boost/math/special_functions/owens_t.hpp
// Copyright Benjamin Sobotta 2012
// 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)
#ifndef BOOST_OWENS_T_HPP
#define BOOST_OWENS_T_HPP
// Reference:
// Mike Patefield, David Tandy
// FAST AND ACCURATE CALCULATION OF OWEN'S T-FUNCTION
// Journal of Statistical Software, 5 (5), 1-25
#ifdef _MSC_VER
# pragma once
#endif
#include <boost/math/special_functions/math_fwd.hpp>
#include <boost/math/special_functions/erf.hpp>
#include <boost/math/special_functions/expm1.hpp>
#include <boost/math/tools/throw_exception.hpp>
#include <boost/math/tools/assert.hpp>
#include <boost/math/constants/constants.hpp>
#include <boost/math/tools/big_constant.hpp>
#include <stdexcept>
#include <cmath>
#ifdef _MSC_VER
#pragma warning(push)
#pragma warning(disable:4127)
#endif
#if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
//
// This is the only way we can avoid
// warning: non-standard suffix on floating constant [-Wpedantic]
// when building with -Wall -pedantic. Neither __extension__
// nor #pragma diagnostic ignored work :(
//
#pragma GCC system_header
#endif
namespace boost
{
namespace math
{
namespace detail
{
// owens_t_znorm1(x) = P(-oo<Z<=x)-0.5 with Z being normally distributed.
template<typename RealType, class Policy>
inline RealType owens_t_znorm1(const RealType x, const Policy& pol)
{
using namespace boost::math::constants;
return boost::math::erf(x*one_div_root_two<RealType>(), pol)*half<RealType>();
} // RealType owens_t_znorm1(const RealType x)
// owens_t_znorm2(x) = P(x<=Z<oo) with Z being normally distributed.
template<typename RealType, class Policy>
inline RealType owens_t_znorm2(const RealType x, const Policy& pol)
{
using namespace boost::math::constants;
return boost::math::erfc(x*one_div_root_two<RealType>(), pol)*half<RealType>();
} // RealType owens_t_znorm2(const RealType x)
// Auxiliary function, it computes an array key that is used to determine
// the specific computation method for Owen's T and the order thereof
// used in owens_t_dispatch.
template<typename RealType>
inline unsigned short owens_t_compute_code(const RealType h, const RealType a)
{
static const RealType hrange[] =
{ 0.02f, 0.06f, 0.09f, 0.125f, 0.26f, 0.4f, 0.6f, 1.6f, 1.7f, 2.33f, 2.4f, 3.36f, 3.4f, 4.8f };
static const RealType arange[] = { 0.025f, 0.09f, 0.15f, 0.36f, 0.5f, 0.9f, 0.99999f };
/*
original select array from paper:
1, 1, 2,13,13,13,13,13,13,13,13,16,16,16, 9
1, 2, 2, 3, 3, 5, 5,14,14,15,15,16,16,16, 9
2, 2, 3, 3, 3, 5, 5,15,15,15,15,16,16,16,10
2, 2, 3, 5, 5, 5, 5, 7, 7,16,16,16,16,16,10
2, 3, 3, 5, 5, 6, 6, 8, 8,17,17,17,12,12,11
2, 3, 5, 5, 5, 6, 6, 8, 8,17,17,17,12,12,12
2, 3, 4, 4, 6, 6, 8, 8,17,17,17,17,17,12,12
2, 3, 4, 4, 6, 6,18,18,18,18,17,17,17,12,12
*/
// subtract one because the array is written in FORTRAN in mind - in C arrays start @ zero
static const unsigned short select[] =
{
0, 0 , 1 , 12 ,12 , 12 , 12 , 12 , 12 , 12 , 12 , 15 , 15 , 15 , 8,
0 , 1 , 1 , 2 , 2 , 4 , 4 , 13 , 13 , 14 , 14 , 15 , 15 , 15 , 8,
1 , 1 , 2 , 2 , 2 , 4 , 4 , 14 , 14 , 14 , 14 , 15 , 15 , 15 , 9,
1 , 1 , 2 , 4 , 4 , 4 , 4 , 6 , 6 , 15 , 15 , 15 , 15 , 15 , 9,
1 , 2 , 2 , 4 , 4 , 5 , 5 , 7 , 7 , 16 ,16 , 16 , 11 , 11 , 10,
1 , 2 , 4 , 4 , 4 , 5 , 5 , 7 , 7 , 16 , 16 , 16 , 11 , 11 , 11,
1 , 2 , 3 , 3 , 5 , 5 , 7 , 7 , 16 , 16 , 16 , 16 , 16 , 11 , 11,
1 , 2 , 3 , 3 , 5 , 5 , 17 , 17 , 17 , 17 , 16 , 16 , 16 , 11 , 11
};
unsigned short ihint = 14, iaint = 7;
for(unsigned short i = 0; i != 14; i++)
{
if( h <= hrange[i] )
{
ihint = i;
break;
}
} // for(unsigned short i = 0; i != 14; i++)
for(unsigned short i = 0; i != 7; i++)
{
if( a <= arange[i] )
{
iaint = i;
break;
}
} // for(unsigned short i = 0; i != 7; i++)
// interpret select array as 8x15 matrix
return select[iaint*15 + ihint];
} // unsigned short owens_t_compute_code(const RealType h, const RealType a)
template<typename RealType>
inline unsigned short owens_t_get_order_imp(const unsigned short icode, RealType, const std::integral_constant<int, 53>&)
{
static const unsigned short ord[] = {2, 3, 4, 5, 7, 10, 12, 18, 10, 20, 30, 0, 4, 7, 8, 20, 0, 0}; // 18 entries
BOOST_MATH_ASSERT(icode<18);
return ord[icode];
} // unsigned short owens_t_get_order(const unsigned short icode, RealType, std::integral_constant<int, 53> const&)
template<typename RealType>
inline unsigned short owens_t_get_order_imp(const unsigned short icode, RealType, const std::integral_constant<int, 64>&)
{
// method ================>>> {1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6}
static const unsigned short ord[] = {3, 4, 5, 6, 8, 11, 13, 19, 10, 20, 30, 0, 7, 10, 11, 23, 0, 0}; // 18 entries
BOOST_MATH_ASSERT(icode<18);
return ord[icode];
} // unsigned short owens_t_get_order(const unsigned short icode, RealType, std::integral_constant<int, 64> const&)
template<typename RealType, typename Policy>
inline unsigned short owens_t_get_order(const unsigned short icode, RealType r, const Policy&)
{
typedef typename policies::precision<RealType, Policy>::type precision_type;
typedef std::integral_constant<int,
precision_type::value <= 0 ? 64 :
precision_type::value <= 53 ? 53 : 64
> tag_type;
return owens_t_get_order_imp(icode, r, tag_type());
}
// compute the value of Owen's T function with method T1 from the reference paper
template<typename RealType, typename Policy>
inline RealType owens_t_T1(const RealType h, const RealType a, const unsigned short m, const Policy& pol)
{
BOOST_MATH_STD_USING
using namespace boost::math::constants;
const RealType hs = -h*h*half<RealType>();
const RealType dhs = exp( hs );
const RealType as = a*a;
unsigned short j=1;
RealType jj = 1;
RealType aj = a * one_div_two_pi<RealType>();
RealType dj = boost::math::expm1( hs, pol);
RealType gj = hs*dhs;
RealType val = atan( a ) * one_div_two_pi<RealType>();
while( true )
{
val += dj*aj/jj;
if( m <= j )
break;
j++;
jj += static_cast<RealType>(2);
aj *= as;
dj = gj - dj;
gj *= hs / static_cast<RealType>(j);
} // while( true )
return val;
} // RealType owens_t_T1(const RealType h, const RealType a, const unsigned short m)
// compute the value of Owen's T function with method T2 from the reference paper
template<typename RealType, class Policy>
inline RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah, const Policy& pol, const std::false_type&)
{
BOOST_MATH_STD_USING
using namespace boost::math::constants;
const unsigned short maxii = m+m+1;
const RealType hs = h*h;
const RealType as = -a*a;
const RealType y = static_cast<RealType>(1) / hs;
unsigned short ii = 1;
RealType val = 0;
RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
RealType z = owens_t_znorm1(ah, pol)/h;
while( true )
{
val += z;
if( maxii <= ii )
{
val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
break;
} // if( maxii <= ii )
z = y * ( vi - static_cast<RealType>(ii) * z );
vi *= as;
ii += 2;
} // while( true )
return val;
} // RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah)
// compute the value of Owen's T function with method T3 from the reference paper
template<typename RealType, class Policy>
inline RealType owens_t_T3_imp(const RealType h, const RealType a, const RealType ah, const std::integral_constant<int, 53>&, const Policy& pol)
{
BOOST_MATH_STD_USING
using namespace boost::math::constants;
const unsigned short m = 20;
static const RealType c2[] =
{
static_cast<RealType>(0.99999999999999987510),
static_cast<RealType>(-0.99999999999988796462), static_cast<RealType>(0.99999999998290743652),
static_cast<RealType>(-0.99999999896282500134), static_cast<RealType>(0.99999996660459362918),
static_cast<RealType>(-0.99999933986272476760), static_cast<RealType>(0.99999125611136965852),
static_cast<RealType>(-0.99991777624463387686), static_cast<RealType>(0.99942835555870132569),
static_cast<RealType>(-0.99697311720723000295), static_cast<RealType>(0.98751448037275303682),
static_cast<RealType>(-0.95915857980572882813), static_cast<RealType>(0.89246305511006708555),
static_cast<RealType>(-0.76893425990463999675), static_cast<RealType>(0.58893528468484693250),
static_cast<RealType>(-0.38380345160440256652), static_cast<RealType>(0.20317601701045299653),
static_cast<RealType>(-0.82813631607004984866E-01), static_cast<RealType>(0.24167984735759576523E-01),
static_cast<RealType>(-0.44676566663971825242E-02), static_cast<RealType>(0.39141169402373836468E-03)
};
const RealType as = a*a;
const RealType hs = h*h;
const RealType y = static_cast<RealType>(1)/hs;
RealType ii = 1;
unsigned short i = 0;
RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
RealType zi = owens_t_znorm1(ah, pol)/h;
RealType val = 0;
while( true )
{
BOOST_MATH_ASSERT(i < 21);
val += zi*c2[i];
if( m <= i ) // if( m < i+1 )
{
val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
break;
} // if( m < i )
zi = y * (ii*zi - vi);
vi *= as;
ii += 2;
i++;
} // while( true )
return val;
} // RealType owens_t_T3(const RealType h, const RealType a, const RealType ah)
// compute the value of Owen's T function with method T3 from the reference paper
template<class RealType, class Policy>
inline RealType owens_t_T3_imp(const RealType h, const RealType a, const RealType ah, const std::integral_constant<int, 64>&, const Policy& pol)
{
BOOST_MATH_STD_USING
using namespace boost::math::constants;
const unsigned short m = 30;
static const RealType c2[] =
{
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999999999999999729978162447266851932041876728736094298092917625009873),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.99999999999999999999467056379678391810626533251885323416799874878563998732905968),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999999999824849349313270659391127814689133077036298754586814091034842536),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999999999997703859616213643405880166422891953033591551179153879839440241685),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999998394883415238173334565554173013941245103172035286759201504179038147),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999999993063616095509371081203145247992197457263066869044528823599399470977),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999999999797336340409464429599229870590160411238245275855903767652432017766116267),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.999999999574958412069046680119051639753412378037565521359444170241346845522403274),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999999933226234193375324943920160947158239076786103108097456617750134812033362048),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999188923242461073033481053037468263536806742737922476636768006622772762168467),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999992195143483674402853783549420883055129680082932629160081128947764415749728967),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.999993935137206712830997921913316971472227199741857386575097250553105958772041501),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99996135597690552745362392866517133091672395614263398912807169603795088421057688716),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.99979556366513946026406788969630293820987757758641211293079784585126692672425362469),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.999092789629617100153486251423850590051366661947344315423226082520411961968929483),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.996593837411918202119308620432614600338157335862888580671450938858935084316004769854),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.98910017138386127038463510314625339359073956513420458166238478926511821146316469589567),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.970078558040693314521331982203762771512160168582494513347846407314584943870399016019),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.92911438683263187495758525500033707204091967947532160289872782771388170647150321633673),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.8542058695956156057286980736842905011429254735181323743367879525470479126968822863),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.73796526033030091233118357742803709382964420335559408722681794195743240930748630755),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.58523469882837394570128599003785154144164680587615878645171632791404210655891158),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.415997776145676306165661663581868460503874205343014196580122174949645271353372263),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.2588210875241943574388730510317252236407805082485246378222935376279663808416534365),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.1375535825163892648504646951500265585055789019410617565727090346559210218472356689),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.0607952766325955730493900985022020434830339794955745989150270485056436844239206648),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.0216337683299871528059836483840390514275488679530797294557060229266785853764115),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.00593405693455186729876995814181203900550014220428843483927218267309209471516256),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.0011743414818332946510474576182739210553333860106811865963485870668929503649964142),
BOOST_MATH_BIG_CONSTANT(RealType, 260, -1.489155613350368934073453260689881330166342484405529981510694514036264969925132e-4),
BOOST_MATH_BIG_CONSTANT(RealType, 260, 9.072354320794357587710929507988814669454281514268844884841547607134260303118208e-6)
};
const RealType as = a*a;
const RealType hs = h*h;
const RealType y = 1 / hs;
RealType ii = 1;
unsigned short i = 0;
RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
RealType zi = owens_t_znorm1(ah, pol)/h;
RealType val = 0;
while( true )
{
BOOST_MATH_ASSERT(i < 31);
val += zi*c2[i];
if( m <= i ) // if( m < i+1 )
{
val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
break;
} // if( m < i )
zi = y * (ii*zi - vi);
vi *= as;
ii += 2;
i++;
} // while( true )
return val;
} // RealType owens_t_T3(const RealType h, const RealType a, const RealType ah)
template<class RealType, class Policy>
inline RealType owens_t_T3(const RealType h, const RealType a, const RealType ah, const Policy& pol)
{
typedef typename policies::precision<RealType, Policy>::type precision_type;
typedef std::integral_constant<int,
precision_type::value <= 0 ? 64 :
precision_type::value <= 53 ? 53 : 64
> tag_type;
return owens_t_T3_imp(h, a, ah, tag_type(), pol);
}
// compute the value of Owen's T function with method T4 from the reference paper
template<typename RealType>
inline RealType owens_t_T4(const RealType h, const RealType a, const unsigned short m)
{
BOOST_MATH_STD_USING
using namespace boost::math::constants;
const unsigned short maxii = m+m+1;
const RealType hs = h*h;
const RealType as = -a*a;
unsigned short ii = 1;
RealType ai = a * exp( -hs*(static_cast<RealType>(1)-as)*half<RealType>() ) * one_div_two_pi<RealType>();
RealType yi = 1;
RealType val = 0;
while( true )
{
val += ai*yi;
if( maxii <= ii )
break;
ii += 2;
yi = (static_cast<RealType>(1)-hs*yi) / static_cast<RealType>(ii);
ai *= as;
} // while( true )
return val;
} // RealType owens_t_T4(const RealType h, const RealType a, const unsigned short m)
// compute the value of Owen's T function with method T5 from the reference paper
template<typename RealType>
inline RealType owens_t_T5_imp(const RealType h, const RealType a, const std::integral_constant<int, 53>&)
{
BOOST_MATH_STD_USING
/*
NOTICE:
- The pts[] array contains the squares (!) of the abscissas, i.e. the roots of the Legendre
polynomial P_n(x), instead of the plain roots as required in Gauss-Legendre
quadrature, because T5(h,a,m) contains only x^2 terms.
- The wts[] array contains the weights for Gauss-Legendre quadrature scaled with a factor
of 1/(2*pi) according to T5(h,a,m).
*/
const unsigned short m = 13;
static const RealType pts[] = {
static_cast<RealType>(0.35082039676451715489E-02),
static_cast<RealType>(0.31279042338030753740E-01), static_cast<RealType>(0.85266826283219451090E-01),
static_cast<RealType>(0.16245071730812277011), static_cast<RealType>(0.25851196049125434828),
static_cast<RealType>(0.36807553840697533536), static_cast<RealType>(0.48501092905604697475),
static_cast<RealType>(0.60277514152618576821), static_cast<RealType>(0.71477884217753226516),
static_cast<RealType>(0.81475510988760098605), static_cast<RealType>(0.89711029755948965867),
static_cast<RealType>(0.95723808085944261843), static_cast<RealType>(0.99178832974629703586) };
static const RealType wts[] = {
static_cast<RealType>(0.18831438115323502887E-01),
static_cast<RealType>(0.18567086243977649478E-01), static_cast<RealType>(0.18042093461223385584E-01),
static_cast<RealType>(0.17263829606398753364E-01), static_cast<RealType>(0.16243219975989856730E-01),
static_cast<RealType>(0.14994592034116704829E-01), static_cast<RealType>(0.13535474469662088392E-01),
static_cast<RealType>(0.11886351605820165233E-01), static_cast<RealType>(0.10070377242777431897E-01),
static_cast<RealType>(0.81130545742299586629E-02), static_cast<RealType>(0.60419009528470238773E-02),
static_cast<RealType>(0.38862217010742057883E-02), static_cast<RealType>(0.16793031084546090448E-02) };
const RealType as = a*a;
const RealType hs = -h*h*boost::math::constants::half<RealType>();
RealType val = 0;
for(unsigned short i = 0; i < m; ++i)
{
BOOST_MATH_ASSERT(i < 13);
const RealType r = static_cast<RealType>(1) + as*pts[i];
val += wts[i] * exp( hs*r ) / r;
} // for(unsigned short i = 0; i < m; ++i)
return val*a;
} // RealType owens_t_T5(const RealType h, const RealType a)
// compute the value of Owen's T function with method T5 from the reference paper
template<typename RealType>
inline RealType owens_t_T5_imp(const RealType h, const RealType a, const std::integral_constant<int, 64>&)
{
BOOST_MATH_STD_USING
/*
NOTICE:
- The pts[] array contains the squares (!) of the abscissas, i.e. the roots of the Legendre
polynomial P_n(x), instead of the plain roots as required in Gauss-Legendre
quadrature, because T5(h,a,m) contains only x^2 terms.
- The wts[] array contains the weights for Gauss-Legendre quadrature scaled with a factor
of 1/(2*pi) according to T5(h,a,m).
*/
const unsigned short m = 19;
static const RealType pts[] = {
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0016634282895983227941),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.014904509242697054183),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.04103478879005817919),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.079359853513391511008),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.1288612130237615133),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.18822336642448518856),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.25586876186122962384),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.32999972011807857222),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.40864620815774761438),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.48971819306044782365),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.57106118513245543894),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.6505134942981533829),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.72596367859928091618),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.79540665919549865924),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.85699701386308739244),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.90909804422384697594),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.95032536436570154409),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.97958418733152273717),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.99610366384229088321)
};
static const RealType wts[] = {
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012975111395684900835),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012888764187499150078),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012716644398857307844),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012459897461364705691),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012120231988292330388),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.011699908404856841158),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.011201723906897224448),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.010628993848522759853),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0099855296835573320047),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0092756136096132857933),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0085039700881139589055),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0076757344408814561254),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0067964187616556459109),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.005871875456524750363),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0049082589542498110071),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0039119870792519721409),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0028897090921170700834),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0018483371329504443947),
BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.00079623320100438873578)
};
const RealType as = a*a;
const RealType hs = -h*h*boost::math::constants::half<RealType>();
RealType val = 0;
for(unsigned short i = 0; i < m; ++i)
{
BOOST_MATH_ASSERT(i < 19);
const RealType r = 1 + as*pts[i];
val += wts[i] * exp( hs*r ) / r;
} // for(unsigned short i = 0; i < m; ++i)
return val*a;
} // RealType owens_t_T5(const RealType h, const RealType a)
template<class RealType, class Policy>
inline RealType owens_t_T5(const RealType h, const RealType a, const Policy&)
{
typedef typename policies::precision<RealType, Policy>::type precision_type;
typedef std::integral_constant<int,
precision_type::value <= 0 ? 64 :
precision_type::value <= 53 ? 53 : 64
> tag_type;
return owens_t_T5_imp(h, a, tag_type());
}
// compute the value of Owen's T function with method T6 from the reference paper
template<typename RealType, class Policy>
inline RealType owens_t_T6(const RealType h, const RealType a, const Policy& pol)
{
BOOST_MATH_STD_USING
using namespace boost::math::constants;
const RealType normh = owens_t_znorm2(h, pol);
const RealType y = static_cast<RealType>(1) - a;
const RealType r = atan2(y, static_cast<RealType>(1 + a) );
RealType val = normh * ( static_cast<RealType>(1) - normh ) * half<RealType>();
if( r != 0 )
val -= r * exp( -y*h*h*half<RealType>()/r ) * one_div_two_pi<RealType>();
return val;
} // RealType owens_t_T6(const RealType h, const RealType a, const unsigned short m)
template <class T, class Policy>
std::pair<T, T> owens_t_T1_accelerated(T h, T a, const Policy& pol)
{
//
// This is the same series as T1, but:
// * The Taylor series for atan has been combined with that for T1,
// reducing but not eliminating cancellation error.
// * The resulting alternating series is then accelerated using method 1
// from H. Cohen, F. Rodriguez Villegas, D. Zagier,
// "Convergence acceleration of alternating series", Bonn, (1991).
//
BOOST_MATH_STD_USING
static const char* function = "boost::math::owens_t<%1%>(%1%, %1%)";
T half_h_h = h * h / 2;
T a_pow = a;
T aa = a * a;
T exp_term = exp(-h * h / 2);
T one_minus_dj_sum = exp_term;
T sum = a_pow * exp_term;
T dj_pow = exp_term;
T term = sum;
T abs_err;
int j = 1;
//
// Normally with this form of series acceleration we can calculate
// up front how many terms will be required - based on the assumption
// that each term decreases in size by a factor of 3. However,
// that assumption does not apply here, as the underlying T1 series can
// go quite strongly divergent in the early terms, before strongly
// converging later. Various "guesstimates" have been tried to take account
// of this, but they don't always work.... so instead set "n" to the
// largest value that won't cause overflow later, and abort iteration
// when the last accelerated term was small enough...
//
int n;
#ifndef BOOST_MATH_NO_EXCEPTIONS
try
{
#endif
n = itrunc(T(tools::log_max_value<T>() / 6));
#ifndef BOOST_MATH_NO_EXCEPTIONS
}
catch(...)
{
n = (std::numeric_limits<int>::max)();
}
#endif
n = (std::min)(n, 1500);
T d = pow(3 + sqrt(T(8)), T(n));
d = (d + 1 / d) / 2;
T b = -1;
T c = -d;
c = b - c;
sum *= c;
b = -n * n * b * 2;
abs_err = ldexp(fabs(sum), -tools::digits<T>());
while(j < n)
{
a_pow *= aa;
dj_pow *= half_h_h / j;
one_minus_dj_sum += dj_pow;
term = one_minus_dj_sum * a_pow / (2 * j + 1);
c = b - c;
sum += c * term;
abs_err += ldexp((std::max)(T(fabs(sum)), T(fabs(c*term))), -tools::digits<T>());
b = (j + n) * (j - n) * b / ((j + T(0.5)) * (j + 1));
++j;
//
// Include an escape route to prevent calculating too many terms:
//
if((j > 10) && (fabs(sum * tools::epsilon<T>()) > fabs(c * term)))
break;
}
abs_err += fabs(c * term);
if(sum < 0) // sum must always be positive, if it's negative something really bad has happened:
policies::raise_evaluation_error(function, 0, T(0), pol);
return std::pair<T, T>((sum / d) / boost::math::constants::two_pi<T>(), abs_err / sum);
}
template<typename RealType, class Policy>
inline RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah, const Policy& pol, const std::true_type&)
{
BOOST_MATH_STD_USING
using namespace boost::math::constants;
const unsigned short maxii = m+m+1;
const RealType hs = h*h;
const RealType as = -a*a;
const RealType y = static_cast<RealType>(1) / hs;
unsigned short ii = 1;
RealType val = 0;
RealType vi = a * exp( -ah*ah*half<RealType>() ) / root_two_pi<RealType>();
RealType z = owens_t_znorm1(ah, pol)/h;
RealType last_z = fabs(z);
RealType lim = policies::get_epsilon<RealType, Policy>();
while( true )
{
val += z;
//
// This series stops converging after a while, so put a limit
// on how far we go before returning our best guess:
//
if((fabs(lim * val) > fabs(z)) || ((ii > maxii) && (fabs(z) > last_z)) || (z == 0))
{
val *= exp( -hs*half<RealType>() ) / root_two_pi<RealType>();
break;
} // if( maxii <= ii )
last_z = fabs(z);
z = y * ( vi - static_cast<RealType>(ii) * z );
vi *= as;
ii += 2;
} // while( true )
return val;
} // RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah)
template<typename RealType, class Policy>
inline std::pair<RealType, RealType> owens_t_T2_accelerated(const RealType h, const RealType a, const RealType ah, const Policy& pol)
{
//
// This is the same series as T2, but with acceleration applied.
// Note that we have to be *very* careful to check that nothing bad
// has happened during evaluation - this series will go divergent
// and/or fail to alternate at a drop of a hat! :-(
//
BOOST_MATH_STD_USING
using namespace boost::math::constants;
const RealType hs = h*h;
const RealType as = -a*a;
const RealType y = static_cast<RealType>(1) / hs;
unsigned short ii = 1;
RealType val = 0;
RealType vi = a * exp( -ah*ah*half<RealType>() ) / root_two_pi<RealType>();
RealType z = boost::math::detail::owens_t_znorm1(ah, pol)/h;
RealType last_z = fabs(z);
//
// Normally with this form of series acceleration we can calculate
// up front how many terms will be required - based on the assumption
// that each term decreases in size by a factor of 3. However,
// that assumption does not apply here, as the underlying T1 series can
// go quite strongly divergent in the early terms, before strongly
// converging later. Various "guesstimates" have been tried to take account
// of this, but they don't always work.... so instead set "n" to the
// largest value that won't cause overflow later, and abort iteration
// when the last accelerated term was small enough...
//
int n;
#ifndef BOOST_MATH_NO_EXCEPTIONS
try
{
#endif
n = itrunc(RealType(tools::log_max_value<RealType>() / 6));
#ifndef BOOST_MATH_NO_EXCEPTIONS
}
catch(...)
{
n = (std::numeric_limits<int>::max)();
}
#endif
n = (std::min)(n, 1500);
RealType d = pow(3 + sqrt(RealType(8)), RealType(n));
d = (d + 1 / d) / 2;
RealType b = -1;
RealType c = -d;
int s = 1;
for(int k = 0; k < n; ++k)
{
//
// Check for both convergence and whether the series has gone bad:
//
if(
(fabs(z) > last_z) // Series has gone divergent, abort
|| (fabs(val) * tools::epsilon<RealType>() > fabs(c * s * z)) // Convergence!
|| (z * s < 0) // Series has stopped alternating - all bets are off - abort.
)
{
break;
}
c = b - c;
val += c * s * z;
b = (k + n) * (k - n) * b / ((k + RealType(0.5)) * (k + 1));
last_z = fabs(z);
s = -s;
z = y * ( vi - static_cast<RealType>(ii) * z );
vi *= as;
ii += 2;
} // while( true )
RealType err = fabs(c * z) / val;
return std::pair<RealType, RealType>(val * exp( -hs*half<RealType>() ) / (d * root_two_pi<RealType>()), err);
} // RealType owens_t_T2_accelerated(const RealType h, const RealType a, const RealType ah, const Policy&)
template<typename RealType, typename Policy>
inline RealType T4_mp(const RealType h, const RealType a, const Policy& pol)
{
BOOST_MATH_STD_USING
const RealType hs = h*h;
const RealType as = -a*a;
unsigned short ii = 1;
RealType ai = constants::one_div_two_pi<RealType>() * a * exp( -0.5*hs*(1.0-as) );
RealType yi = 1.0;
RealType val = 0.0;
RealType lim = boost::math::policies::get_epsilon<RealType, Policy>();
while( true )
{
RealType term = ai*yi;
val += term;
if((yi != 0) && (fabs(val * lim) > fabs(term)))
break;
ii += 2;
yi = (1.0-hs*yi) / static_cast<RealType>(ii);
ai *= as;
if(ii > (std::min)(1500, (int)policies::get_max_series_iterations<Policy>()))
policies::raise_evaluation_error("boost::math::owens_t<%1%>", 0, val, pol);
} // while( true )
return val;
} // arg_type owens_t_T4(const arg_type h, const arg_type a, const unsigned short m)
// This routine dispatches the call to one of six subroutines, depending on the values
// of h and a.
// preconditions: h >= 0, 0<=a<=1, ah=a*h
//
// Note there are different versions for different precisions....
template<typename RealType, typename Policy>
inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, std::integral_constant<int, 64> const&)
{
// Simple main case for 64-bit precision or less, this is as per the Patefield-Tandy paper:
BOOST_MATH_STD_USING
//
// Handle some special cases first, these are from
// page 1077 of Owen's original paper:
//
if(h == 0)
{
return atan(a) * constants::one_div_two_pi<RealType>();
}
if(a == 0)
{
return 0;
}
if(a == 1)
{
return owens_t_znorm2(RealType(-h), pol) * owens_t_znorm2(h, pol) / 2;
}
if(a >= tools::max_value<RealType>())
{
return owens_t_znorm2(RealType(fabs(h)), pol);
}
RealType val = 0; // avoid compiler warnings, 0 will be overwritten in any case
const unsigned short icode = owens_t_compute_code(h, a);
const unsigned short m = owens_t_get_order(icode, val /* just a dummy for the type */, pol);
static const unsigned short meth[] = {1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6}; // 18 entries
// determine the appropriate method, T1 ... T6
switch( meth[icode] )
{
case 1: // T1
val = owens_t_T1(h,a,m,pol);
break;
case 2: // T2
typedef typename policies::precision<RealType, Policy>::type precision_type;
typedef std::integral_constant<bool, (precision_type::value == 0) || (precision_type::value > 64)> tag_type;
val = owens_t_T2(h, a, m, ah, pol, tag_type());
break;
case 3: // T3
val = owens_t_T3(h,a,ah, pol);
break;
case 4: // T4
val = owens_t_T4(h,a,m);
break;
case 5: // T5
val = owens_t_T5(h,a, pol);
break;
case 6: // T6
val = owens_t_T6(h,a, pol);
break;
default:
val = policies::raise_evaluation_error<RealType>("boost::math::owens_t", "selection routine in Owen's T function failed with h = %1%", h, pol);
}
return val;
}
template<typename RealType, typename Policy>
inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, const std::integral_constant<int, 65>&)
{
// Arbitrary precision version:
BOOST_MATH_STD_USING
//
// Handle some special cases first, these are from
// page 1077 of Owen's original paper:
//
if(h == 0)
{
return atan(a) * constants::one_div_two_pi<RealType>();
}
if(a == 0)
{
return 0;
}
if(a == 1)
{
return owens_t_znorm2(RealType(-h), pol) * owens_t_znorm2(h, pol) / 2;
}
if(a >= tools::max_value<RealType>())
{
return owens_t_znorm2(RealType(fabs(h)), pol);
}
// Attempt arbitrary precision code, this will throw if it goes wrong:
typedef typename boost::math::policies::normalise<Policy, boost::math::policies::evaluation_error<> >::type forwarding_policy;
std::pair<RealType, RealType> p1(0, tools::max_value<RealType>()), p2(0, tools::max_value<RealType>());
RealType target_precision = policies::get_epsilon<RealType, Policy>() * 1000;
bool have_t1(false), have_t2(false);
if(ah < 3)
{
#ifndef BOOST_MATH_NO_EXCEPTIONS
try
{
#endif
have_t1 = true;
p1 = owens_t_T1_accelerated(h, a, forwarding_policy());
if(p1.second < target_precision)
return p1.first;
#ifndef BOOST_MATH_NO_EXCEPTIONS
}
catch(const boost::math::evaluation_error&){} // T1 may fail and throw, that's OK
#endif
}
if(ah > 1)
{
#ifndef BOOST_MATH_NO_EXCEPTIONS
try
{
#endif
have_t2 = true;
p2 = owens_t_T2_accelerated(h, a, ah, forwarding_policy());
if(p2.second < target_precision)
return p2.first;
#ifndef BOOST_MATH_NO_EXCEPTIONS
}
catch(const boost::math::evaluation_error&){} // T2 may fail and throw, that's OK
#endif
}
//
// If we haven't tried T1 yet, do it now - sometimes it succeeds and the number of iterations
// is fairly low compared to T4.
//
if(!have_t1)
{
#ifndef BOOST_MATH_NO_EXCEPTIONS
try
{
#endif
have_t1 = true;
p1 = owens_t_T1_accelerated(h, a, forwarding_policy());
if(p1.second < target_precision)
return p1.first;
#ifndef BOOST_MATH_NO_EXCEPTIONS
}
catch(const boost::math::evaluation_error&){} // T1 may fail and throw, that's OK
#endif
}
//
// If we haven't tried T2 yet, do it now - sometimes it succeeds and the number of iterations
// is fairly low compared to T4.
//
if(!have_t2)
{
#ifndef BOOST_MATH_NO_EXCEPTIONS
try
{
#endif
have_t2 = true;
p2 = owens_t_T2_accelerated(h, a, ah, forwarding_policy());
if(p2.second < target_precision)
return p2.first;
#ifndef BOOST_MATH_NO_EXCEPTIONS
}
catch(const boost::math::evaluation_error&){} // T2 may fail and throw, that's OK
#endif
}
//
// OK, nothing left to do but try the most expensive option which is T4,
// this is often slow to converge, but when it does converge it tends to
// be accurate:
#ifndef BOOST_MATH_NO_EXCEPTIONS
try
{
#endif
return T4_mp(h, a, pol);
#ifndef BOOST_MATH_NO_EXCEPTIONS
}
catch(const boost::math::evaluation_error&){} // T4 may fail and throw, that's OK
#endif
//
// Now look back at the results from T1 and T2 and see if either gave better
// results than we could get from the 64-bit precision versions.
//
if((std::min)(p1.second, p2.second) < RealType(1e-20))
{
return p1.second < p2.second ? p1.first : p2.first;
}
//
// We give up - no arbitrary precision versions succeeded!
//
return owens_t_dispatch(h, a, ah, pol, std::integral_constant<int, 64>());
} // RealType owens_t_dispatch(RealType h, RealType a, RealType ah)
template<typename RealType, typename Policy>
inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, const std::integral_constant<int, 0>&)
{
// We don't know what the precision is until runtime:
if(tools::digits<RealType>() <= 64)
return owens_t_dispatch(h, a, ah, pol, std::integral_constant<int, 64>());
return owens_t_dispatch(h, a, ah, pol, std::integral_constant<int, 65>());
}
template<typename RealType, typename Policy>
inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol)
{
// Figure out the precision and forward to the correct version:
typedef typename policies::precision<RealType, Policy>::type precision_type;
typedef std::integral_constant<int,
precision_type::value <= 0 ? 0 :
precision_type::value <= 64 ? 64 : 65
> tag_type;
return owens_t_dispatch(h, a, ah, pol, tag_type());
}
// compute Owen's T function, T(h,a), for arbitrary values of h and a
template<typename RealType, class Policy>
inline RealType owens_t(RealType h, RealType a, const Policy& pol)
{
BOOST_MATH_STD_USING
// exploit that T(-h,a) == T(h,a)
h = fabs(h);
// Use equation (2) in the paper to remap the arguments
// such that h>=0 and 0<=a<=1 for the call of the actual
// computation routine.
const RealType fabs_a = fabs(a);
const RealType fabs_ah = fabs_a*h;
RealType val = static_cast<RealType>(0.0f); // avoid compiler warnings, 0.0 will be overwritten in any case
if(fabs_a <= 1)
{
val = owens_t_dispatch(h, fabs_a, fabs_ah, pol);
} // if(fabs_a <= 1.0)
else
{
if( h <= RealType(0.67) )
{
const RealType normh = owens_t_znorm1(h, pol);
const RealType normah = owens_t_znorm1(fabs_ah, pol);
val = static_cast<RealType>(1)/static_cast<RealType>(4) - normh*normah -
owens_t_dispatch(fabs_ah, static_cast<RealType>(1 / fabs_a), h, pol);
} // if( h <= 0.67 )
else
{
const RealType normh = detail::owens_t_znorm2(h, pol);
const RealType normah = detail::owens_t_znorm2(fabs_ah, pol);
val = constants::half<RealType>()*(normh+normah) - normh*normah -
owens_t_dispatch(fabs_ah, static_cast<RealType>(1 / fabs_a), h, pol);
} // else [if( h <= 0.67 )]
} // else [if(fabs_a <= 1)]
// exploit that T(h,-a) == -T(h,a)
if(a < 0)
{
return -val;
} // if(a < 0)
return val;
} // RealType owens_t(RealType h, RealType a)
template <class T, class Policy, class tag>
struct owens_t_initializer
{
struct init
{
init()
{
do_init(tag());
}
template <int N>
static void do_init(const std::integral_constant<int, N>&){}
static void do_init(const std::integral_constant<int, 64>&)
{
boost::math::owens_t(static_cast<T>(7), static_cast<T>(0.96875), Policy());
boost::math::owens_t(static_cast<T>(2), static_cast<T>(0.5), Policy());
}
void force_instantiate()const{}
};
static const init initializer;
static void force_instantiate()
{
initializer.force_instantiate();
}
};
template <class T, class Policy, class tag>
const typename owens_t_initializer<T, Policy, tag>::init owens_t_initializer<T, Policy, tag>::initializer;
} // namespace detail
template <class T1, class T2, class Policy>
inline typename tools::promote_args<T1, T2>::type owens_t(T1 h, T2 a, const Policy& pol)
{
typedef typename tools::promote_args<T1, T2>::type result_type;
typedef typename policies::evaluation<result_type, Policy>::type value_type;
typedef typename policies::precision<value_type, Policy>::type precision_type;
typedef std::integral_constant<int,
precision_type::value <= 0 ? 0 :
precision_type::value <= 64 ? 64 : 65
> tag_type;
detail::owens_t_initializer<result_type, Policy, tag_type>::force_instantiate();
return policies::checked_narrowing_cast<result_type, Policy>(detail::owens_t(static_cast<value_type>(h), static_cast<value_type>(a), pol), "boost::math::owens_t<%1%>(%1%,%1%)");
}
template <class T1, class T2>
inline typename tools::promote_args<T1, T2>::type owens_t(T1 h, T2 a)
{
return owens_t(h, a, policies::policy<>());
}
} // namespace math
} // namespace boost
#ifdef _MSC_VER
#pragma warning(pop)
#endif
#endif
// EOF