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/config/no_tr1/cmath.hpp>
21 #include <boost/math/special_functions/erf.hpp>
22 #include <boost/math/special_functions/expm1.hpp>
23 #include <boost/throw_exception.hpp>
24 #include <boost/assert.hpp>
25 #include <boost/math/constants/constants.hpp>
26 #include <boost/math/tools/big_constant.hpp>
32 #pragma warning(disable:4127)
41 // owens_t_znorm1(x) = P(-oo<Z<=x)-0.5 with Z being normally distributed.
42 template<typename RealType>
43 inline RealType owens_t_znorm1(const RealType x)
45 using namespace boost::math::constants;
46 return erf(x*one_div_root_two<RealType>())*half<RealType>();
47 } // RealType owens_t_znorm1(const RealType x)
49 // owens_t_znorm2(x) = P(x<=Z<oo) with Z being normally distributed.
50 template<typename RealType>
51 inline RealType owens_t_znorm2(const RealType x)
53 using namespace boost::math::constants;
54 return erfc(x*one_div_root_two<RealType>())*half<RealType>();
55 } // RealType owens_t_znorm2(const RealType x)
57 // Auxiliary function, it computes an array key that is used to determine
58 // the specific computation method for Owen's T and the order thereof
59 // used in owens_t_dispatch.
60 template<typename RealType>
61 inline unsigned short owens_t_compute_code(const RealType h, const RealType a)
63 static const RealType hrange[] =
64 { 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 };
66 static const RealType arange[] = { 0.025f, 0.09f, 0.15f, 0.36f, 0.5f, 0.9f, 0.99999f };
68 original select array from paper:
69 1, 1, 2,13,13,13,13,13,13,13,13,16,16,16, 9
70 1, 2, 2, 3, 3, 5, 5,14,14,15,15,16,16,16, 9
71 2, 2, 3, 3, 3, 5, 5,15,15,15,15,16,16,16,10
72 2, 2, 3, 5, 5, 5, 5, 7, 7,16,16,16,16,16,10
73 2, 3, 3, 5, 5, 6, 6, 8, 8,17,17,17,12,12,11
74 2, 3, 5, 5, 5, 6, 6, 8, 8,17,17,17,12,12,12
75 2, 3, 4, 4, 6, 6, 8, 8,17,17,17,17,17,12,12
76 2, 3, 4, 4, 6, 6,18,18,18,18,17,17,17,12,12
78 // subtract one because the array is written in FORTRAN in mind - in C arrays start @ zero
79 static const unsigned short select[] =
81 0, 0 , 1 , 12 ,12 , 12 , 12 , 12 , 12 , 12 , 12 , 15 , 15 , 15 , 8,
82 0 , 1 , 1 , 2 , 2 , 4 , 4 , 13 , 13 , 14 , 14 , 15 , 15 , 15 , 8,
83 1 , 1 , 2 , 2 , 2 , 4 , 4 , 14 , 14 , 14 , 14 , 15 , 15 , 15 , 9,
84 1 , 1 , 2 , 4 , 4 , 4 , 4 , 6 , 6 , 15 , 15 , 15 , 15 , 15 , 9,
85 1 , 2 , 2 , 4 , 4 , 5 , 5 , 7 , 7 , 16 ,16 , 16 , 11 , 11 , 10,
86 1 , 2 , 4 , 4 , 4 , 5 , 5 , 7 , 7 , 16 , 16 , 16 , 11 , 11 , 11,
87 1 , 2 , 3 , 3 , 5 , 5 , 7 , 7 , 16 , 16 , 16 , 16 , 16 , 11 , 11,
88 1 , 2 , 3 , 3 , 5 , 5 , 17 , 17 , 17 , 17 , 16 , 16 , 16 , 11 , 11
91 unsigned short ihint = 14, iaint = 7;
92 for(unsigned short i = 0; i != 14; i++)
99 } // for(unsigned short i = 0; i != 14; i++)
101 for(unsigned short i = 0; i != 7; i++)
108 } // for(unsigned short i = 0; i != 7; i++)
110 // interprete select array as 8x15 matrix
111 return select[iaint*15 + ihint];
113 } // unsigned short owens_t_compute_code(const RealType h, const RealType a)
115 template<typename RealType>
116 inline unsigned short owens_t_get_order_imp(const unsigned short icode, RealType, const mpl::int_<53>&)
118 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
120 BOOST_ASSERT(icode<18);
123 } // unsigned short owens_t_get_order(const unsigned short icode, RealType, mpl::int<53> const&)
125 template<typename RealType>
126 inline unsigned short owens_t_get_order_imp(const unsigned short icode, RealType, const mpl::int_<64>&)
128 // method ================>>> {1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6}
129 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
131 BOOST_ASSERT(icode<18);
134 } // unsigned short owens_t_get_order(const unsigned short icode, RealType, mpl::int<64> const&)
136 template<typename RealType, typename Policy>
137 inline unsigned short owens_t_get_order(const unsigned short icode, RealType r, const Policy&)
139 typedef typename policies::precision<RealType, Policy>::type precision_type;
140 typedef typename mpl::if_<
142 mpl::less_equal<precision_type, mpl::int_<0> >,
143 mpl::greater<precision_type, mpl::int_<53> >
149 return owens_t_get_order_imp(icode, r, tag_type());
152 // compute the value of Owen's T function with method T1 from the reference paper
153 template<typename RealType, typename Policy>
154 inline RealType owens_t_T1(const RealType h, const RealType a, const unsigned short m, const Policy& pol)
157 using namespace boost::math::constants;
159 const RealType hs = -h*h*half<RealType>();
160 const RealType dhs = exp( hs );
161 const RealType as = a*a;
165 RealType aj = a * one_div_two_pi<RealType>();
166 RealType dj = boost::math::expm1( hs, pol);
167 RealType gj = hs*dhs;
169 RealType val = atan( a ) * one_div_two_pi<RealType>();
179 jj += static_cast<RealType>(2);
182 gj *= hs / static_cast<RealType>(j);
186 } // RealType owens_t_T1(const RealType h, const RealType a, const unsigned short m)
188 // compute the value of Owen's T function with method T2 from the reference paper
189 template<typename RealType, class Policy>
190 inline RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah, const Policy&, const mpl::false_&)
193 using namespace boost::math::constants;
195 const unsigned short maxii = m+m+1;
196 const RealType hs = h*h;
197 const RealType as = -a*a;
198 const RealType y = static_cast<RealType>(1) / hs;
200 unsigned short ii = 1;
202 RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
203 RealType z = owens_t_znorm1(ah)/h;
210 val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
212 } // if( maxii <= ii )
213 z = y * ( vi - static_cast<RealType>(ii) * z );
219 } // RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah)
221 // compute the value of Owen's T function with method T3 from the reference paper
222 template<typename RealType>
223 inline RealType owens_t_T3_imp(const RealType h, const RealType a, const RealType ah, const mpl::int_<53>&)
226 using namespace boost::math::constants;
228 const unsigned short m = 20;
230 static const RealType c2[] =
232 static_cast<RealType>(0.99999999999999987510),
233 static_cast<RealType>(-0.99999999999988796462), static_cast<RealType>(0.99999999998290743652),
234 static_cast<RealType>(-0.99999999896282500134), static_cast<RealType>(0.99999996660459362918),
235 static_cast<RealType>(-0.99999933986272476760), static_cast<RealType>(0.99999125611136965852),
236 static_cast<RealType>(-0.99991777624463387686), static_cast<RealType>(0.99942835555870132569),
237 static_cast<RealType>(-0.99697311720723000295), static_cast<RealType>(0.98751448037275303682),
238 static_cast<RealType>(-0.95915857980572882813), static_cast<RealType>(0.89246305511006708555),
239 static_cast<RealType>(-0.76893425990463999675), static_cast<RealType>(0.58893528468484693250),
240 static_cast<RealType>(-0.38380345160440256652), static_cast<RealType>(0.20317601701045299653),
241 static_cast<RealType>(-0.82813631607004984866E-01), static_cast<RealType>(0.24167984735759576523E-01),
242 static_cast<RealType>(-0.44676566663971825242E-02), static_cast<RealType>(0.39141169402373836468E-03)
245 const RealType as = a*a;
246 const RealType hs = h*h;
247 const RealType y = static_cast<RealType>(1)/hs;
250 unsigned short i = 0;
251 RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
252 RealType zi = owens_t_znorm1(ah)/h;
257 BOOST_ASSERT(i < 21);
259 if( m <= i ) // if( m < i+1 )
261 val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
264 zi = y * (ii*zi - vi);
271 } // RealType owens_t_T3(const RealType h, const RealType a, const RealType ah)
273 // compute the value of Owen's T function with method T3 from the reference paper
274 template<class RealType>
275 inline RealType owens_t_T3_imp(const RealType h, const RealType a, const RealType ah, const mpl::int_<64>&)
278 using namespace boost::math::constants;
280 const unsigned short m = 30;
282 static const RealType c2[] =
284 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999999999999999729978162447266851932041876728736094298092917625009873),
285 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.99999999999999999999467056379678391810626533251885323416799874878563998732905968),
286 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999999999824849349313270659391127814689133077036298754586814091034842536),
287 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999999999997703859616213643405880166422891953033591551179153879839440241685),
288 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999998394883415238173334565554173013941245103172035286759201504179038147),
289 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999999993063616095509371081203145247992197457263066869044528823599399470977),
290 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999999999797336340409464429599229870590160411238245275855903767652432017766116267),
291 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.999999999574958412069046680119051639753412378037565521359444170241346845522403274),
292 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999999933226234193375324943920160947158239076786103108097456617750134812033362048),
293 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999188923242461073033481053037468263536806742737922476636768006622772762168467),
294 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999992195143483674402853783549420883055129680082932629160081128947764415749728967),
295 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.999993935137206712830997921913316971472227199741857386575097250553105958772041501),
296 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99996135597690552745362392866517133091672395614263398912807169603795088421057688716),
297 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.99979556366513946026406788969630293820987757758641211293079784585126692672425362469),
298 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.999092789629617100153486251423850590051366661947344315423226082520411961968929483),
299 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.996593837411918202119308620432614600338157335862888580671450938858935084316004769854),
300 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.98910017138386127038463510314625339359073956513420458166238478926511821146316469589567),
301 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.970078558040693314521331982203762771512160168582494513347846407314584943870399016019),
302 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.92911438683263187495758525500033707204091967947532160289872782771388170647150321633673),
303 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.8542058695956156057286980736842905011429254735181323743367879525470479126968822863),
304 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.73796526033030091233118357742803709382964420335559408722681794195743240930748630755),
305 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.58523469882837394570128599003785154144164680587615878645171632791404210655891158),
306 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.415997776145676306165661663581868460503874205343014196580122174949645271353372263),
307 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.2588210875241943574388730510317252236407805082485246378222935376279663808416534365),
308 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.1375535825163892648504646951500265585055789019410617565727090346559210218472356689),
309 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.0607952766325955730493900985022020434830339794955745989150270485056436844239206648),
310 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.0216337683299871528059836483840390514275488679530797294557060229266785853764115),
311 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.00593405693455186729876995814181203900550014220428843483927218267309209471516256),
312 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.0011743414818332946510474576182739210553333860106811865963485870668929503649964142),
313 BOOST_MATH_BIG_CONSTANT(RealType, 260, -1.489155613350368934073453260689881330166342484405529981510694514036264969925132e-4),
314 BOOST_MATH_BIG_CONSTANT(RealType, 260, 9.072354320794357587710929507988814669454281514268844884841547607134260303118208e-6)
317 const RealType as = a*a;
318 const RealType hs = h*h;
319 const RealType y = 1 / hs;
322 unsigned short i = 0;
323 RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
324 RealType zi = owens_t_znorm1(ah)/h;
329 BOOST_ASSERT(i < 31);
331 if( m <= i ) // if( m < i+1 )
333 val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
336 zi = y * (ii*zi - vi);
343 } // RealType owens_t_T3(const RealType h, const RealType a, const RealType ah)
345 template<class RealType, class Policy>
346 inline RealType owens_t_T3(const RealType h, const RealType a, const RealType ah, const Policy&)
348 typedef typename policies::precision<RealType, Policy>::type precision_type;
349 typedef typename mpl::if_<
351 mpl::less_equal<precision_type, mpl::int_<0> >,
352 mpl::greater<precision_type, mpl::int_<53> >
358 return owens_t_T3_imp(h, a, ah, tag_type());
361 // compute the value of Owen's T function with method T4 from the reference paper
362 template<typename RealType>
363 inline RealType owens_t_T4(const RealType h, const RealType a, const unsigned short m)
366 using namespace boost::math::constants;
368 const unsigned short maxii = m+m+1;
369 const RealType hs = h*h;
370 const RealType as = -a*a;
372 unsigned short ii = 1;
373 RealType ai = a * exp( -hs*(static_cast<RealType>(1)-as)*half<RealType>() ) * one_div_two_pi<RealType>();
383 yi = (static_cast<RealType>(1)-hs*yi) / static_cast<RealType>(ii);
388 } // RealType owens_t_T4(const RealType h, const RealType a, const unsigned short m)
390 // compute the value of Owen's T function with method T5 from the reference paper
391 template<typename RealType>
392 inline RealType owens_t_T5_imp(const RealType h, const RealType a, const mpl::int_<53>&)
397 - The pts[] array contains the squares (!) of the abscissas, i.e. the roots of the Legendre
398 polynomial P_n(x), instead of the plain roots as required in Gauss-Legendre
399 quadrature, because T5(h,a,m) contains only x^2 terms.
400 - The wts[] array contains the weights for Gauss-Legendre quadrature scaled with a factor
401 of 1/(2*pi) according to T5(h,a,m).
404 const unsigned short m = 13;
405 static const RealType pts[] = {
406 static_cast<RealType>(0.35082039676451715489E-02),
407 static_cast<RealType>(0.31279042338030753740E-01), static_cast<RealType>(0.85266826283219451090E-01),
408 static_cast<RealType>(0.16245071730812277011), static_cast<RealType>(0.25851196049125434828),
409 static_cast<RealType>(0.36807553840697533536), static_cast<RealType>(0.48501092905604697475),
410 static_cast<RealType>(0.60277514152618576821), static_cast<RealType>(0.71477884217753226516),
411 static_cast<RealType>(0.81475510988760098605), static_cast<RealType>(0.89711029755948965867),
412 static_cast<RealType>(0.95723808085944261843), static_cast<RealType>(0.99178832974629703586) };
413 static const RealType wts[] = {
414 static_cast<RealType>(0.18831438115323502887E-01),
415 static_cast<RealType>(0.18567086243977649478E-01), static_cast<RealType>(0.18042093461223385584E-01),
416 static_cast<RealType>(0.17263829606398753364E-01), static_cast<RealType>(0.16243219975989856730E-01),
417 static_cast<RealType>(0.14994592034116704829E-01), static_cast<RealType>(0.13535474469662088392E-01),
418 static_cast<RealType>(0.11886351605820165233E-01), static_cast<RealType>(0.10070377242777431897E-01),
419 static_cast<RealType>(0.81130545742299586629E-02), static_cast<RealType>(0.60419009528470238773E-02),
420 static_cast<RealType>(0.38862217010742057883E-02), static_cast<RealType>(0.16793031084546090448E-02) };
422 const RealType as = a*a;
423 const RealType hs = -h*h*boost::math::constants::half<RealType>();
426 for(unsigned short i = 0; i < m; ++i)
428 BOOST_ASSERT(i < 13);
429 const RealType r = static_cast<RealType>(1) + as*pts[i];
430 val += wts[i] * exp( hs*r ) / r;
431 } // for(unsigned short i = 0; i < m; ++i)
434 } // RealType owens_t_T5(const RealType h, const RealType a)
436 // compute the value of Owen's T function with method T5 from the reference paper
437 template<typename RealType>
438 inline RealType owens_t_T5_imp(const RealType h, const RealType a, const mpl::int_<64>&)
443 - The pts[] array contains the squares (!) of the abscissas, i.e. the roots of the Legendre
444 polynomial P_n(x), instead of the plain roots as required in Gauss-Legendre
445 quadrature, because T5(h,a,m) contains only x^2 terms.
446 - The wts[] array contains the weights for Gauss-Legendre quadrature scaled with a factor
447 of 1/(2*pi) according to T5(h,a,m).
450 const unsigned short m = 19;
451 static const RealType pts[] = {
452 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0016634282895983227941),
453 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.014904509242697054183),
454 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.04103478879005817919),
455 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.079359853513391511008),
456 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.1288612130237615133),
457 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.18822336642448518856),
458 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.25586876186122962384),
459 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.32999972011807857222),
460 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.40864620815774761438),
461 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.48971819306044782365),
462 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.57106118513245543894),
463 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.6505134942981533829),
464 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.72596367859928091618),
465 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.79540665919549865924),
466 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.85699701386308739244),
467 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.90909804422384697594),
468 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.95032536436570154409),
469 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.97958418733152273717),
470 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.99610366384229088321)
472 static const RealType wts[] = {
473 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012975111395684900835),
474 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012888764187499150078),
475 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012716644398857307844),
476 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012459897461364705691),
477 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012120231988292330388),
478 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.011699908404856841158),
479 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.011201723906897224448),
480 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.010628993848522759853),
481 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0099855296835573320047),
482 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0092756136096132857933),
483 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0085039700881139589055),
484 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0076757344408814561254),
485 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0067964187616556459109),
486 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.005871875456524750363),
487 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0049082589542498110071),
488 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0039119870792519721409),
489 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0028897090921170700834),
490 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0018483371329504443947),
491 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.00079623320100438873578)
494 const RealType as = a*a;
495 const RealType hs = -h*h*boost::math::constants::half<RealType>();
498 for(unsigned short i = 0; i < m; ++i)
500 BOOST_ASSERT(i < 19);
501 const RealType r = 1 + as*pts[i];
502 val += wts[i] * exp( hs*r ) / r;
503 } // for(unsigned short i = 0; i < m; ++i)
506 } // RealType owens_t_T5(const RealType h, const RealType a)
508 template<class RealType, class Policy>
509 inline RealType owens_t_T5(const RealType h, const RealType a, const Policy&)
511 typedef typename policies::precision<RealType, Policy>::type precision_type;
512 typedef typename mpl::if_<
514 mpl::less_equal<precision_type, mpl::int_<0> >,
515 mpl::greater<precision_type, mpl::int_<53> >
521 return owens_t_T5_imp(h, a, tag_type());
525 // compute the value of Owen's T function with method T6 from the reference paper
526 template<typename RealType>
527 inline RealType owens_t_T6(const RealType h, const RealType a)
530 using namespace boost::math::constants;
532 const RealType normh = owens_t_znorm2( h );
533 const RealType y = static_cast<RealType>(1) - a;
534 const RealType r = atan2(y, static_cast<RealType>(1 + a) );
536 RealType val = normh * ( static_cast<RealType>(1) - normh ) * half<RealType>();
539 val -= r * exp( -y*h*h*half<RealType>()/r ) * one_div_two_pi<RealType>();
542 } // RealType owens_t_T6(const RealType h, const RealType a, const unsigned short m)
544 template <class T, class Policy>
545 std::pair<T, T> owens_t_T1_accelerated(T h, T a, const Policy& pol)
548 // This is the same series as T1, but:
549 // * The Taylor series for atan has been combined with that for T1,
550 // reducing but not eliminating cancellation error.
551 // * The resulting alternating series is then accelerated using method 1
552 // from H. Cohen, F. Rodriguez Villegas, D. Zagier,
553 // "Convergence acceleration of alternating series", Bonn, (1991).
556 static const char* function = "boost::math::owens_t<%1%>(%1%, %1%)";
557 T half_h_h = h * h / 2;
560 T exp_term = exp(-h * h / 2);
561 T one_minus_dj_sum = exp_term;
562 T sum = a_pow * exp_term;
569 // Normally with this form of series acceleration we can calculate
570 // up front how many terms will be required - based on the assumption
571 // that each term decreases in size by a factor of 3. However,
572 // that assumption does not apply here, as the underlying T1 series can
573 // go quite strongly divergent in the early terms, before strongly
574 // converging later. Various "guestimates" have been tried to take account
575 // of this, but they don't always work.... so instead set "n" to the
576 // largest value that won't cause overflow later, and abort iteration
577 // when the last accelerated term was small enough...
580 #ifndef BOOST_NO_EXCEPTIONS
584 n = itrunc(T(tools::log_max_value<T>() / 6));
585 #ifndef BOOST_NO_EXCEPTIONS
589 n = (std::numeric_limits<int>::max)();
592 n = (std::min)(n, 1500);
593 T d = pow(3 + sqrt(T(8)), n);
600 abs_err = ldexp(fabs(sum), -tools::digits<T>());
605 dj_pow *= half_h_h / j;
606 one_minus_dj_sum += dj_pow;
607 term = one_minus_dj_sum * a_pow / (2 * j + 1);
610 abs_err += ldexp((std::max)(T(fabs(sum)), T(fabs(c*term))), -tools::digits<T>());
611 b = (j + n) * (j - n) * b / ((j + T(0.5)) * (j + 1));
614 // Include an escape route to prevent calculating too many terms:
616 if((j > 10) && (fabs(sum * tools::epsilon<T>()) > fabs(c * term)))
619 abs_err += fabs(c * term);
620 if(sum < 0) // sum must always be positive, if it's negative something really bad has happend:
621 policies::raise_evaluation_error(function, 0, T(0), pol);
622 return std::pair<T, T>((sum / d) / boost::math::constants::two_pi<T>(), abs_err / sum);
625 template<typename RealType, class Policy>
626 inline RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah, const Policy&, const mpl::true_&)
629 using namespace boost::math::constants;
631 const unsigned short maxii = m+m+1;
632 const RealType hs = h*h;
633 const RealType as = -a*a;
634 const RealType y = static_cast<RealType>(1) / hs;
636 unsigned short ii = 1;
638 RealType vi = a * exp( -ah*ah*half<RealType>() ) / root_two_pi<RealType>();
639 RealType z = owens_t_znorm1(ah)/h;
640 RealType last_z = fabs(z);
641 RealType lim = policies::get_epsilon<RealType, Policy>();
647 // This series stops converging after a while, so put a limit
648 // on how far we go before returning our best guess:
650 if((fabs(lim * val) > fabs(z)) || ((ii > maxii) && (fabs(z) > last_z)) || (z == 0))
652 val *= exp( -hs*half<RealType>() ) / root_two_pi<RealType>();
654 } // if( maxii <= ii )
656 z = y * ( vi - static_cast<RealType>(ii) * z );
662 } // RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah)
664 template<typename RealType, class Policy>
665 inline std::pair<RealType, RealType> owens_t_T2_accelerated(const RealType h, const RealType a, const RealType ah, const Policy&)
668 // This is the same series as T2, but with acceleration applied.
669 // Note that we have to be *very* careful to check that nothing bad
670 // has happened during evaluation - this series will go divergent
671 // and/or fail to alternate at a drop of a hat! :-(
674 using namespace boost::math::constants;
676 const RealType hs = h*h;
677 const RealType as = -a*a;
678 const RealType y = static_cast<RealType>(1) / hs;
680 unsigned short ii = 1;
682 RealType vi = a * exp( -ah*ah*half<RealType>() ) / root_two_pi<RealType>();
683 RealType z = boost::math::detail::owens_t_znorm1(ah)/h;
684 RealType last_z = fabs(z);
687 // Normally with this form of series acceleration we can calculate
688 // up front how many terms will be required - based on the assumption
689 // that each term decreases in size by a factor of 3. However,
690 // that assumption does not apply here, as the underlying T1 series can
691 // go quite strongly divergent in the early terms, before strongly
692 // converging later. Various "guestimates" have been tried to take account
693 // of this, but they don't always work.... so instead set "n" to the
694 // largest value that won't cause overflow later, and abort iteration
695 // when the last accelerated term was small enough...
698 #ifndef BOOST_NO_EXCEPTIONS
702 n = itrunc(RealType(tools::log_max_value<RealType>() / 6));
703 #ifndef BOOST_NO_EXCEPTIONS
707 n = (std::numeric_limits<int>::max)();
710 n = (std::min)(n, 1500);
711 RealType d = pow(3 + sqrt(RealType(8)), n);
717 for(int k = 0; k < n; ++k)
720 // Check for both convergence and whether the series has gone bad:
723 (fabs(z) > last_z) // Series has gone divergent, abort
724 || (fabs(val) * tools::epsilon<RealType>() > fabs(c * s * z)) // Convergence!
725 || (z * s < 0) // Series has stopped alternating - all bets are off - abort.
732 b = (k + n) * (k - n) * b / ((k + RealType(0.5)) * (k + 1));
735 z = y * ( vi - static_cast<RealType>(ii) * z );
739 RealType err = fabs(c * z) / val;
740 return std::pair<RealType, RealType>(val * exp( -hs*half<RealType>() ) / (d * root_two_pi<RealType>()), err);
741 } // RealType owens_t_T2_accelerated(const RealType h, const RealType a, const RealType ah, const Policy&)
743 template<typename RealType, typename Policy>
744 inline RealType T4_mp(const RealType h, const RealType a, const Policy& pol)
748 const RealType hs = h*h;
749 const RealType as = -a*a;
751 unsigned short ii = 1;
752 RealType ai = constants::one_div_two_pi<RealType>() * a * exp( -0.5*hs*(1.0-as) );
756 RealType lim = boost::math::policies::get_epsilon<RealType, Policy>();
760 RealType term = ai*yi;
762 if((yi != 0) && (fabs(val * lim) > fabs(term)))
765 yi = (1.0-hs*yi) / static_cast<RealType>(ii);
767 if(ii > (std::min)(1500, (int)policies::get_max_series_iterations<Policy>()))
768 policies::raise_evaluation_error("boost::math::owens_t<%1%>", 0, val, pol);
772 } // arg_type owens_t_T4(const arg_type h, const arg_type a, const unsigned short m)
775 // This routine dispatches the call to one of six subroutines, depending on the values
777 // preconditions: h >= 0, 0<=a<=1, ah=a*h
779 // Note there are different versions for different precisions....
780 template<typename RealType, typename Policy>
781 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, mpl::int_<64> const&)
783 // Simple main case for 64-bit precision or less, this is as per the Patefield-Tandy paper:
786 // Handle some special cases first, these are from
787 // page 1077 of Owen's original paper:
791 return atan(a) * constants::one_div_two_pi<RealType>();
799 return owens_t_znorm2(RealType(-h)) * owens_t_znorm2(h) / 2;
801 if(a >= tools::max_value<RealType>())
803 return owens_t_znorm2(RealType(fabs(h)));
805 RealType val = 0; // avoid compiler warnings, 0 will be overwritten in any case
806 const unsigned short icode = owens_t_compute_code(h, a);
807 const unsigned short m = owens_t_get_order(icode, val /* just a dummy for the type */, pol);
808 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
810 // determine the appropriate method, T1 ... T6
811 switch( meth[icode] )
814 val = owens_t_T1(h,a,m,pol);
817 typedef typename policies::precision<RealType, Policy>::type precision_type;
818 typedef mpl::bool_<(precision_type::value == 0) || (precision_type::value > 64)> tag_type;
819 val = owens_t_T2(h, a, m, ah, pol, tag_type());
822 val = owens_t_T3(h,a,ah, pol);
825 val = owens_t_T4(h,a,m);
828 val = owens_t_T5(h,a, pol);
831 val = owens_t_T6(h,a);
834 BOOST_THROW_EXCEPTION(std::logic_error("selection routine in Owen's T function failed"));
839 template<typename RealType, typename Policy>
840 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, const mpl::int_<65>&)
842 // Arbitrary precision version:
845 // Handle some special cases first, these are from
846 // page 1077 of Owen's original paper:
850 return atan(a) * constants::one_div_two_pi<RealType>();
858 return owens_t_znorm2(RealType(-h)) * owens_t_znorm2(h) / 2;
860 if(a >= tools::max_value<RealType>())
862 return owens_t_znorm2(RealType(fabs(h)));
864 // Attempt arbitrary precision code, this will throw if it goes wrong:
865 typedef typename boost::math::policies::normalise<Policy, boost::math::policies::evaluation_error<> >::type forwarding_policy;
866 std::pair<RealType, RealType> p1(0, tools::max_value<RealType>()), p2(0, tools::max_value<RealType>());
867 RealType target_precision = policies::get_epsilon<RealType, Policy>() * 1000;
868 bool have_t1(false), have_t2(false);
871 #ifndef BOOST_NO_EXCEPTIONS
876 p1 = owens_t_T1_accelerated(h, a, forwarding_policy());
877 if(p1.second < target_precision)
879 #ifndef BOOST_NO_EXCEPTIONS
881 catch(const boost::math::evaluation_error&){} // T1 may fail and throw, that's OK
886 #ifndef BOOST_NO_EXCEPTIONS
891 p2 = owens_t_T2_accelerated(h, a, ah, forwarding_policy());
892 if(p2.second < target_precision)
894 #ifndef BOOST_NO_EXCEPTIONS
896 catch(const boost::math::evaluation_error&){} // T2 may fail and throw, that's OK
900 // If we haven't tried T1 yet, do it now - sometimes it succeeds and the number of iterations
901 // is fairly low compared to T4.
905 #ifndef BOOST_NO_EXCEPTIONS
910 p1 = owens_t_T1_accelerated(h, a, forwarding_policy());
911 if(p1.second < target_precision)
913 #ifndef BOOST_NO_EXCEPTIONS
915 catch(const boost::math::evaluation_error&){} // T1 may fail and throw, that's OK
919 // If we haven't tried T2 yet, do it now - sometimes it succeeds and the number of iterations
920 // is fairly low compared to T4.
924 #ifndef BOOST_NO_EXCEPTIONS
929 p2 = owens_t_T2_accelerated(h, a, ah, forwarding_policy());
930 if(p2.second < target_precision)
932 #ifndef BOOST_NO_EXCEPTIONS
934 catch(const boost::math::evaluation_error&){} // T2 may fail and throw, that's OK
938 // OK, nothing left to do but try the most expensive option which is T4,
939 // this is often slow to converge, but when it does converge it tends to
941 #ifndef BOOST_NO_EXCEPTIONS
945 return T4_mp(h, a, pol);
946 #ifndef BOOST_NO_EXCEPTIONS
948 catch(const boost::math::evaluation_error&){} // T4 may fail and throw, that's OK
951 // Now look back at the results from T1 and T2 and see if either gave better
952 // results than we could get from the 64-bit precision versions.
954 if((std::min)(p1.second, p2.second) < 1e-20)
956 return p1.second < p2.second ? p1.first : p2.first;
959 // We give up - no arbitrary precision versions succeeded!
961 return owens_t_dispatch(h, a, ah, pol, mpl::int_<64>());
962 } // RealType owens_t_dispatch(RealType h, RealType a, RealType ah)
963 template<typename RealType, typename Policy>
964 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, const mpl::int_<0>&)
966 // We don't know what the precision is until runtime:
967 if(tools::digits<RealType>() <= 64)
968 return owens_t_dispatch(h, a, ah, pol, mpl::int_<64>());
969 return owens_t_dispatch(h, a, ah, pol, mpl::int_<65>());
971 template<typename RealType, typename Policy>
972 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol)
974 // Figure out the precision and forward to the correct version:
975 typedef typename policies::precision<RealType, Policy>::type precision_type;
976 typedef typename mpl::if_c<
977 precision_type::value == 0,
980 precision_type::value <= 64,
985 return owens_t_dispatch(h, a, ah, pol, tag_type());
987 // compute Owen's T function, T(h,a), for arbitrary values of h and a
988 template<typename RealType, class Policy>
989 inline RealType owens_t(RealType h, RealType a, const Policy& pol)
992 // exploit that T(-h,a) == T(h,a)
995 // Use equation (2) in the paper to remap the arguments
996 // such that h>=0 and 0<=a<=1 for the call of the actual
997 // computation routine.
999 const RealType fabs_a = fabs(a);
1000 const RealType fabs_ah = fabs_a*h;
1002 RealType val = 0.0; // avoid compiler warnings, 0.0 will be overwritten in any case
1006 val = owens_t_dispatch(h, fabs_a, fabs_ah, pol);
1007 } // if(fabs_a <= 1.0)
1012 const RealType normh = owens_t_znorm1(h);
1013 const RealType normah = owens_t_znorm1(fabs_ah);
1014 val = static_cast<RealType>(1)/static_cast<RealType>(4) - normh*normah -
1015 owens_t_dispatch(fabs_ah, static_cast<RealType>(1 / fabs_a), h, pol);
1016 } // if( h <= 0.67 )
1019 const RealType normh = detail::owens_t_znorm2(h);
1020 const RealType normah = detail::owens_t_znorm2(fabs_ah);
1021 val = constants::half<RealType>()*(normh+normah) - normh*normah -
1022 owens_t_dispatch(fabs_ah, static_cast<RealType>(1 / fabs_a), h, pol);
1023 } // else [if( h <= 0.67 )]
1024 } // else [if(fabs_a <= 1)]
1026 // exploit that T(h,-a) == -T(h,a)
1033 } // RealType owens_t(RealType h, RealType a)
1035 template <class T, class Policy, class tag>
1036 struct owens_t_initializer
1045 static void do_init(const mpl::int_<N>&){}
1046 static void do_init(const mpl::int_<64>&)
1048 boost::math::owens_t(static_cast<T>(7), static_cast<T>(0.96875), Policy());
1049 boost::math::owens_t(static_cast<T>(2), static_cast<T>(0.5), Policy());
1051 void force_instantiate()const{}
1053 static const init initializer;
1054 static void force_instantiate()
1056 initializer.force_instantiate();
1060 template <class T, class Policy, class tag>
1061 const typename owens_t_initializer<T, Policy, tag>::init owens_t_initializer<T, Policy, tag>::initializer;
1063 } // namespace detail
1065 template <class T1, class T2, class Policy>
1066 inline typename tools::promote_args<T1, T2>::type owens_t(T1 h, T2 a, const Policy& pol)
1068 typedef typename tools::promote_args<T1, T2>::type result_type;
1069 typedef typename policies::evaluation<result_type, Policy>::type value_type;
1070 typedef typename policies::precision<value_type, Policy>::type precision_type;
1071 typedef typename mpl::if_c<
1072 precision_type::value == 0,
1075 precision_type::value <= 64,
1081 detail::owens_t_initializer<result_type, Policy, tag_type>::force_instantiate();
1083 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%)");
1086 template <class T1, class T2>
1087 inline typename tools::promote_args<T1, T2>::type owens_t(T1 h, T2 a)
1089 return owens_t(h, a, policies::policy<>());
1094 } // namespace boost
1097 #pragma warning(pop)