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/config/no_tr1/cmath.hpp> #include <boost/math/special_functions/erf.hpp> #include <boost/math/special_functions/expm1.hpp> #include <boost/throw_exception.hpp> #include <boost/assert.hpp> #include <boost/math/constants/constants.hpp> #include <boost/math/tools/big_constant.hpp> #include <stdexcept> namespace boost { namespace math { namespace detail { // owens_t_znorm1(x) = P(-oo<Z<=x)-0.5 with Z being normally distributed. template<typename RealType> inline RealType owens_t_znorm1(const RealType x) { using namespace boost::math::constants; return erf(x*one_div_root_two<RealType>())*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> inline RealType owens_t_znorm2(const RealType x) { using namespace boost::math::constants; return erfc(x*one_div_root_two<RealType>())*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[] = {0.02, 0.06, 0.09, 0.125, 0.26, 0.4, 0.6, 1.6, 1.7, 2.33, 2.4, 3.36, 3.4, 4.8}; static const RealType arange[] = {0.025, 0.09, 0.15, 0.36, 0.5, 0.9, 0.99999}; /* 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++) // interprete 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 mpl::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_ASSERT(icode<18); return ord[icode]; } // unsigned short owens_t_get_order(const unsigned short icode, RealType, mpl::int<53> const&) template<typename RealType> inline unsigned short owens_t_get_order_imp(const unsigned short icode, RealType, const mpl::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_ASSERT(icode<18); return ord[icode]; } // unsigned short owens_t_get_order(const unsigned short icode, RealType, mpl::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 typename mpl::if_< mpl::or_< mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<53> > >, mpl::int_<64>, mpl::int_<53> >::type 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> inline RealType owens_t_T1(const RealType h, const RealType a, const unsigned short m) { 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 = expm1( hs ); 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&, const mpl::false_&) { 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)/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> inline RealType owens_t_T3_imp(const RealType h, const RealType a, const RealType ah, const mpl::int_<53>&) { BOOST_MATH_STD_USING using namespace boost::math::constants; const unsigned short m = 20; static const RealType c2[] = { 0.99999999999999987510, -0.99999999999988796462, 0.99999999998290743652, -0.99999999896282500134, 0.99999996660459362918, -0.99999933986272476760, 0.99999125611136965852, -0.99991777624463387686, 0.99942835555870132569, -0.99697311720723000295, 0.98751448037275303682, -0.95915857980572882813, 0.89246305511006708555, -0.76893425990463999675, 0.58893528468484693250, -0.38380345160440256652, 0.20317601701045299653, -0.82813631607004984866E-01, 0.24167984735759576523E-01, -0.44676566663971825242E-02, 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)/h; RealType val = 0; while( true ) { BOOST_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> inline RealType owens_t_T3_imp(const RealType h, const RealType a, const RealType ah, const mpl::int_<64>&) { 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)/h; RealType val = 0; while( true ) { BOOST_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&) { typedef typename policies::precision<RealType, Policy>::type precision_type; typedef typename mpl::if_< mpl::or_< mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<53> > >, mpl::int_<64>, mpl::int_<53> >::type tag_type; return owens_t_T3_imp(h, a, ah, tag_type()); } // 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 mpl::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[] = {0.35082039676451715489E-02, 0.31279042338030753740E-01, 0.85266826283219451090E-01, 0.16245071730812277011, 0.25851196049125434828, 0.36807553840697533536, 0.48501092905604697475, 0.60277514152618576821, 0.71477884217753226516, 0.81475510988760098605, 0.89711029755948965867, 0.95723808085944261843, 0.99178832974629703586}; static const RealType wts[] = { 0.18831438115323502887E-01, 0.18567086243977649478E-01, 0.18042093461223385584E-01, 0.17263829606398753364E-01, 0.16243219975989856730E-01, 0.14994592034116704829E-01, 0.13535474469662088392E-01, 0.11886351605820165233E-01, 0.10070377242777431897E-01, 0.81130545742299586629E-02, 0.60419009528470238773E-02, 0.38862217010742057883E-02, 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_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 mpl::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_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 typename mpl::if_< mpl::or_< mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<53> > >, mpl::int_<64>, mpl::int_<53> >::type 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> inline RealType owens_t_T6(const RealType h, const RealType a) { BOOST_MATH_STD_USING using namespace boost::math::constants; const RealType normh = owens_t_znorm2( h ); 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 "guestimates" 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; try { n = itrunc(T(tools::log_max_value<T>() / 6)); } catch(...) { n = (std::numeric_limits<int>::max)(); } n = (std::min)(n, 1500); T d = pow(3 + sqrt(T(8)), 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 happend: 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&, const mpl::true_&) { 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)/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&) { // // 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)/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 "guestimates" 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; try { n = itrunc(RealType(tools::log_max_value<RealType>() / 6)); } catch(...) { n = (std::numeric_limits<int>::max)(); } n = (std::min)(n, 1500); RealType d = pow(3 + sqrt(RealType(8)), 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, mpl::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)) * owens_t_znorm2(h) / 2; } if(a >= tools::max_value<RealType>()) { return owens_t_znorm2(RealType(fabs(h))); } 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); break; case 2: // T2 typedef typename policies::precision<RealType, Policy>::type precision_type; typedef mpl::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); break; default: BOOST_THROW_EXCEPTION(std::logic_error("selection routine in Owen's T function failed")); } 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 mpl::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)) * owens_t_znorm2(h) / 2; } if(a >= tools::max_value<RealType>()) { return owens_t_znorm2(RealType(fabs(h))); } // 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) { try { have_t1 = true; p1 = owens_t_T1_accelerated(h, a, forwarding_policy()); if(p1.second < target_precision) return p1.first; } catch(const boost::math::evaluation_error&){} // T1 may fail and throw, that's OK } if(ah > 1) { try { have_t2 = true; p2 = owens_t_T2_accelerated(h, a, ah, forwarding_policy()); if(p2.second < target_precision) return p2.first; } catch(const boost::math::evaluation_error&){} // T2 may fail and throw, that's OK } // // 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) { try { have_t1 = true; p1 = owens_t_T1_accelerated(h, a, forwarding_policy()); if(p1.second < target_precision) return p1.first; } catch(const boost::math::evaluation_error&){} // T1 may fail and throw, that's OK } // // 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) { try { have_t2 = true; p2 = owens_t_T2_accelerated(h, a, ah, forwarding_policy()); if(p2.second < target_precision) return p2.first; } catch(const boost::math::evaluation_error&){} // T2 may fail and throw, that's OK } // // 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: try { return T4_mp(h, a, pol); } catch(const boost::math::evaluation_error&){} // T4 may fail and throw, that's OK // // 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) < 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, mpl::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 mpl::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, mpl::int_<64>()); return owens_t_dispatch(h, a, ah, pol, mpl::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 typename mpl::if_c< precision_type::value == 0, mpl::int_<0>, typename mpl::if_c< precision_type::value <= 64, mpl::int_<64>, mpl::int_<65> >::type >::type 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 = 0.0; // 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 <= 0.67 ) { const RealType normh = owens_t_znorm1(h); const RealType normah = owens_t_znorm1(fabs_ah); 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); const RealType normah = detail::owens_t_znorm2(fabs_ah); 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 mpl::int_<N>&){} static void do_init(const mpl::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 typename mpl::if_c< precision_type::value == 0, mpl::int_<0>, typename mpl::if_c< precision_type::value <= 64, mpl::int_<64>, mpl::int_<65> >::type >::type 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 #endif // EOF