1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
|
// Copyright (C) Benjamin Sobotta 2012
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0. (See accompanying file
// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
#ifndef BOOST_OWENS_T7_HPP
#define BOOST_OWENS_T7_HPP
// Reference:
// Mike Patefield, David Tandy
// FAST AND ACCURATE CALCULATION OF OWEN'S T-FUNCTION
// Journal of Statistical Software, 5 (5), 1-25
#ifdef _MSC_VER
# pragma once
#endif
#include <boost/math/special_functions/owens_t.hpp>
#include <vector>
#include <algorithm> // std::sort
#include <boost/bind.hpp>
#include <boost/bind/placeholders.hpp>
namespace boost
{
namespace math
{
namespace detail
{
template<typename ndx_type, typename data_type>
inline bool owens_t_sort_proxy(const ndx_type n1, const ndx_type n2, const data_type* data) { return (fabs(data[n1]) < fabs(data[n2])); }
// compute the value of Owen's T function with method T7 from the reference paper
template<typename RealType>
inline RealType compute_owens_t_T7(const RealType h, const RealType a)
{
BOOST_MATH_STD_USING
using namespace boost::math::constants;
std::vector<RealType> c2(51);
c2[0]=static_cast<RealType>(0.99999999999999999999999999999999999999868061520482714021759648020474317574987355923862368638038320094484952302876219e0l);
c2[1]=static_cast<RealType>(-0.99999999999999999999999999999999999302462011069054837422425677393793571720981603447930035285521866357878934009159708e0l);
c2[2]=static_cast<RealType>(0.9999999999999999999999999999999938559078263892797131217517182153540396811776160269133372859096210993958152102616265e0l);
c2[3]=static_cast<RealType>(-0.99999999999999999999999999999783757316880746021776281028307338046033395960167168255107746699076325159707117075122598e0l);
c2[4]=static_cast<RealType>(0.99999999999999999999999999959301304409247897363805160078169883190960201161021089063082244377785297302604378738778956e0l);
c2[5]=static_cast<RealType>(-0.99999999999999999999999995245835998362821388118964869664453661644738440917895538883200648244745024569685062708634168e0l);
c2[6]=static_cast<RealType>(0.99999999999999999999999622575084004275114986560329977419595335828482080658911071010175083376987487699799620433312026e0l);
c2[7]=static_cast<RealType>(-0.9999999999999999999997835456196743116277756586095640965807421017047019205312795102010127835367920317933410096951991e0l);
c2[8]=static_cast<RealType>(0.99999999999999999999063044410000386518700820850762955399774821040632831953277713720592510531609301137092442411088318e0l);
c2[9]=static_cast<RealType>(-0.99999999999999999968363056215229124110523314446131391646528327924749429301803531868618645140379927991069880419895362e0l);
c2[10]=static_cast<RealType>(0.99999999999999999145103710789254646427253519282003979608764987525701449150796674610676803428471393828460151017134006e0l);
c2[11]=static_cast<RealType>(-0.99999999999999981131055812853212338067344201452850197384362641710824830755936071166619436310647545618172487913130615e0l);
c2[12]=static_cast<RealType>(0.99999999999999654123165670138354353746969688832625358460636033129592270928035522003120035894014007078745281902561497e0l);
c2[13]=static_cast<RealType>(-0.99999999999994661543772980917477218111196118287898714561621502747335071313305657131504975341280407503235841894095032e0l);
c2[14]=static_cast<RealType>(0.9999999999992981439043273655170176041716503383856785977723125483146102600097712876017125231262708400720094556926045e0l);
c2[15]=static_cast<RealType>(-0.99999999999206265150568436336459788968621075287629010127877385312281458164262755824589807456902954264068398140350718e0l);
c2[16]=static_cast<RealType>(0.99999999992213502163371840879683818667748180414402080554120502891144910024410028330131981216472217247119418222397804e0l);
c2[17]=static_cast<RealType>(-0.99999999933259095924055744571071653706200038213621273711937864999491782183629569409353668088363952580491517608609879e0l);
c2[18]=static_cast<RealType>(0.99999999497015572069563854257687102391713999264054303202862355260549032899913629304955062400529548430606517184769124e0l);
c2[19]=static_cast<RealType>(-0.99999996648712934179852410529668328351470567694495362454934194838547830786603251986748376180326858512474939517328148e0l);
c2[20]=static_cast<RealType>(0.9999998016410566689368417618988041312715506400990955108339485167214078601378203496694653725800913994771609966243554e0l);
c2[21]=static_cast<RealType>(-0.9999989525985184659277351038817797931250021614576921513415321341983217728506335369734522548408960126473437428356652e0l);
c2[22]=static_cast<RealType>(0.99999504755520008311742537842275606593998257673570772153723704887194777830960070883905373411740901379060307967971713e0l);
c2[23]=static_cast<RealType>(-0.99997896168242457050517756812692812892733179258193505494454091938132703728342085231042113297420877668525201728601112e0l);
c2[24]=static_cast<RealType>(0.99991946779137118276149990348538316553731391341330104178195711109963439425910633064752492021322249165299888263409172e0l);
c2[25]=static_cast<RealType>(-0.99972148778885346568502104725962502476360296866194523161605564314646459144266349095901801601276421334075117884929242e0l);
c2[26]=static_cast<RealType>(0.9991276859682010382270359712512423427555252098639083241851998143073842159537070734558612169871757287024196969316524e0l);
c2[27]=static_cast<RealType>(-0.99752026434927262148166061506294447561314364149201350129770769406693673101306930831078789512571673723645189614485884e0l);
c2[28]=static_cast<RealType>(0.99358896672458664010385299342636899693001298318272470058323214841807825867323354755879957618925036849552478542814464e0l);
c2[29]=static_cast<RealType>(-0.98489603971030687415560911591294255813647866616337974199082847337212210737029861747833396359832470961032052017929136e0l);
c2[30]=static_cast<RealType>(0.96751068500593933205520070783164495637686016604265460899685963694355146172759956614360799629964537023691986327884468e0l);
c2[31]=static_cast<RealType>(-0.9360616426988023756922055734494146109996789121506654176278348120618330748262465556674776006234661241542758473118521e0l);
c2[32]=static_cast<RealType>(0.88462086639047762401913298378165508003735156726358522187746603832270096575743334309836954108840694198036767569315108e0l);
c2[33]=static_cast<RealType>(-0.80858877429268840241019894355701184332631739713578890873529014128055757110890871538802016131209054476205332440695604e0l);
c2[34]=static_cast<RealType>(0.70714608761601125771088731886413274896589865282669140675007048926200481310909899767964678381587391145800655465120571e0l);
c2[35]=static_cast<RealType>(-0.58515160223005410186594841297741982997622522778974325847124400853371098492516978008742019478185592841561858134639963e0l);
c2[36]=static_cast<RealType>(0.45317050454923571830339270216379220523529541036307428556285489535205431482111178945999365703295821120700802922701552e0l);
c2[37]=static_cast<RealType>(-0.32503812067644347537685664861030147676006324710175782617307839344555807653091746007670995202936607469695700327677892e0l);
c2[38]=static_cast<RealType>(0.2137534704367539657335116107238861128574908030994099713290249568609752278177667873985860406515292692728315916731017e0l);
c2[39]=static_cast<RealType>(-0.127619403804624293099162792279930050338998502678126917236608595894255012173888208013041488822409228894596510380150144e0l);
c2[40]=static_cast<RealType>(0.06848690630890464573940500271274680442209417344598560366267896451733825978600890476338760194302261371849816636335383e0l);
c2[41]=static_cast<RealType>(-0.032688749970534932162217917343517016881654494605298904939793624525513795530795862058337162798487051632431756396211348e0l);
c2[42]=static_cast<RealType>(0.013715287119044125499391779954972556538977518079926978123084819229658358422347583377635796490450010646568919625741353e0l);
c2[43]=static_cast<RealType>(-0.0049902360281804772302071182896105260677340309700627303663654639589340705520269439056577683691783146342849984775533766e0l);
c2[44]=static_cast<RealType>(0.00154866713536508358511616306566119586713663676744267610952381384231395851333350529444375690335427250570956017759427897e0l);
c2[45]=static_cast<RealType>(-4.0136222792839484462604207322390667014747359362667770129433998717936089480817561247228048128811298286479443203785308e-4l);
c2[46]=static_cast<RealType>(8.4429279202769263712337394035686562275089641577318336414305767736129198590671750045859928722396139236629554169757073e-5l);
c2[47]=static_cast<RealType>(-1.3837048406451028003397758396690378745165135180030260294919960014069368993089617158234925766366971794969129152002478e-5l);
c2[48]=static_cast<RealType>(1.6563766752141156889761471523635738102614710503912266484040958175597362783384628269667016371235475899745552923626272e-6l);
c2[49]=static_cast<RealType>(-1.2874491982958001084373112367555214480376326465550959775329113928120230317885310174458065240563391789639012991976545e-7l);
c2[50]=static_cast<RealType>(4.8740773419420086658863239448043342637632445767226310886251599519774455542302728072876112682268516221911975181091336e-9l);
const RealType hs = h*h;
const RealType as = a*a;
//RealType u = one_div_two_pi<RealType>() * a * exp(-half<RealType>()*hs*(static_cast<RealType>(1)+as));
RealType v = c2[0];
RealType val = v;
RealType last_val = val+1; // last_val must not be the same as val
int k = 0;
std::vector<RealType> memory;
memory.push_back(v);
while((val != last_val) || (k<51)) // use all c2
{
last_val = val;
k++;
const RealType u = pow(as,k);
if(k < static_cast<int>(c2.size()))
{
v = (hs*v + c2[k])/(static_cast<RealType>(2*k+1));
}
else
{
// assume that for numerical purposes c2[k]==0
v = hs*v/(static_cast<RealType>(2*k+1));
}
val += u*v;
if(val >= tools::max_value<RealType>())
break;
memory.push_back(u*v);
} // while(val != last_val)
std::vector<unsigned> ndx_4_sorted_data(memory.size());
for(unsigned i = 0; i != memory.size(); ++i)
{
ndx_4_sorted_data[i]=i;
}
std::sort(ndx_4_sorted_data.begin(), ndx_4_sorted_data.end(),
boost::bind(owens_t_sort_proxy<typename std::vector<RealType>::size_type, RealType>, _1, _2, &memory[0]));
val = memory[ndx_4_sorted_data[0]];
for(unsigned i = 1; i != memory.size(); i++)
{
val+=memory[ndx_4_sorted_data[i]];
}
// split the exponential to avoid values that go below the minimum floating pt value
val *= exp(-half<RealType>()*hs*as);
val *= exp(-half<RealType>()*hs);
val *= one_div_two_pi<RealType>() * a;
return val;
} // RealType compute_owens_t_T7(const RealType h, const RealType a)
// compute Owen's T function, T(h,a), for arbitrary values of h and a
template<typename RealType, class Policy>
inline RealType owens_t_T7(RealType h, RealType a, const Policy&)
{
BOOST_MATH_STD_USING
// exploit that T(-h,a) == T(h,a)
h = fabs(h);
// Use equation (2) in the paper to remap the arguments
// such that h>=0 and 0<=a<=1 for the call of the actual
// computation routine.
const RealType fabs_a = fabs(a);
const RealType fabs_ah = fabs_a*h;
RealType val = 0.0; // avoid compiler warnings, 0.0 will be overwritten in any case
if(fabs_a <= 1)
{
val = compute_owens_t_T7(h, fabs_a);
} // if(fabs_a <= 1.0)
else
{
if( h <= 0.67 )
{
const RealType normh = owens_t_znorm1(h);
const RealType normah = owens_t_znorm1(fabs_ah);
val = static_cast<RealType>(1)/static_cast<RealType>(4) - normh*normah -
compute_owens_t_T7(fabs_ah, static_cast<RealType>(1 / fabs_a));
} // if( h <= 0.67 )
else
{
const RealType normh = owens_t_znorm2(h);
const RealType normah = owens_t_znorm2(fabs_ah);
val = constants::half<RealType>()*(normh+normah) - normh*normah -
compute_owens_t_T7(fabs_ah, static_cast<RealType>(1 / fabs_a));
} // else [if( h <= 0.67 )]
} // else [if(fabs_a <= 1)]
// exploit that T(h,-a) == -T(h,a)
if(a < 0)
{
return -val;
} // if(a < 0)
return val;
} // RealType owens_t(RealType h, RealType a)
} // namespace detail
template <class T1, class T2, class Policy>
inline typename tools::promote_args<T1, T2>::type owens_t_T7(T1 h, T2 a, const Policy& pol)
{
typedef typename tools::promote_args<T1, T2>::type result_type;
typedef typename policies::evaluation<result_type, Policy>::type value_type;
return policies::checked_narrowing_cast<result_type, Policy>(detail::owens_t_T7(static_cast<value_type>(h), static_cast<value_type>(a), pol), "boost::math::owens_t_T7<%1%>(%1%,%1%)");
}
template <class T1, class T2>
inline typename tools::promote_args<T1, T2>::type owens_t_T7(T1 h, T2 a)
{
return owens_t_T7(h, a, policies::policy<>());
}
} // namespace math
} // namespace boost
#endif
|
