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/config/no_tr1/cmath.hpp>
20 #include <boost/math/special_functions/erf.hpp>
21 #include <boost/math/special_functions/expm1.hpp>
22 #include <boost/throw_exception.hpp>
23 #include <boost/assert.hpp>
24 #include <boost/math/constants/constants.hpp>
25 #include <boost/math/tools/big_constant.hpp>
35 // owens_t_znorm1(x) = P(-oo<Z<=x)-0.5 with Z being normally distributed.
36 template<typename RealType>
37 inline RealType owens_t_znorm1(const RealType x)
39 using namespace boost::math::constants;
40 return erf(x*one_div_root_two<RealType>())*half<RealType>();
41 } // RealType owens_t_znorm1(const RealType x)
43 // owens_t_znorm2(x) = P(x<=Z<oo) with Z being normally distributed.
44 template<typename RealType>
45 inline RealType owens_t_znorm2(const RealType x)
47 using namespace boost::math::constants;
48 return erfc(x*one_div_root_two<RealType>())*half<RealType>();
49 } // RealType owens_t_znorm2(const RealType x)
51 // Auxiliary function, it computes an array key that is used to determine
52 // the specific computation method for Owen's T and the order thereof
53 // used in owens_t_dispatch.
54 template<typename RealType>
55 inline unsigned short owens_t_compute_code(const RealType h, const RealType a)
57 static const RealType hrange[] =
58 {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};
60 static const RealType arange[] = {0.025, 0.09, 0.15, 0.36, 0.5, 0.9, 0.99999};
62 original select array from paper:
63 1, 1, 2,13,13,13,13,13,13,13,13,16,16,16, 9
64 1, 2, 2, 3, 3, 5, 5,14,14,15,15,16,16,16, 9
65 2, 2, 3, 3, 3, 5, 5,15,15,15,15,16,16,16,10
66 2, 2, 3, 5, 5, 5, 5, 7, 7,16,16,16,16,16,10
67 2, 3, 3, 5, 5, 6, 6, 8, 8,17,17,17,12,12,11
68 2, 3, 5, 5, 5, 6, 6, 8, 8,17,17,17,12,12,12
69 2, 3, 4, 4, 6, 6, 8, 8,17,17,17,17,17,12,12
70 2, 3, 4, 4, 6, 6,18,18,18,18,17,17,17,12,12
72 // subtract one because the array is written in FORTRAN in mind - in C arrays start @ zero
73 static const unsigned short select[] =
75 0, 0 , 1 , 12 ,12 , 12 , 12 , 12 , 12 , 12 , 12 , 15 , 15 , 15 , 8,
76 0 , 1 , 1 , 2 , 2 , 4 , 4 , 13 , 13 , 14 , 14 , 15 , 15 , 15 , 8,
77 1 , 1 , 2 , 2 , 2 , 4 , 4 , 14 , 14 , 14 , 14 , 15 , 15 , 15 , 9,
78 1 , 1 , 2 , 4 , 4 , 4 , 4 , 6 , 6 , 15 , 15 , 15 , 15 , 15 , 9,
79 1 , 2 , 2 , 4 , 4 , 5 , 5 , 7 , 7 , 16 ,16 , 16 , 11 , 11 , 10,
80 1 , 2 , 4 , 4 , 4 , 5 , 5 , 7 , 7 , 16 , 16 , 16 , 11 , 11 , 11,
81 1 , 2 , 3 , 3 , 5 , 5 , 7 , 7 , 16 , 16 , 16 , 16 , 16 , 11 , 11,
82 1 , 2 , 3 , 3 , 5 , 5 , 17 , 17 , 17 , 17 , 16 , 16 , 16 , 11 , 11
85 unsigned short ihint = 14, iaint = 7;
86 for(unsigned short i = 0; i != 14; i++)
93 } // for(unsigned short i = 0; i != 14; i++)
95 for(unsigned short i = 0; i != 7; i++)
102 } // for(unsigned short i = 0; i != 7; i++)
104 // interprete select array as 8x15 matrix
105 return select[iaint*15 + ihint];
107 } // unsigned short owens_t_compute_code(const RealType h, const RealType a)
109 template<typename RealType>
110 inline unsigned short owens_t_get_order_imp(const unsigned short icode, RealType, const mpl::int_<53>&)
112 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
114 BOOST_ASSERT(icode<18);
117 } // unsigned short owens_t_get_order(const unsigned short icode, RealType, mpl::int<53> const&)
119 template<typename RealType>
120 inline unsigned short owens_t_get_order_imp(const unsigned short icode, RealType, const mpl::int_<64>&)
122 // method ================>>> {1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6}
123 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
125 BOOST_ASSERT(icode<18);
128 } // unsigned short owens_t_get_order(const unsigned short icode, RealType, mpl::int<64> const&)
130 template<typename RealType, typename Policy>
131 inline unsigned short owens_t_get_order(const unsigned short icode, RealType r, const Policy&)
133 typedef typename policies::precision<RealType, Policy>::type precision_type;
134 typedef typename mpl::if_<
136 mpl::less_equal<precision_type, mpl::int_<0> >,
137 mpl::greater<precision_type, mpl::int_<53> >
143 return owens_t_get_order_imp(icode, r, tag_type());
146 // compute the value of Owen's T function with method T1 from the reference paper
147 template<typename RealType>
148 inline RealType owens_t_T1(const RealType h, const RealType a, const unsigned short m)
151 using namespace boost::math::constants;
153 const RealType hs = -h*h*half<RealType>();
154 const RealType dhs = exp( hs );
155 const RealType as = a*a;
159 RealType aj = a * one_div_two_pi<RealType>();
160 RealType dj = expm1( hs );
161 RealType gj = hs*dhs;
163 RealType val = atan( a ) * one_div_two_pi<RealType>();
173 jj += static_cast<RealType>(2);
176 gj *= hs / static_cast<RealType>(j);
180 } // RealType owens_t_T1(const RealType h, const RealType a, const unsigned short m)
182 // compute the value of Owen's T function with method T2 from the reference paper
183 template<typename RealType, class Policy>
184 inline RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah, const Policy&, const mpl::false_&)
187 using namespace boost::math::constants;
189 const unsigned short maxii = m+m+1;
190 const RealType hs = h*h;
191 const RealType as = -a*a;
192 const RealType y = static_cast<RealType>(1) / hs;
194 unsigned short ii = 1;
196 RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
197 RealType z = owens_t_znorm1(ah)/h;
204 val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
206 } // if( maxii <= ii )
207 z = y * ( vi - static_cast<RealType>(ii) * z );
213 } // RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah)
215 // compute the value of Owen's T function with method T3 from the reference paper
216 template<typename RealType>
217 inline RealType owens_t_T3_imp(const RealType h, const RealType a, const RealType ah, const mpl::int_<53>&)
220 using namespace boost::math::constants;
222 const unsigned short m = 20;
224 static const RealType c2[] =
226 0.99999999999999987510,
227 -0.99999999999988796462, 0.99999999998290743652,
228 -0.99999999896282500134, 0.99999996660459362918,
229 -0.99999933986272476760, 0.99999125611136965852,
230 -0.99991777624463387686, 0.99942835555870132569,
231 -0.99697311720723000295, 0.98751448037275303682,
232 -0.95915857980572882813, 0.89246305511006708555,
233 -0.76893425990463999675, 0.58893528468484693250,
234 -0.38380345160440256652, 0.20317601701045299653,
235 -0.82813631607004984866E-01, 0.24167984735759576523E-01,
236 -0.44676566663971825242E-02, 0.39141169402373836468E-03
239 const RealType as = a*a;
240 const RealType hs = h*h;
241 const RealType y = static_cast<RealType>(1)/hs;
244 unsigned short i = 0;
245 RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
246 RealType zi = owens_t_znorm1(ah)/h;
251 BOOST_ASSERT(i < 21);
253 if( m <= i ) // if( m < i+1 )
255 val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
258 zi = y * (ii*zi - vi);
265 } // RealType owens_t_T3(const RealType h, const RealType a, const RealType ah)
267 // compute the value of Owen's T function with method T3 from the reference paper
268 template<class RealType>
269 inline RealType owens_t_T3_imp(const RealType h, const RealType a, const RealType ah, const mpl::int_<64>&)
272 using namespace boost::math::constants;
274 const unsigned short m = 30;
276 static const RealType c2[] =
278 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999999999999999729978162447266851932041876728736094298092917625009873),
279 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.99999999999999999999467056379678391810626533251885323416799874878563998732905968),
280 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999999999824849349313270659391127814689133077036298754586814091034842536),
281 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999999999997703859616213643405880166422891953033591551179153879839440241685),
282 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99999999999998394883415238173334565554173013941245103172035286759201504179038147),
283 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999999993063616095509371081203145247992197457263066869044528823599399470977),
284 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999999999797336340409464429599229870590160411238245275855903767652432017766116267),
285 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.999999999574958412069046680119051639753412378037565521359444170241346845522403274),
286 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999999933226234193375324943920160947158239076786103108097456617750134812033362048),
287 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.9999999188923242461073033481053037468263536806742737922476636768006622772762168467),
288 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.9999992195143483674402853783549420883055129680082932629160081128947764415749728967),
289 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.999993935137206712830997921913316971472227199741857386575097250553105958772041501),
290 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.99996135597690552745362392866517133091672395614263398912807169603795088421057688716),
291 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.99979556366513946026406788969630293820987757758641211293079784585126692672425362469),
292 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.999092789629617100153486251423850590051366661947344315423226082520411961968929483),
293 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.996593837411918202119308620432614600338157335862888580671450938858935084316004769854),
294 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.98910017138386127038463510314625339359073956513420458166238478926511821146316469589567),
295 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.970078558040693314521331982203762771512160168582494513347846407314584943870399016019),
296 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.92911438683263187495758525500033707204091967947532160289872782771388170647150321633673),
297 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.8542058695956156057286980736842905011429254735181323743367879525470479126968822863),
298 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.73796526033030091233118357742803709382964420335559408722681794195743240930748630755),
299 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.58523469882837394570128599003785154144164680587615878645171632791404210655891158),
300 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.415997776145676306165661663581868460503874205343014196580122174949645271353372263),
301 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.2588210875241943574388730510317252236407805082485246378222935376279663808416534365),
302 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.1375535825163892648504646951500265585055789019410617565727090346559210218472356689),
303 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.0607952766325955730493900985022020434830339794955745989150270485056436844239206648),
304 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.0216337683299871528059836483840390514275488679530797294557060229266785853764115),
305 BOOST_MATH_BIG_CONSTANT(RealType, 260, -0.00593405693455186729876995814181203900550014220428843483927218267309209471516256),
306 BOOST_MATH_BIG_CONSTANT(RealType, 260, 0.0011743414818332946510474576182739210553333860106811865963485870668929503649964142),
307 BOOST_MATH_BIG_CONSTANT(RealType, 260, -1.489155613350368934073453260689881330166342484405529981510694514036264969925132e-4),
308 BOOST_MATH_BIG_CONSTANT(RealType, 260, 9.072354320794357587710929507988814669454281514268844884841547607134260303118208e-6)
311 const RealType as = a*a;
312 const RealType hs = h*h;
313 const RealType y = 1 / hs;
316 unsigned short i = 0;
317 RealType vi = a * exp( -ah*ah*half<RealType>() ) * one_div_root_two_pi<RealType>();
318 RealType zi = owens_t_znorm1(ah)/h;
323 BOOST_ASSERT(i < 31);
325 if( m <= i ) // if( m < i+1 )
327 val *= exp( -hs*half<RealType>() ) * one_div_root_two_pi<RealType>();
330 zi = y * (ii*zi - vi);
337 } // RealType owens_t_T3(const RealType h, const RealType a, const RealType ah)
339 template<class RealType, class Policy>
340 inline RealType owens_t_T3(const RealType h, const RealType a, const RealType ah, const Policy&)
342 typedef typename policies::precision<RealType, Policy>::type precision_type;
343 typedef typename mpl::if_<
345 mpl::less_equal<precision_type, mpl::int_<0> >,
346 mpl::greater<precision_type, mpl::int_<53> >
352 return owens_t_T3_imp(h, a, ah, tag_type());
355 // compute the value of Owen's T function with method T4 from the reference paper
356 template<typename RealType>
357 inline RealType owens_t_T4(const RealType h, const RealType a, const unsigned short m)
360 using namespace boost::math::constants;
362 const unsigned short maxii = m+m+1;
363 const RealType hs = h*h;
364 const RealType as = -a*a;
366 unsigned short ii = 1;
367 RealType ai = a * exp( -hs*(static_cast<RealType>(1)-as)*half<RealType>() ) * one_div_two_pi<RealType>();
377 yi = (static_cast<RealType>(1)-hs*yi) / static_cast<RealType>(ii);
382 } // RealType owens_t_T4(const RealType h, const RealType a, const unsigned short m)
384 // compute the value of Owen's T function with method T5 from the reference paper
385 template<typename RealType>
386 inline RealType owens_t_T5_imp(const RealType h, const RealType a, const mpl::int_<53>&)
391 - The pts[] array contains the squares (!) of the abscissas, i.e. the roots of the Legendre
392 polynomial P_n(x), instead of the plain roots as required in Gauss-Legendre
393 quadrature, because T5(h,a,m) contains only x^2 terms.
394 - The wts[] array contains the weights for Gauss-Legendre quadrature scaled with a factor
395 of 1/(2*pi) according to T5(h,a,m).
398 const unsigned short m = 13;
399 static const RealType pts[] = {0.35082039676451715489E-02,
400 0.31279042338030753740E-01, 0.85266826283219451090E-01,
401 0.16245071730812277011, 0.25851196049125434828,
402 0.36807553840697533536, 0.48501092905604697475,
403 0.60277514152618576821, 0.71477884217753226516,
404 0.81475510988760098605, 0.89711029755948965867,
405 0.95723808085944261843, 0.99178832974629703586};
406 static const RealType wts[] = { 0.18831438115323502887E-01,
407 0.18567086243977649478E-01, 0.18042093461223385584E-01,
408 0.17263829606398753364E-01, 0.16243219975989856730E-01,
409 0.14994592034116704829E-01, 0.13535474469662088392E-01,
410 0.11886351605820165233E-01, 0.10070377242777431897E-01,
411 0.81130545742299586629E-02, 0.60419009528470238773E-02,
412 0.38862217010742057883E-02, 0.16793031084546090448E-02};
414 const RealType as = a*a;
415 const RealType hs = -h*h*boost::math::constants::half<RealType>();
418 for(unsigned short i = 0; i < m; ++i)
420 BOOST_ASSERT(i < 13);
421 const RealType r = static_cast<RealType>(1) + as*pts[i];
422 val += wts[i] * exp( hs*r ) / r;
423 } // for(unsigned short i = 0; i < m; ++i)
426 } // RealType owens_t_T5(const RealType h, const RealType a)
428 // compute the value of Owen's T function with method T5 from the reference paper
429 template<typename RealType>
430 inline RealType owens_t_T5_imp(const RealType h, const RealType a, const mpl::int_<64>&)
435 - The pts[] array contains the squares (!) of the abscissas, i.e. the roots of the Legendre
436 polynomial P_n(x), instead of the plain roots as required in Gauss-Legendre
437 quadrature, because T5(h,a,m) contains only x^2 terms.
438 - The wts[] array contains the weights for Gauss-Legendre quadrature scaled with a factor
439 of 1/(2*pi) according to T5(h,a,m).
442 const unsigned short m = 19;
443 static const RealType pts[] = {
444 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0016634282895983227941),
445 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.014904509242697054183),
446 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.04103478879005817919),
447 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.079359853513391511008),
448 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.1288612130237615133),
449 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.18822336642448518856),
450 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.25586876186122962384),
451 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.32999972011807857222),
452 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.40864620815774761438),
453 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.48971819306044782365),
454 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.57106118513245543894),
455 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.6505134942981533829),
456 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.72596367859928091618),
457 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.79540665919549865924),
458 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.85699701386308739244),
459 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.90909804422384697594),
460 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.95032536436570154409),
461 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.97958418733152273717),
462 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.99610366384229088321)
464 static const RealType wts[] = {
465 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012975111395684900835),
466 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012888764187499150078),
467 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012716644398857307844),
468 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012459897461364705691),
469 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.012120231988292330388),
470 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.011699908404856841158),
471 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.011201723906897224448),
472 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.010628993848522759853),
473 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0099855296835573320047),
474 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0092756136096132857933),
475 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0085039700881139589055),
476 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0076757344408814561254),
477 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0067964187616556459109),
478 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.005871875456524750363),
479 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0049082589542498110071),
480 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0039119870792519721409),
481 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0028897090921170700834),
482 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.0018483371329504443947),
483 BOOST_MATH_BIG_CONSTANT(RealType, 64, 0.00079623320100438873578)
486 const RealType as = a*a;
487 const RealType hs = -h*h*boost::math::constants::half<RealType>();
490 for(unsigned short i = 0; i < m; ++i)
492 BOOST_ASSERT(i < 19);
493 const RealType r = 1 + as*pts[i];
494 val += wts[i] * exp( hs*r ) / r;
495 } // for(unsigned short i = 0; i < m; ++i)
498 } // RealType owens_t_T5(const RealType h, const RealType a)
500 template<class RealType, class Policy>
501 inline RealType owens_t_T5(const RealType h, const RealType a, const Policy&)
503 typedef typename policies::precision<RealType, Policy>::type precision_type;
504 typedef typename mpl::if_<
506 mpl::less_equal<precision_type, mpl::int_<0> >,
507 mpl::greater<precision_type, mpl::int_<53> >
513 return owens_t_T5_imp(h, a, tag_type());
517 // compute the value of Owen's T function with method T6 from the reference paper
518 template<typename RealType>
519 inline RealType owens_t_T6(const RealType h, const RealType a)
522 using namespace boost::math::constants;
524 const RealType normh = owens_t_znorm2( h );
525 const RealType y = static_cast<RealType>(1) - a;
526 const RealType r = atan2(y, static_cast<RealType>(1 + a) );
528 RealType val = normh * ( static_cast<RealType>(1) - normh ) * half<RealType>();
531 val -= r * exp( -y*h*h*half<RealType>()/r ) * one_div_two_pi<RealType>();
534 } // RealType owens_t_T6(const RealType h, const RealType a, const unsigned short m)
536 template <class T, class Policy>
537 std::pair<T, T> owens_t_T1_accelerated(T h, T a, const Policy& pol)
540 // This is the same series as T1, but:
541 // * The Taylor series for atan has been combined with that for T1,
542 // reducing but not eliminating cancellation error.
543 // * The resulting alternating series is then accelerated using method 1
544 // from H. Cohen, F. Rodriguez Villegas, D. Zagier,
545 // "Convergence acceleration of alternating series", Bonn, (1991).
548 static const char* function = "boost::math::owens_t<%1%>(%1%, %1%)";
549 T half_h_h = h * h / 2;
552 T exp_term = exp(-h * h / 2);
553 T one_minus_dj_sum = exp_term;
554 T sum = a_pow * exp_term;
561 // Normally with this form of series acceleration we can calculate
562 // up front how many terms will be required - based on the assumption
563 // that each term decreases in size by a factor of 3. However,
564 // that assumption does not apply here, as the underlying T1 series can
565 // go quite strongly divergent in the early terms, before strongly
566 // converging later. Various "guestimates" have been tried to take account
567 // of this, but they don't always work.... so instead set "n" to the
568 // largest value that won't cause overflow later, and abort iteration
569 // when the last accelerated term was small enough...
574 n = itrunc(T(tools::log_max_value<T>() / 6));
578 n = (std::numeric_limits<int>::max)();
580 n = (std::min)(n, 1500);
581 T d = pow(3 + sqrt(T(8)), n);
588 abs_err = ldexp(fabs(sum), -tools::digits<T>());
593 dj_pow *= half_h_h / j;
594 one_minus_dj_sum += dj_pow;
595 term = one_minus_dj_sum * a_pow / (2 * j + 1);
598 abs_err += ldexp((std::max)(T(fabs(sum)), T(fabs(c*term))), -tools::digits<T>());
599 b = (j + n) * (j - n) * b / ((j + T(0.5)) * (j + 1));
602 // Include an escape route to prevent calculating too many terms:
604 if((j > 10) && (fabs(sum * tools::epsilon<T>()) > fabs(c * term)))
607 abs_err += fabs(c * term);
608 if(sum < 0) // sum must always be positive, if it's negative something really bad has happend:
609 policies::raise_evaluation_error(function, 0, T(0), pol);
610 return std::pair<T, T>((sum / d) / boost::math::constants::two_pi<T>(), abs_err / sum);
613 template<typename RealType, class Policy>
614 inline RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah, const Policy&, const mpl::true_&)
617 using namespace boost::math::constants;
619 const unsigned short maxii = m+m+1;
620 const RealType hs = h*h;
621 const RealType as = -a*a;
622 const RealType y = static_cast<RealType>(1) / hs;
624 unsigned short ii = 1;
626 RealType vi = a * exp( -ah*ah*half<RealType>() ) / root_two_pi<RealType>();
627 RealType z = owens_t_znorm1(ah)/h;
628 RealType last_z = fabs(z);
629 RealType lim = policies::get_epsilon<RealType, Policy>();
635 // This series stops converging after a while, so put a limit
636 // on how far we go before returning our best guess:
638 if((fabs(lim * val) > fabs(z)) || ((ii > maxii) && (fabs(z) > last_z)) || (z == 0))
640 val *= exp( -hs*half<RealType>() ) / root_two_pi<RealType>();
642 } // if( maxii <= ii )
644 z = y * ( vi - static_cast<RealType>(ii) * z );
650 } // RealType owens_t_T2(const RealType h, const RealType a, const unsigned short m, const RealType ah)
652 template<typename RealType, class Policy>
653 inline std::pair<RealType, RealType> owens_t_T2_accelerated(const RealType h, const RealType a, const RealType ah, const Policy&)
656 // This is the same series as T2, but with acceleration applied.
657 // Note that we have to be *very* careful to check that nothing bad
658 // has happened during evaluation - this series will go divergent
659 // and/or fail to alternate at a drop of a hat! :-(
662 using namespace boost::math::constants;
664 const RealType hs = h*h;
665 const RealType as = -a*a;
666 const RealType y = static_cast<RealType>(1) / hs;
668 unsigned short ii = 1;
670 RealType vi = a * exp( -ah*ah*half<RealType>() ) / root_two_pi<RealType>();
671 RealType z = boost::math::detail::owens_t_znorm1(ah)/h;
672 RealType last_z = fabs(z);
675 // Normally with this form of series acceleration we can calculate
676 // up front how many terms will be required - based on the assumption
677 // that each term decreases in size by a factor of 3. However,
678 // that assumption does not apply here, as the underlying T1 series can
679 // go quite strongly divergent in the early terms, before strongly
680 // converging later. Various "guestimates" have been tried to take account
681 // of this, but they don't always work.... so instead set "n" to the
682 // largest value that won't cause overflow later, and abort iteration
683 // when the last accelerated term was small enough...
688 n = itrunc(RealType(tools::log_max_value<RealType>() / 6));
692 n = (std::numeric_limits<int>::max)();
694 n = (std::min)(n, 1500);
695 RealType d = pow(3 + sqrt(RealType(8)), n);
701 for(int k = 0; k < n; ++k)
704 // Check for both convergence and whether the series has gone bad:
707 (fabs(z) > last_z) // Series has gone divergent, abort
708 || (fabs(val) * tools::epsilon<RealType>() > fabs(c * s * z)) // Convergence!
709 || (z * s < 0) // Series has stopped alternating - all bets are off - abort.
716 b = (k + n) * (k - n) * b / ((k + RealType(0.5)) * (k + 1));
719 z = y * ( vi - static_cast<RealType>(ii) * z );
723 RealType err = fabs(c * z) / val;
724 return std::pair<RealType, RealType>(val * exp( -hs*half<RealType>() ) / (d * root_two_pi<RealType>()), err);
725 } // RealType owens_t_T2_accelerated(const RealType h, const RealType a, const RealType ah, const Policy&)
727 template<typename RealType, typename Policy>
728 inline RealType T4_mp(const RealType h, const RealType a, const Policy& pol)
732 const RealType hs = h*h;
733 const RealType as = -a*a;
735 unsigned short ii = 1;
736 RealType ai = constants::one_div_two_pi<RealType>() * a * exp( -0.5*hs*(1.0-as) );
740 RealType lim = boost::math::policies::get_epsilon<RealType, Policy>();
744 RealType term = ai*yi;
746 if((yi != 0) && (fabs(val * lim) > fabs(term)))
749 yi = (1.0-hs*yi) / static_cast<RealType>(ii);
751 if(ii > (std::min)(1500, (int)policies::get_max_series_iterations<Policy>()))
752 policies::raise_evaluation_error("boost::math::owens_t<%1%>", 0, val, pol);
756 } // arg_type owens_t_T4(const arg_type h, const arg_type a, const unsigned short m)
759 // This routine dispatches the call to one of six subroutines, depending on the values
761 // preconditions: h >= 0, 0<=a<=1, ah=a*h
763 // Note there are different versions for different precisions....
764 template<typename RealType, typename Policy>
765 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, mpl::int_<64> const&)
767 // Simple main case for 64-bit precision or less, this is as per the Patefield-Tandy paper:
770 // Handle some special cases first, these are from
771 // page 1077 of Owen's original paper:
775 return atan(a) * constants::one_div_two_pi<RealType>();
783 return owens_t_znorm2(RealType(-h)) * owens_t_znorm2(h) / 2;
785 if(a >= tools::max_value<RealType>())
787 return owens_t_znorm2(RealType(fabs(h)));
789 RealType val = 0; // avoid compiler warnings, 0 will be overwritten in any case
790 const unsigned short icode = owens_t_compute_code(h, a);
791 const unsigned short m = owens_t_get_order(icode, val /* just a dummy for the type */, pol);
792 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
794 // determine the appropriate method, T1 ... T6
795 switch( meth[icode] )
798 val = owens_t_T1(h,a,m);
801 typedef typename policies::precision<RealType, Policy>::type precision_type;
802 typedef mpl::bool_<(precision_type::value == 0) || (precision_type::value > 64)> tag_type;
803 val = owens_t_T2(h, a, m, ah, pol, tag_type());
806 val = owens_t_T3(h,a,ah, pol);
809 val = owens_t_T4(h,a,m);
812 val = owens_t_T5(h,a, pol);
815 val = owens_t_T6(h,a);
818 BOOST_THROW_EXCEPTION(std::logic_error("selection routine in Owen's T function failed"));
823 template<typename RealType, typename Policy>
824 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, const mpl::int_<65>&)
826 // Arbitrary precision version:
829 // Handle some special cases first, these are from
830 // page 1077 of Owen's original paper:
834 return atan(a) * constants::one_div_two_pi<RealType>();
842 return owens_t_znorm2(RealType(-h)) * owens_t_znorm2(h) / 2;
844 if(a >= tools::max_value<RealType>())
846 return owens_t_znorm2(RealType(fabs(h)));
848 // Attempt arbitrary precision code, this will throw if it goes wrong:
849 typedef typename boost::math::policies::normalise<Policy, boost::math::policies::evaluation_error<> >::type forwarding_policy;
850 std::pair<RealType, RealType> p1(0, tools::max_value<RealType>()), p2(0, tools::max_value<RealType>());
851 RealType target_precision = policies::get_epsilon<RealType, Policy>() * 1000;
852 bool have_t1(false), have_t2(false);
858 p1 = owens_t_T1_accelerated(h, a, forwarding_policy());
859 if(p1.second < target_precision)
862 catch(const boost::math::evaluation_error&){} // T1 may fail and throw, that's OK
869 p2 = owens_t_T2_accelerated(h, a, ah, forwarding_policy());
870 if(p2.second < target_precision)
873 catch(const boost::math::evaluation_error&){} // T2 may fail and throw, that's OK
876 // If we haven't tried T1 yet, do it now - sometimes it succeeds and the number of iterations
877 // is fairly low compared to T4.
884 p1 = owens_t_T1_accelerated(h, a, forwarding_policy());
885 if(p1.second < target_precision)
888 catch(const boost::math::evaluation_error&){} // T1 may fail and throw, that's OK
891 // If we haven't tried T2 yet, do it now - sometimes it succeeds and the number of iterations
892 // is fairly low compared to T4.
899 p2 = owens_t_T2_accelerated(h, a, ah, forwarding_policy());
900 if(p2.second < target_precision)
903 catch(const boost::math::evaluation_error&){} // T2 may fail and throw, that's OK
906 // OK, nothing left to do but try the most expensive option which is T4,
907 // this is often slow to converge, but when it does converge it tends to
911 return T4_mp(h, a, pol);
913 catch(const boost::math::evaluation_error&){} // T4 may fail and throw, that's OK
915 // Now look back at the results from T1 and T2 and see if either gave better
916 // results than we could get from the 64-bit precision versions.
918 if((std::min)(p1.second, p2.second) < 1e-20)
920 return p1.second < p2.second ? p1.first : p2.first;
923 // We give up - no arbitrary precision versions succeeded!
925 return owens_t_dispatch(h, a, ah, pol, mpl::int_<64>());
926 } // RealType owens_t_dispatch(RealType h, RealType a, RealType ah)
927 template<typename RealType, typename Policy>
928 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol, const mpl::int_<0>&)
930 // We don't know what the precision is until runtime:
931 if(tools::digits<RealType>() <= 64)
932 return owens_t_dispatch(h, a, ah, pol, mpl::int_<64>());
933 return owens_t_dispatch(h, a, ah, pol, mpl::int_<65>());
935 template<typename RealType, typename Policy>
936 inline RealType owens_t_dispatch(const RealType h, const RealType a, const RealType ah, const Policy& pol)
938 // Figure out the precision and forward to the correct version:
939 typedef typename policies::precision<RealType, Policy>::type precision_type;
940 typedef typename mpl::if_c<
941 precision_type::value == 0,
944 precision_type::value <= 64,
949 return owens_t_dispatch(h, a, ah, pol, tag_type());
951 // compute Owen's T function, T(h,a), for arbitrary values of h and a
952 template<typename RealType, class Policy>
953 inline RealType owens_t(RealType h, RealType a, const Policy& pol)
956 // exploit that T(-h,a) == T(h,a)
959 // Use equation (2) in the paper to remap the arguments
960 // such that h>=0 and 0<=a<=1 for the call of the actual
961 // computation routine.
963 const RealType fabs_a = fabs(a);
964 const RealType fabs_ah = fabs_a*h;
966 RealType val = 0.0; // avoid compiler warnings, 0.0 will be overwritten in any case
970 val = owens_t_dispatch(h, fabs_a, fabs_ah, pol);
971 } // if(fabs_a <= 1.0)
976 const RealType normh = owens_t_znorm1(h);
977 const RealType normah = owens_t_znorm1(fabs_ah);
978 val = static_cast<RealType>(1)/static_cast<RealType>(4) - normh*normah -
979 owens_t_dispatch(fabs_ah, static_cast<RealType>(1 / fabs_a), h, pol);
983 const RealType normh = detail::owens_t_znorm2(h);
984 const RealType normah = detail::owens_t_znorm2(fabs_ah);
985 val = constants::half<RealType>()*(normh+normah) - normh*normah -
986 owens_t_dispatch(fabs_ah, static_cast<RealType>(1 / fabs_a), h, pol);
987 } // else [if( h <= 0.67 )]
988 } // else [if(fabs_a <= 1)]
990 // exploit that T(h,-a) == -T(h,a)
997 } // RealType owens_t(RealType h, RealType a)
999 template <class T, class Policy, class tag>
1000 struct owens_t_initializer
1009 static void do_init(const mpl::int_<N>&){}
1010 static void do_init(const mpl::int_<64>&)
1012 boost::math::owens_t(static_cast<T>(7), static_cast<T>(0.96875), Policy());
1013 boost::math::owens_t(static_cast<T>(2), static_cast<T>(0.5), Policy());
1015 void force_instantiate()const{}
1017 static const init initializer;
1018 static void force_instantiate()
1020 initializer.force_instantiate();
1024 template <class T, class Policy, class tag>
1025 const typename owens_t_initializer<T, Policy, tag>::init owens_t_initializer<T, Policy, tag>::initializer;
1027 } // namespace detail
1029 template <class T1, class T2, class Policy>
1030 inline typename tools::promote_args<T1, T2>::type owens_t(T1 h, T2 a, const Policy& pol)
1032 typedef typename tools::promote_args<T1, T2>::type result_type;
1033 typedef typename policies::evaluation<result_type, Policy>::type value_type;
1034 typedef typename policies::precision<value_type, Policy>::type precision_type;
1035 typedef typename mpl::if_c<
1036 precision_type::value == 0,
1039 precision_type::value <= 64,
1045 detail::owens_t_initializer<result_type, Policy, tag_type>::force_instantiate();
1047 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%)");
1050 template <class T1, class T2>
1051 inline typename tools::promote_args<T1, T2>::type owens_t(T1 h, T2 a)
1053 return owens_t(h, a, policies::policy<>());
1058 } // namespace boost