222242685145e-07),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 4.92094986131253986838697503e-09),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 6.83465258979924922633502182e-11),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 7.59405830675154933645967137e-13),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 6.90369179710633344508897178e-15),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 5.23003610041709452814262671e-17),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 3.35291901027762552549170038e-19),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.83991379419781823063672109e-21),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 8.87732714140192556332037815e-24),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 3.32120654663773147206454247e-26),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.95294659305369207813486871e-28)
+ };
+ T a = x * x / 4;
+ T Q[3] = { 1, 0.5f, boost::math::tools::evaluate_polynomial(P, a) };
+ return x * boost::math::tools::evaluate_polynomial(Q, a) / 2;
+ }
+ else if(x < 20)
+ {
+ // Max error in interpolated form: 4.258e-20
+ // Max Error found at float80 precision = Poly: 2.851105e-19
+ // Maximum Deviation Found : 3.887e-20
+ // Expected Error Term : 3.887e-20
+ // Maximum Relative Change in Control Points : 1.681e-04
+ static const T P[] = {
+ BOOST_MATH_BIG_CONSTANT(T, 64, 3.98942260530218897338680e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -1.49599542849073670179540e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -4.70492865454119188276875e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -3.12389893307392002405869e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.49696126385202602071197e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -3.84206507612717711565967e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.14748094784412558689584e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -7.70652726663596993005669e+04),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.01659736164815617174439e+06),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -4.04740659606466305607544e+07),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 6.38383394696382837263656e+08),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -8.00779638649147623107378e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 8.02338237858684714480491e+10),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -6.41198553664947312995879e+11),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 4.05915186909564986897554e+12),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -2.00907636964168581116181e+13),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 7.60855263982359981275199e+13),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -2.12901817219239205393806e+14),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 4.14861794397709807823575e+14),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -5.02808138522587680348583e+14),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.85505477056514919387171e+14)
+ };
+ return exp(x) * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
+ }
+ else if(x < 100)
+ {
+ // Bessel I0 over [15, 50]
+ // Maximum Deviation Found: 2.444e-20
+ // Expected Error Term : 2.438e-20
+ // Maximum Relative Change in Control Points : 2.101e-03
+ // Max Error found at float80 precision = Poly : 6.029974e-20
+
+ static const T P[] = {
+ BOOST_MATH_BIG_CONSTANT(T, 64, 3.98942280401431675205845e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -1.49603355149968887210170e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -4.67510486284376330257260e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -4.09071458907089270559464e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -5.75278280327696940044714e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -1.10591299500956620739254e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -2.77061766699949309115618e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -5.42683771801837596371638e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -9.17021412070404158464316e+00),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.04154379346763380543310e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -1.43462345357478348323006e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 9.98109660274422449523837e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -3.74438822767781410362757e+04)
+ };
+ return exp(x) * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
+ }
+ else
+ {
+ // Bessel I0 over[100, INF]
+ // Max error in interpolated form: 2.456e-20
+ // Max Error found at float80 precision = Poly: 5.446356e-20
+ static const T P[] = {
+ BOOST_MATH_BIG_CONSTANT(T, 64, 3.98942280401432677958445e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -1.49603355150537411254359e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -4.67510484842456251368526e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -4.09071676503922479645155e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -5.75256179814881566010606e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -1.10754910257965227825040e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -2.67858639515616079840294e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -9.17266479586791298924367e-01)
+ };
+ T ex = exp(x / 2);
+ T result = ex * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
+ result *= ex;
+ return result;
+ }
+}
+
+template <typename T>
+T bessel_i1_imp(const T& x, const mpl::int_<113>&)
+{
+ BOOST_MATH_STD_USING
+ if(x < 7.75)
+ {
+ // Bessel I0 over[10 ^ -34, 7.75]
+ // Max error in interpolated form: 1.835e-35
+ // Max Error found at float128 precision = Poly: 1.645036e-34
+
+ static const T P[] = {
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.3333333333333333333333333333333331804098e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.9444444444444444444444444444445418303082e-03),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.4722222222222222222222222222119082346591e-04),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.1574074074074074074074074078415867655987e-05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.7557319223985890652557318255143448192453e-07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 4.9209498614260519022423916850415000626427e-09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.8346525853139609753354247043900442393686e-11),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.5940584281266233060080535940234144302217e-13),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.9036894801151120925605467963949641957095e-15),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.2300677879659941472662086395055636394839e-17),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.3526075563884539394691458717439115962233e-19),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.8420920639497841692288943167036233338434e-21),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.7718669711748690065381181691546032291365e-24),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.6549445715236427401845636880769861424730e-26),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.3437296196812697924703896979250126739676e-28),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 4.3912734588619073883015937023564978854893e-31),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.2839967682792395867255384448052781306897e-33),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.3790094235693528861015312806394354114982e-36),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.0423861671932104308662362292359563970482e-39),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.7493858979396446292135661268130281652945e-41),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.2786079392547776769387921361408303035537e-44),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.2335693685833531118863552173880047183822e-47)
+ };
+ T a = x * x / 4;
+ T Q[3] = { 1, 0.5f, boost::math::tools::evaluate_polynomial(P, a) };
+ return x * boost::math::tools::evaluate_polynomial(Q, a) / 2;
+ }
+ else if(x < 11)
+ {
+ // Max error in interpolated form: 8.574e-36
+ // Maximum Deviation Found : 4.689e-36
+ // Expected Error Term : 3.760e-36
+ // Maximum Relative Change in Control Points : 5.204e-03
+ // Max Error found at float128 precision = Poly : 2.882561e-34
+
+ static const T P[] = {
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.333333333333333326889717360850080939e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.944444444444444511272790848815114507e-03),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.472222222222221892451965054394153443e-04),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.157407407407408437378868534321538798e-05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.755731922398566216824909767320161880e-07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 4.920949861426434829568192525456800388e-09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.834652585308926245465686943255486934e-11),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.594058428179852047689599244015979196e-13),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.903689479655006062822949671528763738e-15),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.230067791254403974475987777406992984e-17),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.352607536815161679702105115200693346e-19),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.842092161364672561828681848278567885e-21),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.771862912600611801856514076709932773e-24),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.654958704184380914803366733193713605e-26),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.343688672071130980471207297730607625e-28),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 4.392252844664709532905868749753463950e-31),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.282086786672692641959912811902298600e-33),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.408812012322547015191398229942864809e-36),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.681220437734066258673404589233009892e-39),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.072417451640733785626701738789290055e-41),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.352218520142636864158849446833681038e-44),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.407918492276267527897751358794783640e-46)
+ };
+ T a = x * x / 4;
+ T Q[3] = { 1, 0.5f, boost::math::tools::evaluate_polynomial(P, a) };
+ return x * boost::math::tools::evaluate_polynomial(Q, a) / 2;
+ }
+ else if(x < 15)
+ {
+ //Max error in interpolated form: 7.599e-36
+ // Maximum Deviation Found : 1.766e-35
+ // Expected Error Term : 1.021e-35
+ // Maximum Relative Change in Control Points : 6.228e-03
+ static const T P[] = {
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.333333333333255774414858563409941233e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.944444444444897867884955912228700291e-03),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.472222222220954970397343617150959467e-04),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.157407407409660682751155024932538578e-05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.755731922369973706427272809014190998e-07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 4.920949861702265600960449699129258153e-09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.834652583208361401197752793379677147e-11),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.594058441128280500819776168239988143e-13),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.903689413939268702265479276217647209e-15),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.230068069012898202890718644753625569e-17),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.352606552027491657204243201021677257e-19),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.842095100698532984651921750204843362e-21),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.771789051329870174925649852681844169e-24),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.655114381199979536997025497438385062e-26),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.343415732516712339472538688374589373e-28),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 4.396177019032432392793591204647901390e-31),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.277563309255167951005939802771456315e-33),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.449201419305514579791370198046544736e-36),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.415430703400740634202379012388035255e-39),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.195458831864936225409005027914934499e-41),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.829726762743879793396637797534668039e-45),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.698302711685624490806751012380215488e-46),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.062520475425422618494185821587228317e-49),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.732372906742845717148185173723304360e-52)
+ };
+ T a = x * x / 4;
+ T Q[3] = { 1, 0.5f, boost::math::tools::evaluate_polynomial(P, a) };
+ return x * boost::math::tools::evaluate_polynomial(Q, a) / 2;
+ }
+ else if(x < 20)
+ {
+ // Max error in interpolated form: 8.864e-36
+ // Max Error found at float128 precision = Poly: 8.522841e-35
+ static const T P[] = {
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.989422793693152031514179994954750043e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.496029423752889591425633234009799670e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.682975926820553021482820043377990241e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -3.138871171577224532369979905856458929e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -8.765350219426341341990447005798111212e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.321389275507714530941178258122955540e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.727748393898888756515271847678850411e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.123040820686242586086564998713862335e+05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -3.784112378374753535335272752884808068e+06),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.054920416060932189433079126269416563e+08),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.450129415468060676827180524327749553e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 4.758831882046487398739784498047935515e+10),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -7.736936520262204842199620784338052937e+11),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.051128683324042629513978256179115439e+13),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.188008285959794869092624343537262342e+14),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.108530004906954627420484180793165669e+15),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -8.441516828490144766650287123765318484e+15),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.158251664797753450664499268756393535e+16),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.467314522709016832128790443932896401e+17),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.896222045367960462945885220710294075e+17),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.273382139594876997203657902425653079e+18),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.669871448568623680543943144842394531e+18),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.813923031370708069940575240509912588e+18)
+ };
+ return exp(x) * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
+ }
+ else if(x < 35)
+ {
+ // Max error in interpolated form: 6.028e-35
+ // Max Error found at float128 precision = Poly: 1.368313e-34
+
+ static const T P[] = {
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.989422804012941975429616956496046931e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.496033550576049830976679315420681402e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.675107835141866009896710750800622147e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.090104965125365961928716504473692957e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -5.842241652296980863361375208605487570e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.063604828033747303936724279018650633e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -9.113375972811586130949401996332817152e+00),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.334748570425075872639817839399823709e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -3.759150758768733692594821032784124765e+04),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.863672813448915255286274382558526321e+06),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -7.798248643371718775489178767529282534e+07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.769963173932801026451013022000669267e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -8.381780137198278741566746511015220011e+10),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.163891337116820832871382141011952931e+12),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.764325864671438675151635117936912390e+13),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.925668307403332887856809510525154955e+14),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.416692606589060039334938090985713641e+16),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.892398600219306424294729851605944429e+17),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.107232903741874160308537145391245060e+18),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.930223393531877588898224144054112045e+19),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.427759576167665663373350433236061007e+20),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.306019279465532835530812122374386654e+20),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -3.653753000392125229440044977239174472e+21),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.140760686989511568435076842569804906e+22),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.249149337812510200795436107962504749e+22),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.101619088427348382058085685849420866e+22)
+ };
+ return exp(x) * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
+ }
+ else if(x < 100)
+ {
+ // Max error in interpolated form: 5.494e-35
+ // Max Error found at float128 precision = Poly: 1.214651e-34
+
+ static const T P[] = {
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.989422804014326779399307367861631577e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.496033551505372542086590873271571919e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.675104848454290286276466276677172664e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.090716742397105403027549796269213215e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -5.752570419098513588311026680089351230e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.107369803696534592906420980901195808e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.699214194000085622941721628134575121e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -7.953006169077813678478720427604462133e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.746618809476524091493444128605380593e+00),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.084446249943196826652788161656973391e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -5.020325182518980633783194648285500554e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.510195971266257573425196228564489134e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -5.241661863814900938075696173192225056e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.323374362891993686413568398575539777e+05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.112838452096066633754042734723911040e+06),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 9.369270194978310081563767560113534023e+07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.704295412488936504389347368131134993e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.320829576277038198439987439508754886e+10),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.258818139077875493434420764260185306e+11),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.396791306321498426110315039064592443e+12),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.217617301585849875301440316301068439e+12)
+ };
+ return exp(x) * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
+ }
+ else
+ {
+ // Bessel I0 over[100, INF]
+ // Max error in interpolated form: 6.081e-35
+ // Max Error found at float128 precision = Poly: 1.407151e-34
+ static const T P[] = {
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.9894228040143267793994605993438200208417e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.4960335515053725422747977247811372936584e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.6751048484542891946087411826356811991039e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.0907167423975030452875828826630006305665e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -5.7525704189964886494791082898669060345483e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.1073698056568248642163476807108190176386e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.6992139012879749064623499618582631684228e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -7.9530409594026597988098934027440110587905e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.7462844478733532517044536719240098183686e+00),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.0870711340681926669381449306654104739256e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.8510175413216969245241059608553222505228e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -2.4094682286011573747064907919522894740063e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.3128845936764406865199641778959502795443e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -8.1655901321962541203257516341266838487359e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -3.8019591025686295090160445920753823994556e+04),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -6.7008089049178178697338128837158732831105e+05)
+ };
+ T ex = exp(x / 2);
+ T result = ex * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
+ result *= ex;
+ return result;
+ }
+}
+
+template <typename T>
+T bessel_i1_imp(const T& x, const mpl::int_<0>&)
+{
+ if(boost::math::tools::digits<T>() <= 24)
+ return bessel_i1_imp(x, mpl::int_<24>());
+ else if(boost::math::tools::digits<T>() <= 53)
+ return bessel_i1_imp(x, mpl::int_<53>());
+ else if(boost::math::tools::digits<T>() <= 64)
+ return bessel_i1_imp(x, mpl::int_<64>());
+ else if(boost::math::tools::digits<T>() <= 113)
+ return bessel_i1_imp(x, mpl::int_<113>());
+ BOOST_ASSERT(0);
+ return 0;
+}
template <typename T>
-T bessel_i1(T x)
+inline T bessel_i1(const T& x)
{
+ typedef mpl::int_<
+ std::numeric_limits<T>::digits == 0 ?
+ 0 :
+ std::numeric_limits<T>::digits <= 24 ?
+ 24 :
+ std::numeric_limits<T>::digits <= 53 ?
+ 53 :
+ std::numeric_limits<T>::digits <= 64 ?
+ 64 :
+ std::numeric_limits<T>::digits <= 113 ?
+ 113 : -1
+ > tag_type;
- bessel_i1_initializer<T>::force_instantiate();
-
- static const T P1[] = {
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4577180278143463643e+15)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7732037840791591320e+14)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9876779648010090070e+12)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.3357437682275493024e+11)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4828267606612366099e+09)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0588550724769347106e+07)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.1894091982308017540e+04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8225946631657315931e+02)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.7207090827310162436e-01)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.1746443287817501309e-04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.3466829827635152875e-06)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4831904935994647675e-09)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1928788903603238754e-12)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.5245515583151902910e-16)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9705291802535139930e-19)),
- };
- static const T Q1[] = {
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.9154360556286927285e+15)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.7887501377547640438e+12)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4386907088588283434e+10)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1594225856856884006e+07)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.1326864679904189920e+03)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
- };
- static const T P2[] = {
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4582087408985668208e-05)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9359825138577646443e-04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9204895411257790122e-02)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.4198728018058047439e-01)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960118277609544334e+00)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9746376087200685843e+00)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.5591872901933459000e-01)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.0437159056137599999e-02)),
- };
- static const T Q2[] = {
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7510433111922824643e-05)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2835624489492512649e-03)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.4212010813186530069e-02)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.5017476463217924408e-01)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.2593714889036996297e+00)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.8806586721556593450e+00)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
- };
- T value, factor, r, w;
-
- BOOST_MATH_STD_USING
- using namespace boost::math::tools;
-
- BOOST_ASSERT(x >= 0); // negative x is handled before we get here
- w = abs(x);
- if (x == 0)
- {
- return static_cast<T>(0);
- }
- if (w <= 15) // w in (0, 15]
- {
- T y = x * x;
- r = evaluate_polynomial(P1, y) / evaluate_polynomial(Q1, y);
- factor = w;
- value = factor * r;
- }
- else // w in (15, \infty)
- {
- T y = 1 / w - T(1) / 15;
- r = evaluate_polynomial(P2, y) / evaluate_polynomial(Q2, y);
- factor = exp(w) / sqrt(w);
- value = factor * r;
- }
-
- return value;
+ bessel_i1_initializer<T, tag_type>::force_instantiate();
+ return bessel_i1_imp(x, tag_type());
}
}}} // namespaces