1 // Copyright Benjamin Sobotta 2012
3 // Use, modification and distribution are subject to the
4 // Boost Software License, Version 1.0. (See accompanying file
5 // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
7 #ifndef BOOST_OWENS_T_HPP
8 #define BOOST_OWENS_T_HPP
11 // Mike Patefield, David Tandy
12 // FAST AND ACCURATE CALCULATION OF OWEN'S T-FUNCTION
13 // Journal of Statistical Software, 5 (5), 1-25
19 #include <boost/math/special_functions/math_fwd.hpp>
20 #include <boost/math/special_functions/erf.hpp>
21 #include <boost/math/special_functions/expm1.hpp>
22 #include <boost/math/tools/throw_exception.hpp>
23 #include <boost/math/tools/assert.hpp>
24 #include <boost/math/constants/constants.hpp>
25 #include <boost/math/tools/big_constant.hpp>
32 #pragma warning(disable:4127)
35 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
37 // This is the only way we can avoid
38 // warning: non-standard suffix on floating constant [-Wpedantic]
39 // when building with -Wall -pedantic. Neither __extension__
40 // nor #pragma diagnostic ignored work :(
42 #pragma GCC system_header
51 // owens_t_znorm1(x) = P(-oo<Z<=x)-0.5 with Z being normally distributed.
52 template<typename RealType, class Policy>
53 inline RealType owens_t_znorm1(const RealType x, const Policy& pol)
55 using namespace boost::math::constants;
56 return boost::math::erf(x*one_div_root_two<RealType>(), pol)*half<RealType>();
57 } // RealType owens_t_znorm1(const RealType x)
59 // owens_t_znorm2(x) = P(x<=Z<oo) with Z being normally distributed.
60 template<typename RealType, class Policy>
61 inline RealType owens_t_znorm2(const RealType x, const Policy& pol)
63 using namespace boost::math::constants;
64 return boost::math::erfc(x*one_div_root_two<RealType>(), pol)*half<RealType>();
65 } // RealType owens_t_znorm2(const RealType x)
67 // Auxiliary function, it computes an array key that is used to determine
68 // the specific computation method for Owen's T and the order thereof
69 // used in owens_t_dispatch.
70 template<typename RealType>
71 inline unsigned short owens_t_compute_code(const RealType h, const RealType a)
73 static const RealType hrange[] =
74 { 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 };
76 static const RealType arange[] = { 0.025f, 0.09f, 0.15f, 0.36f, 0.5f, 0.9f, 0.99999f };
78 original select array from paper:
79 1, 1, 2,13,13,13,13,13,13,13,13,16,16,16, 9
80 1, 2, 2, 3, 3, 5, 5,14,14,15,15,16,16,16, 9
81 2, 2, 3, 3, 3, 5, 5,15,15,15,15,16,16,16,10
82 2, 2, 3, 5, 5, 5, 5, 7, 7,16,16,16,16,16,10
83 2, 3, 3, 5, 5, 6, 6, 8, 8,17,17,17,12,12,11
84 2, 3, 5, 5, 5, 6, 6, 8, 8,17,17,17,12,12,12
85 2, 3, 4, 4, 6, 6, 8, 8,17,17,17,17,17,12,12
86 2, 3, 4, 4, 6, 6,18,18,18,18,17,17,17,12,12
88 // subtract one because the array is written in FORTRAN in mind - in C arrays start @ zero
89 static const unsigned short select[] =
91 0, 0 , 1 , 12 ,12 , 12 , 12 , 12 , 12 , 12 , 12 , 15 , 15 , 15 , 8,
92 0 , 1 , 1 , 2 , 2 , 4 , 4 , 13 , 13 , 14 , 14 , 15 , 15 , 15 , 8,
93 1 , 1 , 2 , 2 , 2 , 4 , 4 , 14 , 14 , 14 , 14 , 15 , 15 , 15 , 9,
94 1 , 1 , 2 , 4 , 4 , 4 , 4 , 6 , 6 , 15 , 15 , 15 , 15 , 15 , 9,
95 1 , 2 , 2 , 4 , 4 , 5 , 5 , 7 , 7 , 16 ,16 , 16 , 11 , 11 , 10,
96 1 , 2 , 4 , 4 , 4 , 5 , 5 , 7 , 7 , 16 , 16 , 16 , 11 , 11 , 11,
97 1 , 2 , 3 , 3 , 5 , 5 , 7 , 7 , 16 , 16 , 16 , 16 , 16 , 11 , 11,
98 1 , 2 , 3 , 3 , 5 , 5 , 17 , 17 , 17 , 17 , 16 , 16 , 16 , 11 , 11
101 unsigned short ihint = 14, iaint = 7;
102 for(unsigned short i = 0; i != 14; i++)
109 } // for(unsigned short i = 0; i != 14; i++)
111 for(unsigned short i = 0; i != 7; i++)
118 } // for(unsigned short i = 0; i != 7; i++)
120 // interpret select array as 8x15 matrix
121 return select[iaint*15 + ihint];
123 } // unsigned short owens_t_compute_code(const RealType h, const RealType a)
125 template<typename RealType>
126 inline unsigned short owens_t_get_order_imp(const unsigned short icode, RealType, const std::integral_constant<int, 53>&)
128 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
130 BOOST_MATH_ASSERT(icode<18);
133 } // unsigned short owens_t_get_order(const unsigned short icode, RealType, std::integral_constant<int, 53> const&)
135 template<typename RealType>
136 inline unsigned short owens_t_get_order_imp(const unsigned short icode, RealType, const std::integral_constant<int, 64>&)
138 // method ================>>> {1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6}
139 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
141 BOOST_MATH_ASSERT(icode<18);
144 } // unsigned short owens_t_get_order(const unsigned short icode, RealType, std::integral_constant<int, 64> const&)
146 template<typename RealType, typename Policy>
147 inline unsigned short owens_t_get_order(const unsigned short icode, RealType r, const Policy&)
149 typedef typename policies::precision<RealType, Policy>::type precision_type;
150 typedef std::integral_constant<int,
151 precision_type::value <= 0 ? 64 :
152 precision_type::value <= 53 ? 53 : 64
155 return owens_t_get_order_imp(icode, r, tag_type());
158 // compute the value of Owen's T function with method T1 from the reference paper
159 template<typename RealType, typename Policy>
160 inline RealType owens_t_T1(const RealType h, const RealType a, const unsigned short m, const Policy& pol)
163 using namespace boost::math::constants;
165 const RealType hs = -h*h*half<RealType>();
166 const RealType dhs = exp( hs );
167 const RealType as = a*a;
171 RealType aj = a * one_div_two_pi<RealType>();
172 RealType dj = boost::math::expm1( hs, pol);
173 RealType gj = hs*dhs;
175 RealType val = atan( a ) * one_div_two_pi<RealType>();
185 jj += static_cast<RealType>(2);
188 gj *= hs / static_cast<RealType>(j);
192 } // RealType owens_t_T1(const RealType h, const RealType a, const unsigned short m)
194 // compute the value of Owen's T function with method T2 from the reference paper
195 template<typename RealType, class Policy>
196 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&)
199 using namespace boost::math::constants;
201 const unsigned short maxii = m+m+1;
202 const RealType hs = h*h;
203 const RealType as = -a*a;
204 const RealType y = static_cast<RealType>(1) / hs;
206 unsigned short ii = 1;
208 RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
209 RealType z = owens_t_znorm1(ah, pol)/h;
216 val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
218 } // if( maxii <= ii )
219 z = y * ( vi - static_cast<RealType>(ii) * z );
225 } // RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah)
227 // compute the value of Owen's T function with method T3 from the reference paper
228 template<typename RealType, class Policy>
229 inline RealType owens_t_T3_imp(const RealType h, const RealType a, const RealType ah, const std::integral_constant<int, 53>&, const Policy& pol)
232 using namespace boost::math::constants;
234 const unsigned short m = 20;
236 static const RealType c2[] =
238 static_cast<RealType>(0.99999999999999987510),
239 static_cast<RealType>(-0.99999999999988796462), static_cast<RealType>(0.99999999998290743652),
240 static_cast<RealType>(-0.99999999896282500134), static_cast<RealType>(0.99999996660459362918),
241 static_cast<RealType>(-0.99999933986272476760), static_cast<RealType>(0.99999125611136965852),
242 static_cast<RealType>(-0.99991777624463387686), static_cast<RealType>(0.99942835555870132569),
243 static_cast<RealType>(-0.99697311720723000295), static_cast<RealType>(0.98751448037275303682),
244 static_cast<RealType>(-0.95915857980572882813), static_cast<RealType>(0.89246305511006708555),
245 static_cast<RealType>(-0.76893425990463999675), static_cast<RealType>(0.58893528468484693250),
246 static_cast<RealType>(-0.38380345160440256652), static_cast<RealType>(0.20317601701045299653),
247 static_cast<RealType>(-0.82813631607004984866E-01), static_cast<RealType>(0.24167984735759576523E-01),
248 static_cast<RealType>(-0.44676566663971825242E-02), static_cast<RealType>(0.39141169402373836468E-03)
251 const RealType as = a*a;
252 const RealType hs = h*h;
253 const RealType y = static_cast<RealType>(1)/hs;
256 unsigned short i = 0;
257 RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
258 RealType zi = owens_t_znorm1(ah, pol)/h;
263 BOOST_MATH_ASSERT(i < 21);
265 if( m <= i ) // if( m < i+1 )
267 val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
270 zi = y * (ii*zi - vi);
277 } // RealType owens_t_T3(const RealType h, const RealType a, const RealType ah)
279 // compute the value of Owen's T function with method T3 from the reference paper
280 template<class RealType, class Policy>
281 inline RealType owens_t_T3_imp(const RealType h, const RealType a, const RealType ah, const std::integral_constant<int, 64>&, const Policy& pol)
284 using namespace boost::math::constants;
286 const unsigned short m = 30;
288 static const RealType c2[] =
290 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999999999999999729978162447266851932041876728736094298092917625009873),
291 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.99999999999999999999467056379678391810626533251885323416799874878563998732905968),
292 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999999999824849349313270659391127814689133077036298754586814091034842536),
293 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999999999997703859616213643405880166422891953033591551179153879839440241685),
294 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999998394883415238173334565554173013941245103172035286759201504179038147),
295 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999999993063616095509371081203145247992197457263066869044528823599399470977),
296 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999999999797336340409464429599229870590160411238245275855903767652432017766116267),
297 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.999999999574958412069046680119051639753412378037565521359444170241346845522403274),
298 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999999933226234193375324943920160947158239076786103108097456617750134812033362048),
299 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999188923242461073033481053037468263536806742737922476636768006622772762168467),
300 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999992195143483674402853783549420883055129680082932629160081128947764415749728967),
301 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.999993935137206712830997921913316971472227199741857386575097250553105958772041501),
302 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99996135597690552745362392866517133091672395614263398912807169603795088421057688716),
303 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.99979556366513946026406788969630293820987757758641211293079784585126692672425362469),
304 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.999092789629617100153486251423850590051366661947344315423226082520411961968929483),
305 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.996593837411918202119308620432614600338157335862888580671450938858935084316004769854),
306 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.98910017138386127038463510314625339359073956513420458166238478926511821146316469589567),
307 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.970078558040693314521331982203762771512160168582494513347846407314584943870399016019),
308 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.92911438683263187495758525500033707204091967947532160289872782771388170647150321633673),
309 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.8542058695956156057286980736842905011429254735181323743367879525470479126968822863),
310 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.73796526033030091233118357742803709382964420335559408722681794195743240930748630755),
311 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.58523469882837394570128599003785154144164680587615878645171632791404210655891158),
312 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.415997776145676306165661663581868460503874205343014196580122174949645271353372263),
313 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.2588210875241943574388730510317252236407805082485246378222935376279663808416534365),
314 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.1375535825163892648504646951500265585055789019410617565727090346559210218472356689),
315 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.0607952766325955730493900985022020434830339794955745989150270485056436844239206648),
316 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.0216337683299871528059836483840390514275488679530797294557060229266785853764115),
317 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.00593405693455186729876995814181203900550014220428843483927218267309209471516256),
318 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.0011743414818332946510474576182739210553333860106811865963485870668929503649964142),
319 BOOST_MATH_BIG_CONSTANT(RealType, 260, -1.489155613350368934073453260689881330166342484405529981510694514036264969925132e-4),
320 BOOST_MATH_BIG_CONSTANT(RealType, 260, 9.072354320794357587710929507988814669454281514268844884841547607134260303118208e-6)
323 const RealType as = a*a;
324 const RealType hs = h*h;
325 const RealType y = 1 / hs;
328 unsigned short i = 0;
329 RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
330 RealType zi = owens_t_znorm1(ah, pol)/h;
335 BOOST_MATH_ASSERT(i < 31);
337 if( m <= i ) // if( m < i+1 )
339 val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
342 zi = y * (ii*zi - vi);
349 } // RealType owens_t_T3(const RealType h, const RealType a, const RealType ah)
351 template<class RealType, class Policy>
352 inline RealType owens_t_T3(const RealType h, const RealType a, const RealType ah, const Policy& pol)
354 typedef typename policies::precision<RealType, Policy>::type precision_type;
355 typedef std::integral_constant<int,
356 precision_type::value <= 0 ? 64 :
357 precision_type::value <= 53 ? 53 : 64
360 return owens_t_T3_imp(h, a, ah, tag_type(), pol);
363 // compute the value of Owen's T function with method T4 from the reference paper
364 template<typename RealType>
365 inline RealType owens_t_T4(const RealType h, const RealType a, const unsigned short m)
368 using namespace boost::math::constants;
370 const unsigned short maxii = m+m+1;
371 const RealType hs = h*h;
372 const RealType as = -a*a;
374 unsigned short ii = 1;
375 RealType ai = a * exp( -hs*(static_cast<RealType>(1)-as)*half<RealType>() ) * one_div_two_pi<RealType>();
385 yi = (static_cast<RealType>(1)-hs*yi) / static_cast<RealType>(ii);
390 } // RealType owens_t_T4(const RealType h, const RealType a, const unsigned short m)
392 // compute the value of Owen's T function with method T5 from the reference paper
393 template<typename RealType>
394 inline RealType owens_t_T5_imp(const RealType h, const RealType a, const std::integral_constant<int, 53>&)
399 - The pts[] array contains the squares (!) of the abscissas, i.e. the roots of the Legendre
400 polynomial P_n(x), instead of the plain roots as required in Gauss-Legendre
401 quadrature, because T5(h,a,m) contains only x^2 terms.
402 - The wts[] array contains the weights for Gauss-Legendre quadrature scaled with a factor
403 of 1/(2*pi) according to T5(h,a,m).
406 const unsigned short m = 13;
407 static const RealType pts[] = {
408 static_cast<RealType>(0.35082039676451715489E-02),
409 static_cast<RealType>(0.31279042338030753740E-01), static_cast<RealType>(0.85266826283219451090E-01),
410 static_cast<RealType>(0.16245071730812277011), static_cast<RealType>(0.25851196049125434828),
411 static_cast<RealType>(0.36807553840697533536), static_cast<RealType>(0.48501092905604697475),
412 static_cast<RealType>(0.60277514152618576821), static_cast<RealType>(0.71477884217753226516),
413 static_cast<RealType>(0.81475510988760098605), static_cast<RealType>(0.89711029755948965867),
414 static_cast<RealType>(0.95723808085944261843), static_cast<RealType>(0.99178832974629703586) };
415 static const RealType wts[] = {
416 static_cast<RealType>(0.18831438115323502887E-01),
417 static_cast<RealType>(0.18567086243977649478E-01), static_cast<RealType>(0.18042093461223385584E-01),
418 static_cast<RealType>(0.17263829606398753364E-01), static_cast<RealType>(0.16243219975989856730E-01),
419 static_cast<RealType>(0.14994592034116704829E-01), static_cast<RealType>(0.13535474469662088392E-01),
420 static_cast<RealType>(0.11886351605820165233E-01), static_cast<RealType>(0.10070377242777431897E-01),
421 static_cast<RealType>(0.81130545742299586629E-02), static_cast<RealType>(0.60419009528470238773E-02),
422 static_cast<RealType>(0.38862217010742057883E-02), static_cast<RealType>(0.16793031084546090448E-02) };
424 const RealType as = a*a;
425 const RealType hs = -h*h*boost::math::constants::half<RealType>();
428 for(unsigned short i = 0; i < m; ++i)
430 BOOST_MATH_ASSERT(i < 13);
431 const RealType r = static_cast<RealType>(1) + as*pts[i];
432 val += wts[i] * exp( hs*r ) / r;
433 } // for(unsigned short i = 0; i < m; ++i)
436 } // RealType owens_t_T5(const RealType h, const RealType a)
438 // compute the value of Owen's T function with method T5 from the reference paper
439 template<typename RealType>
440 inline RealType owens_t_T5_imp(const RealType h, const RealType a, const std::integral_constant<int, 64>&)
445 - The pts[] array contains the squares (!) of the abscissas, i.e. the roots of the Legendre
446 polynomial P_n(x), instead of the plain roots as required in Gauss-Legendre
447 quadrature, because T5(h,a,m) contains only x^2 terms.
448 - The wts[] array contains the weights for Gauss-Legendre quadrature scaled with a factor
449 of 1/(2*pi) according to T5(h,a,m).
452 const unsigned short m = 19;
453 static const RealType pts[] = {
454 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0016634282895983227941),
455 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.014904509242697054183),
456 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.04103478879005817919),
457 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.079359853513391511008),
458 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.1288612130237615133),
459 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.18822336642448518856),
460 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.25586876186122962384),
461 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.32999972011807857222),
462 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.40864620815774761438),
463 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.48971819306044782365),
464 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.57106118513245543894),
465 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.6505134942981533829),
466 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.72596367859928091618),
467 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.79540665919549865924),
468 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.85699701386308739244),
469 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.90909804422384697594),
470 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.95032536436570154409),
471 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.97958418733152273717),
472 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.99610366384229088321)
474 static const RealType wts[] = {
475 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012975111395684900835),
476 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012888764187499150078),
477 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012716644398857307844),
478 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012459897461364705691),
479 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012120231988292330388),
480 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.011699908404856841158),
481 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.011201723906897224448),
482 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.010628993848522759853),
483 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0099855296835573320047),
484 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0092756136096132857933),
485 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0085039700881139589055),
486 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0076757344408814561254),
487 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0067964187616556459109),
488 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.005871875456524750363),
489 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0049082589542498110071),
490 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0039119870792519721409),
491 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0028897090921170700834),
492 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0018483371329504443947),
493 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.00079623320100438873578)
496 const RealType as = a*a;
497 const RealType hs = -h*h*boost::math::constants::half<RealType>();
500 for(unsigned short i = 0; i < m; ++i)
502 BOOST_MATH_ASSERT(i < 19);
503 const RealType r = 1 + as*pts[i];
504 val += wts[i] * exp( hs*r ) / r;
505 } // for(unsigned short i = 0; i < m; ++i)
508 } // RealType owens_t_T5(const RealType h, const RealType a)
510 template<class RealType, class Policy>
511 inline RealType owens_t_T5(const RealType h, const RealType a, const Policy&)
513 typedef typename policies::precision<RealType, Policy>::type precision_type;
514 typedef std::integral_constant<int,
515 precision_type::value <= 0 ? 64 :
516 precision_type::value <= 53 ? 53 : 64
519 return owens_t_T5_imp(h, a, tag_type());
523 // compute the value of Owen's T function with method T6 from the reference paper
524 template<typename RealType, class Policy>
525 inline RealType owens_t_T6(const RealType h, const RealType a, const Policy& pol)
528 using namespace boost::math::constants;
530 const RealType normh = owens_t_znorm2(h, pol);
531 const RealType y = static_cast<RealType>(1) - a;
532 const RealType r = atan2(y, static_cast<RealType>(1 + a) );
534 RealType val = normh * ( static_cast<RealType>(1) - normh ) * half<RealType>();
537 val -= r * exp( -y*h*h*half<RealType>()/r ) * one_div_two_pi<RealType>();
540 } // RealType owens_t_T6(const RealType h, const RealType a, const unsigned short m)
542 template <class T, class Policy>
543 std::pair<T, T> owens_t_T1_accelerated(T h, T a, const Policy& pol)
546 // This is the same series as T1, but:
547 // * The Taylor series for atan has been combined with that for T1,
548 // reducing but not eliminating cancellation error.
549 // * The resulting alternating series is then accelerated using method 1
550 // from H. Cohen, F. Rodriguez Villegas, D. Zagier,
551 // "Convergence acceleration of alternating series", Bonn, (1991).
554 static const char* function = "boost::math::owens_t<%1%>(%1%, %1%)";
555 T half_h_h = h * h / 2;
558 T exp_term = exp(-h * h / 2);
559 T one_minus_dj_sum = exp_term;
560 T sum = a_pow * exp_term;
567 // Normally with this form of series acceleration we can calculate
568 // up front how many terms will be required - based on the assumption
569 // that each term decreases in size by a factor of 3. However,
570 // that assumption does not apply here, as the underlying T1 series can
571 // go quite strongly divergent in the early terms, before strongly
572 // converging later. Various "guesstimates" have been tried to take account
573 // of this, but they don't always work.... so instead set "n" to the
574 // largest value that won't cause overflow later, and abort iteration
575 // when the last accelerated term was small enough...
578 #ifndef BOOST_NO_EXCEPTIONS
582 n = itrunc(T(tools::log_max_value<T>() / 6));
583 #ifndef BOOST_NO_EXCEPTIONS
587 n = (std::numeric_limits<int>::max)();
590 n = (std::min)(n, 1500);
591 T d = pow(3 + sqrt(T(8)), n);
598 abs_err = ldexp(fabs(sum), -tools::digits<T>());
603 dj_pow *= half_h_h / j;
604 one_minus_dj_sum += dj_pow;
605 term = one_minus_dj_sum * a_pow / (2 * j + 1);
608 abs_err += ldexp((std::max)(T(fabs(sum)), T(fabs(c*term))), -tools::digits<T>());
609 b = (j + n) * (j - n) * b / ((j + T(0.5)) * (j + 1));
612 // Include an escape route to prevent calculating too many terms:
614 if((j > 10) && (fabs(sum * tools::epsilon<T>()) > fabs(c * term)))
617 abs_err += fabs(c * term);
618 if(sum < 0) // sum must always be positive, if it's negative something really bad has happened:
619 policies::raise_evaluation_error(function, 0, T(0), pol);
620 return std::pair<T, T>((sum / d) / boost::math::constants::two_pi<T>(), abs_err / sum);
623 template<typename RealType, class Policy>
624 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&)
627 using namespace boost::math::constants;
629 const unsigned short maxii = m+m+1;
630 const RealType hs = h*h;
631 const RealType as = -a*a;
632 const RealType y = static_cast<RealType>(1) / hs;
634 unsigned short ii = 1;
636 RealType vi = a * exp( -ah*ah*half<RealType>() ) / root_two_pi<RealType>();
637 RealType z = owens_t_znorm1(ah, pol)/h;
638 RealType last_z = fabs(z);
639 RealType lim = policies::get_epsilon<RealType, Policy>();
645 // This series stops converging after a while, so put a limit
646 // on how far we go before returning our best guess:
648 if((fabs(lim * val) > fabs(z)) || ((ii > maxii) && (fabs(z) > last_z)) || (z == 0))
650 val *= exp( -hs*half<RealType>() ) / root_two_pi<RealType>();
652 } // if( maxii <= ii )
654 z = y * ( vi - static_cast<RealType>(ii) * z );
660 } // RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah)
662 template<typename RealType, class Policy>
663 inline std::pair<RealType, RealType> owens_t_T2_accelerated(const RealType h, const RealType a, const RealType ah, const Policy& pol)
666 // This is the same series as T2, but with acceleration applied.
667 // Note that we have to be *very* careful to check that nothing bad
668 // has happened during evaluation - this series will go divergent
669 // and/or fail to alternate at a drop of a hat! :-(
672 using namespace boost::math::constants;
674 const RealType hs = h*h;
675 const RealType as = -a*a;
676 const RealType y = static_cast<RealType>(1) / hs;
678 unsigned short ii = 1;
680 RealType vi = a * exp( -ah*ah*half<RealType>() ) / root_two_pi<RealType>();
681 RealType z = boost::math::detail::owens_t_znorm1(ah, pol)/h;
682 RealType last_z = fabs(z);
685 // Normally with this form of series acceleration we can calculate
686 // up front how many terms will be required - based on the assumption
687 // that each term decreases in size by a factor of 3. However,
688 // that assumption does not apply here, as the underlying T1 series can
689 // go quite strongly divergent in the early terms, before strongly
690 // converging later. Various "guesstimates" have been tried to take account
691 // of this, but they don't always work.... so instead set "n" to the
692 // largest value that won't cause overflow later, and abort iteration
693 // when the last accelerated term was small enough...
696 #ifndef BOOST_NO_EXCEPTIONS
700 n = itrunc(RealType(tools::log_max_value<RealType>() / 6));
701 #ifndef BOOST_NO_EXCEPTIONS
705 n = (std::numeric_limits<int>::max)();
708 n = (std::min)(n, 1500);
709 RealType d = pow(3 + sqrt(RealType(8)), n);
715 for(int k = 0; k < n; ++k)
718 // Check for both convergence and whether the series has gone bad:
721 (fabs(z) > last_z) // Series has gone divergent, abort
722 || (fabs(val) * tools::epsilon<RealType>() > fabs(c * s * z)) // Convergence!
723 || (z * s < 0) // Series has stopped alternating - all bets are off - abort.
730 b = (k + n) * (k - n) * b / ((k + RealType(0.5)) * (k + 1));
733 z = y * ( vi - static_cast<RealType>(ii) * z );
737 RealType err = fabs(c * z) / val;
738 return std::pair<RealType, RealType>(val * exp( -hs*half<RealType>() ) / (d * root_two_pi<RealType>()), err);
739 } // RealType owens_t_T2_accelerated(const RealType h, const RealType a, const RealType ah, const Policy&)
741 template<typename RealType, typename Policy>
742 inline RealType T4_mp(const RealType h, const RealType a, const Policy& pol)
746 const RealType hs = h*h;
747 const RealType as = -a*a;
749 unsigned short ii = 1;
750 RealType ai = constants::one_div_two_pi<RealType>() * a * exp( -0.5*hs*(1.0-as) );
754 RealType lim = boost::math::policies::get_epsilon<RealType, Policy>();
758 RealType term = ai*yi;
760 if((yi != 0) && (fabs(val * lim) > fabs(term)))
763 yi = (1.0-hs*yi) / static_cast<RealType>(ii);
765 if(ii > (std::min)(1500, (int)policies::get_max_series_iterations<Policy>()))
766 policies::raise_evaluation_error("boost::math::owens_t<%1%>", 0, val, pol);
770 } // arg_type owens_t_T4(const arg_type h, const arg_type a, const unsigned short m)
773 // This routine dispatches the call to one of six subroutines, depending on the values
775 // preconditions: h >= 0, 0<=a<=1, ah=a*h
777 // Note there are different versions for different precisions....
778 template<typename RealType, typename Policy>
779 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, std::integral_constant<int, 64> const&)
781 // Simple main case for 64-bit precision or less, this is as per the Patefield-Tandy paper:
784 // Handle some special cases first, these are from
785 // page 1077 of Owen's original paper:
789 return atan(a) * constants::one_div_two_pi<RealType>();
797 return owens_t_znorm2(RealType(-h), pol) * owens_t_znorm2(h, pol) / 2;
799 if(a >= tools::max_value<RealType>())
801 return owens_t_znorm2(RealType(fabs(h)), pol);
803 RealType val = 0; // avoid compiler warnings, 0 will be overwritten in any case
804 const unsigned short icode = owens_t_compute_code(h, a);
805 const unsigned short m = owens_t_get_order(icode, val /* just a dummy for the type */, pol);
806 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
808 // determine the appropriate method, T1 ... T6
809 switch( meth[icode] )
812 val = owens_t_T1(h,a,m,pol);
815 typedef typename policies::precision<RealType, Policy>::type precision_type;
816 typedef std::integral_constant<bool, (precision_type::value == 0) || (precision_type::value > 64)> tag_type;
817 val = owens_t_T2(h, a, m, ah, pol, tag_type());
820 val = owens_t_T3(h,a,ah, pol);
823 val = owens_t_T4(h,a,m);
826 val = owens_t_T5(h,a, pol);
829 val = owens_t_T6(h,a, pol);
832 BOOST_MATH_THROW_EXCEPTION(std::logic_error("selection routine in Owen's T function failed"));
837 template<typename RealType, typename Policy>
838 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, const std::integral_constant<int, 65>&)
840 // Arbitrary precision version:
843 // Handle some special cases first, these are from
844 // page 1077 of Owen's original paper:
848 return atan(a) * constants::one_div_two_pi<RealType>();
856 return owens_t_znorm2(RealType(-h), pol) * owens_t_znorm2(h, pol) / 2;
858 if(a >= tools::max_value<RealType>())
860 return owens_t_znorm2(RealType(fabs(h)), pol);
862 // Attempt arbitrary precision code, this will throw if it goes wrong:
863 typedef typename boost::math::policies::normalise<Policy, boost::math::policies::evaluation_error<> >::type forwarding_policy;
864 std::pair<RealType, RealType> p1(0, tools::max_value<RealType>()), p2(0, tools::max_value<RealType>());
865 RealType target_precision = policies::get_epsilon<RealType, Policy>() * 1000;
866 bool have_t1(false), have_t2(false);
869 #ifndef BOOST_NO_EXCEPTIONS
874 p1 = owens_t_T1_accelerated(h, a, forwarding_policy());
875 if(p1.second < target_precision)
877 #ifndef BOOST_NO_EXCEPTIONS
879 catch(const boost::math::evaluation_error&){} // T1 may fail and throw, that's OK
884 #ifndef BOOST_NO_EXCEPTIONS
889 p2 = owens_t_T2_accelerated(h, a, ah, forwarding_policy());
890 if(p2.second < target_precision)
892 #ifndef BOOST_NO_EXCEPTIONS
894 catch(const boost::math::evaluation_error&){} // T2 may fail and throw, that's OK
898 // If we haven't tried T1 yet, do it now - sometimes it succeeds and the number of iterations
899 // is fairly low compared to T4.
903 #ifndef BOOST_NO_EXCEPTIONS
908 p1 = owens_t_T1_accelerated(h, a, forwarding_policy());
909 if(p1.second < target_precision)
911 #ifndef BOOST_NO_EXCEPTIONS
913 catch(const boost::math::evaluation_error&){} // T1 may fail and throw, that's OK
917 // If we haven't tried T2 yet, do it now - sometimes it succeeds and the number of iterations
918 // is fairly low compared to T4.
922 #ifndef BOOST_NO_EXCEPTIONS
927 p2 = owens_t_T2_accelerated(h, a, ah, forwarding_policy());
928 if(p2.second < target_precision)
930 #ifndef BOOST_NO_EXCEPTIONS
932 catch(const boost::math::evaluation_error&){} // T2 may fail and throw, that's OK
936 // OK, nothing left to do but try the most expensive option which is T4,
937 // this is often slow to converge, but when it does converge it tends to
939 #ifndef BOOST_NO_EXCEPTIONS
943 return T4_mp(h, a, pol);
944 #ifndef BOOST_NO_EXCEPTIONS
946 catch(const boost::math::evaluation_error&){} // T4 may fail and throw, that's OK
949 // Now look back at the results from T1 and T2 and see if either gave better
950 // results than we could get from the 64-bit precision versions.
952 if((std::min)(p1.second, p2.second) < 1e-20)
954 return p1.second < p2.second ? p1.first : p2.first;
957 // We give up - no arbitrary precision versions succeeded!
959 return owens_t_dispatch(h, a, ah, pol, std::integral_constant<int, 64>());
960 } // RealType owens_t_dispatch(RealType h, RealType a, RealType ah)
961 template<typename RealType, typename Policy>
962 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, const std::integral_constant<int, 0>&)
964 // We don't know what the precision is until runtime:
965 if(tools::digits<RealType>() <= 64)
966 return owens_t_dispatch(h, a, ah, pol, std::integral_constant<int, 64>());
967 return owens_t_dispatch(h, a, ah, pol, std::integral_constant<int, 65>());
969 template<typename RealType, typename Policy>
970 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol)
972 // Figure out the precision and forward to the correct version:
973 typedef typename policies::precision<RealType, Policy>::type precision_type;
974 typedef std::integral_constant<int,
975 precision_type::value <= 0 ? 0 :
976 precision_type::value <= 64 ? 64 : 65
979 return owens_t_dispatch(h, a, ah, pol, tag_type());
981 // compute Owen's T function, T(h,a), for arbitrary values of h and a
982 template<typename RealType, class Policy>
983 inline RealType owens_t(RealType h, RealType a, const Policy& pol)
986 // exploit that T(-h,a) == T(h,a)
989 // Use equation (2) in the paper to remap the arguments
990 // such that h>=0 and 0<=a<=1 for the call of the actual
991 // computation routine.
993 const RealType fabs_a = fabs(a);
994 const RealType fabs_ah = fabs_a*h;
996 RealType val = 0.0; // avoid compiler warnings, 0.0 will be overwritten in any case
1000 val = owens_t_dispatch(h, fabs_a, fabs_ah, pol);
1001 } // if(fabs_a <= 1.0)
1006 const RealType normh = owens_t_znorm1(h, pol);
1007 const RealType normah = owens_t_znorm1(fabs_ah, pol);
1008 val = static_cast<RealType>(1)/static_cast<RealType>(4) - normh*normah -
1009 owens_t_dispatch(fabs_ah, static_cast<RealType>(1 / fabs_a), h, pol);
1010 } // if( h <= 0.67 )
1013 const RealType normh = detail::owens_t_znorm2(h, pol);
1014 const RealType normah = detail::owens_t_znorm2(fabs_ah, pol);
1015 val = constants::half<RealType>()*(normh+normah) - normh*normah -
1016 owens_t_dispatch(fabs_ah, static_cast<RealType>(1 / fabs_a), h, pol);
1017 } // else [if( h <= 0.67 )]
1018 } // else [if(fabs_a <= 1)]
1020 // exploit that T(h,-a) == -T(h,a)
1027 } // RealType owens_t(RealType h, RealType a)
1029 template <class T, class Policy, class tag>
1030 struct owens_t_initializer
1039 static void do_init(const std::integral_constant<int, N>&){}
1040 static void do_init(const std::integral_constant<int, 64>&)
1042 boost::math::owens_t(static_cast<T>(7), static_cast<T>(0.96875), Policy());
1043 boost::math::owens_t(static_cast<T>(2), static_cast<T>(0.5), Policy());
1045 void force_instantiate()const{}
1047 static const init initializer;
1048 static void force_instantiate()
1050 initializer.force_instantiate();
1054 template <class T, class Policy, class tag>
1055 const typename owens_t_initializer<T, Policy, tag>::init owens_t_initializer<T, Policy, tag>::initializer;
1057 } // namespace detail
1059 template <class T1, class T2, class Policy>
1060 inline typename tools::promote_args<T1, T2>::type owens_t(T1 h, T2 a, const Policy& pol)
1062 typedef typename tools::promote_args<T1, T2>::type result_type;
1063 typedef typename policies::evaluation<result_type, Policy>::type value_type;
1064 typedef typename policies::precision<value_type, Policy>::type precision_type;
1065 typedef std::integral_constant<int,
1066 precision_type::value <= 0 ? 0 :
1067 precision_type::value <= 64 ? 64 : 65
1070 detail::owens_t_initializer<result_type, Policy, tag_type>::force_instantiate();
1072 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%)");
1075 template <class T1, class T2>
1076 inline typename tools::promote_args<T1, T2>::type owens_t(T1 h, T2 a)
1078 return owens_t(h, a, policies::policy<>());
1083 } // namespace boost
1086 #pragma warning(pop)