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1 | // Copyright John Maddock 2007. |
2 | // Copyright Paul A. Bristow 2007 | |
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) | |
6 | ||
7 | #ifndef BOOST_MATH_SF_DETAIL_INV_T_HPP | |
8 | #define BOOST_MATH_SF_DETAIL_INV_T_HPP | |
9 | ||
10 | #ifdef _MSC_VER | |
11 | #pragma once | |
12 | #endif | |
13 | ||
14 | #include <boost/math/special_functions/cbrt.hpp> | |
15 | #include <boost/math/special_functions/round.hpp> | |
16 | #include <boost/math/special_functions/trunc.hpp> | |
17 | ||
18 | namespace boost{ namespace math{ namespace detail{ | |
19 | ||
20 | // | |
21 | // The main method used is due to Hill: | |
22 | // | |
23 | // G. W. Hill, Algorithm 396, Student's t-Quantiles, | |
24 | // Communications of the ACM, 13(10): 619-620, Oct., 1970. | |
25 | // | |
26 | template <class T, class Policy> | |
27 | T inverse_students_t_hill(T ndf, T u, const Policy& pol) | |
28 | { | |
29 | BOOST_MATH_STD_USING | |
30 | BOOST_ASSERT(u <= 0.5); | |
31 | ||
32 | T a, b, c, d, q, x, y; | |
33 | ||
34 | if (ndf > 1e20f) | |
35 | return -boost::math::erfc_inv(2 * u, pol) * constants::root_two<T>(); | |
36 | ||
37 | a = 1 / (ndf - 0.5f); | |
38 | b = 48 / (a * a); | |
39 | c = ((20700 * a / b - 98) * a - 16) * a + 96.36f; | |
40 | d = ((94.5f / (b + c) - 3) / b + 1) * sqrt(a * constants::pi<T>() / 2) * ndf; | |
41 | y = pow(d * 2 * u, 2 / ndf); | |
42 | ||
43 | if (y > (0.05f + a)) | |
44 | { | |
45 | // | |
46 | // Asymptotic inverse expansion about normal: | |
47 | // | |
48 | x = -boost::math::erfc_inv(2 * u, pol) * constants::root_two<T>(); | |
49 | y = x * x; | |
50 | ||
51 | if (ndf < 5) | |
52 | c += 0.3f * (ndf - 4.5f) * (x + 0.6f); | |
53 | c += (((0.05f * d * x - 5) * x - 7) * x - 2) * x + b; | |
54 | y = (((((0.4f * y + 6.3f) * y + 36) * y + 94.5f) / c - y - 3) / b + 1) * x; | |
55 | y = boost::math::expm1(a * y * y, pol); | |
56 | } | |
57 | else | |
58 | { | |
59 | y = static_cast<T>(((1 / (((ndf + 6) / (ndf * y) - 0.089f * d - 0.822f) | |
60 | * (ndf + 2) * 3) + 0.5 / (ndf + 4)) * y - 1) | |
61 | * (ndf + 1) / (ndf + 2) + 1 / y); | |
62 | } | |
63 | q = sqrt(ndf * y); | |
64 | ||
65 | return -q; | |
66 | } | |
67 | // | |
68 | // Tail and body series are due to Shaw: | |
69 | // | |
70 | // www.mth.kcl.ac.uk/~shaww/web_page/papers/Tdistribution06.pdf | |
71 | // | |
72 | // Shaw, W.T., 2006, "Sampling Student's T distribution - use of | |
73 | // the inverse cumulative distribution function." | |
74 | // Journal of Computational Finance, Vol 9 Issue 4, pp 37-73, Summer 2006 | |
75 | // | |
76 | template <class T, class Policy> | |
77 | T inverse_students_t_tail_series(T df, T v, const Policy& pol) | |
78 | { | |
79 | BOOST_MATH_STD_USING | |
80 | // Tail series expansion, see section 6 of Shaw's paper. | |
81 | // w is calculated using Eq 60: | |
82 | T w = boost::math::tgamma_delta_ratio(df / 2, constants::half<T>(), pol) | |
83 | * sqrt(df * constants::pi<T>()) * v; | |
84 | // define some variables: | |
85 | T np2 = df + 2; | |
86 | T np4 = df + 4; | |
87 | T np6 = df + 6; | |
88 | // | |
89 | // Calculate the coefficients d(k), these depend only on the | |
90 | // number of degrees of freedom df, so at least in theory | |
91 | // we could tabulate these for fixed df, see p15 of Shaw: | |
92 | // | |
93 | T d[7] = { 1, }; | |
94 | d[1] = -(df + 1) / (2 * np2); | |
95 | np2 *= (df + 2); | |
96 | d[2] = -df * (df + 1) * (df + 3) / (8 * np2 * np4); | |
97 | np2 *= df + 2; | |
98 | d[3] = -df * (df + 1) * (df + 5) * (((3 * df) + 7) * df -2) / (48 * np2 * np4 * np6); | |
99 | np2 *= (df + 2); | |
100 | np4 *= (df + 4); | |
101 | d[4] = -df * (df + 1) * (df + 7) * | |
102 | ( (((((15 * df) + 154) * df + 465) * df + 286) * df - 336) * df + 64 ) | |
103 | / (384 * np2 * np4 * np6 * (df + 8)); | |
104 | np2 *= (df + 2); | |
105 | d[5] = -df * (df + 1) * (df + 3) * (df + 9) | |
106 | * (((((((35 * df + 452) * df + 1573) * df + 600) * df - 2020) * df) + 928) * df -128) | |
107 | / (1280 * np2 * np4 * np6 * (df + 8) * (df + 10)); | |
108 | np2 *= (df + 2); | |
109 | np4 *= (df + 4); | |
110 | np6 *= (df + 6); | |
111 | d[6] = -df * (df + 1) * (df + 11) | |
112 | * ((((((((((((945 * df) + 31506) * df + 425858) * df + 2980236) * df + 11266745) * df + 20675018) * df + 7747124) * df - 22574632) * df - 8565600) * df + 18108416) * df - 7099392) * df + 884736) | |
113 | / (46080 * np2 * np4 * np6 * (df + 8) * (df + 10) * (df +12)); | |
114 | // | |
115 | // Now bring everthing together to provide the result, | |
116 | // this is Eq 62 of Shaw: | |
117 | // | |
118 | T rn = sqrt(df); | |
119 | T div = pow(rn * w, 1 / df); | |
120 | T power = div * div; | |
121 | T result = tools::evaluate_polynomial<7, T, T>(d, power); | |
122 | result *= rn; | |
123 | result /= div; | |
124 | return -result; | |
125 | } | |
126 | ||
127 | template <class T, class Policy> | |
128 | T inverse_students_t_body_series(T df, T u, const Policy& pol) | |
129 | { | |
130 | BOOST_MATH_STD_USING | |
131 | // | |
132 | // Body series for small N: | |
133 | // | |
134 | // Start with Eq 56 of Shaw: | |
135 | // | |
136 | T v = boost::math::tgamma_delta_ratio(df / 2, constants::half<T>(), pol) | |
137 | * sqrt(df * constants::pi<T>()) * (u - constants::half<T>()); | |
138 | // | |
139 | // Workspace for the polynomial coefficients: | |
140 | // | |
141 | T c[11] = { 0, 1, }; | |
142 | // | |
143 | // Figure out what the coefficients are, note these depend | |
144 | // only on the degrees of freedom (Eq 57 of Shaw): | |
145 | // | |
146 | T in = 1 / df; | |
147 | c[2] = static_cast<T>(0.16666666666666666667 + 0.16666666666666666667 * in); | |
148 | c[3] = static_cast<T>((0.0083333333333333333333 * in | |
149 | + 0.066666666666666666667) * in | |
150 | + 0.058333333333333333333); | |
151 | c[4] = static_cast<T>(((0.00019841269841269841270 * in | |
152 | + 0.0017857142857142857143) * in | |
153 | + 0.026785714285714285714) * in | |
154 | + 0.025198412698412698413); | |
155 | c[5] = static_cast<T>((((2.7557319223985890653e-6 * in | |
156 | + 0.00037477954144620811287) * in | |
157 | - 0.0011078042328042328042) * in | |
158 | + 0.010559964726631393298) * in | |
159 | + 0.012039792768959435626); | |
160 | c[6] = static_cast<T>(((((2.5052108385441718775e-8 * in | |
161 | - 0.000062705427288760622094) * in | |
162 | + 0.00059458674042007375341) * in | |
163 | - 0.0016095979637646304313) * in | |
164 | + 0.0061039211560044893378) * in | |
165 | + 0.0038370059724226390893); | |
166 | c[7] = static_cast<T>((((((1.6059043836821614599e-10 * in | |
167 | + 0.000015401265401265401265) * in | |
168 | - 0.00016376804137220803887) * in | |
169 | + 0.00069084207973096861986) * in | |
170 | - 0.0012579159844784844785) * in | |
171 | + 0.0010898206731540064873) * in | |
172 | + 0.0032177478835464946576); | |
173 | c[8] = static_cast<T>(((((((7.6471637318198164759e-13 * in | |
174 | - 3.9851014346715404916e-6) * in | |
175 | + 0.000049255746366361445727) * in | |
176 | - 0.00024947258047043099953) * in | |
177 | + 0.00064513046951456342991) * in | |
178 | - 0.00076245135440323932387) * in | |
179 | + 0.000033530976880017885309) * in | |
180 | + 0.0017438262298340009980); | |
181 | c[9] = static_cast<T>((((((((2.8114572543455207632e-15 * in | |
182 | + 1.0914179173496789432e-6) * in | |
183 | - 0.000015303004486655377567) * in | |
184 | + 0.000090867107935219902229) * in | |
185 | - 0.00029133414466938067350) * in | |
186 | + 0.00051406605788341121363) * in | |
187 | - 0.00036307660358786885787) * in | |
188 | - 0.00031101086326318780412) * in | |
189 | + 0.00096472747321388644237); | |
190 | c[10] = static_cast<T>(((((((((8.2206352466243297170e-18 * in | |
191 | - 3.1239569599829868045e-7) * in | |
192 | + 4.8903045291975346210e-6) * in | |
193 | - 0.000033202652391372058698) * in | |
194 | + 0.00012645437628698076975) * in | |
195 | - 0.00028690924218514613987) * in | |
196 | + 0.00035764655430568632777) * in | |
197 | - 0.00010230378073700412687) * in | |
198 | - 0.00036942667800009661203) * in | |
199 | + 0.00054229262813129686486); | |
200 | // | |
201 | // The result is then a polynomial in v (see Eq 56 of Shaw): | |
202 | // | |
203 | return tools::evaluate_odd_polynomial<11, T, T>(c, v); | |
204 | } | |
205 | ||
206 | template <class T, class Policy> | |
207 | T inverse_students_t(T df, T u, T v, const Policy& pol, bool* pexact = 0) | |
208 | { | |
209 | // | |
210 | // df = number of degrees of freedom. | |
211 | // u = probablity. | |
212 | // v = 1 - u. | |
213 | // l = lanczos type to use. | |
214 | // | |
215 | BOOST_MATH_STD_USING | |
216 | bool invert = false; | |
217 | T result = 0; | |
218 | if(pexact) | |
219 | *pexact = false; | |
220 | if(u > v) | |
221 | { | |
222 | // function is symmetric, invert it: | |
223 | std::swap(u, v); | |
224 | invert = true; | |
225 | } | |
226 | if((floor(df) == df) && (df < 20)) | |
227 | { | |
228 | // | |
229 | // we have integer degrees of freedom, try for the special | |
230 | // cases first: | |
231 | // | |
232 | T tolerance = ldexp(1.0f, (2 * policies::digits<T, Policy>()) / 3); | |
233 | ||
234 | switch(itrunc(df, Policy())) | |
235 | { | |
236 | case 1: | |
237 | { | |
238 | // | |
239 | // df = 1 is the same as the Cauchy distribution, see | |
240 | // Shaw Eq 35: | |
241 | // | |
242 | if(u == 0.5) | |
243 | result = 0; | |
244 | else | |
245 | result = -cos(constants::pi<T>() * u) / sin(constants::pi<T>() * u); | |
246 | if(pexact) | |
247 | *pexact = true; | |
248 | break; | |
249 | } | |
250 | case 2: | |
251 | { | |
252 | // | |
253 | // df = 2 has an exact result, see Shaw Eq 36: | |
254 | // | |
255 | result =(2 * u - 1) / sqrt(2 * u * v); | |
256 | if(pexact) | |
257 | *pexact = true; | |
258 | break; | |
259 | } | |
260 | case 4: | |
261 | { | |
262 | // | |
263 | // df = 4 has an exact result, see Shaw Eq 38 & 39: | |
264 | // | |
265 | T alpha = 4 * u * v; | |
266 | T root_alpha = sqrt(alpha); | |
267 | T r = 4 * cos(acos(root_alpha) / 3) / root_alpha; | |
268 | T x = sqrt(r - 4); | |
269 | result = u - 0.5f < 0 ? (T)-x : x; | |
270 | if(pexact) | |
271 | *pexact = true; | |
272 | break; | |
273 | } | |
274 | case 6: | |
275 | { | |
276 | // | |
277 | // We get numeric overflow in this area: | |
278 | // | |
279 | if(u < 1e-150) | |
280 | return (invert ? -1 : 1) * inverse_students_t_hill(df, u, pol); | |
281 | // | |
282 | // Newton-Raphson iteration of a polynomial case, | |
283 | // choice of seed value is taken from Shaw's online | |
284 | // supplement: | |
285 | // | |
286 | T a = 4 * (u - u * u);//1 - 4 * (u - 0.5f) * (u - 0.5f); | |
287 | T b = boost::math::cbrt(a); | |
288 | static const T c = static_cast<T>(0.85498797333834849467655443627193); | |
289 | T p = 6 * (1 + c * (1 / b - 1)); | |
290 | T p0; | |
291 | do{ | |
292 | T p2 = p * p; | |
293 | T p4 = p2 * p2; | |
294 | T p5 = p * p4; | |
295 | p0 = p; | |
296 | // next term is given by Eq 41: | |
297 | p = 2 * (8 * a * p5 - 270 * p2 + 2187) / (5 * (4 * a * p4 - 216 * p - 243)); | |
298 | }while(fabs((p - p0) / p) > tolerance); | |
299 | // | |
300 | // Use Eq 45 to extract the result: | |
301 | // | |
302 | p = sqrt(p - df); | |
303 | result = (u - 0.5f) < 0 ? (T)-p : p; | |
304 | break; | |
305 | } | |
306 | #if 0 | |
307 | // | |
308 | // These are Shaw's "exact" but iterative solutions | |
309 | // for even df, the numerical accuracy of these is | |
310 | // rather less than Hill's method, so these are disabled | |
311 | // for now, which is a shame because they are reasonably | |
312 | // quick to evaluate... | |
313 | // | |
314 | case 8: | |
315 | { | |
316 | // | |
317 | // Newton-Raphson iteration of a polynomial case, | |
318 | // choice of seed value is taken from Shaw's online | |
319 | // supplement: | |
320 | // | |
321 | static const T c8 = 0.85994765706259820318168359251872L; | |
322 | T a = 4 * (u - u * u); //1 - 4 * (u - 0.5f) * (u - 0.5f); | |
323 | T b = pow(a, T(1) / 4); | |
324 | T p = 8 * (1 + c8 * (1 / b - 1)); | |
325 | T p0 = p; | |
326 | do{ | |
327 | T p5 = p * p; | |
328 | p5 *= p5 * p; | |
329 | p0 = p; | |
330 | // Next term is given by Eq 42: | |
331 | p = 2 * (3 * p + (640 * (160 + p * (24 + p * (p + 4)))) / (-5120 + p * (-2048 - 960 * p + a * p5))) / 7; | |
332 | }while(fabs((p - p0) / p) > tolerance); | |
333 | // | |
334 | // Use Eq 45 to extract the result: | |
335 | // | |
336 | p = sqrt(p - df); | |
337 | result = (u - 0.5f) < 0 ? -p : p; | |
338 | break; | |
339 | } | |
340 | case 10: | |
341 | { | |
342 | // | |
343 | // Newton-Raphson iteration of a polynomial case, | |
344 | // choice of seed value is taken from Shaw's online | |
345 | // supplement: | |
346 | // | |
347 | static const T c10 = 0.86781292867813396759105692122285L; | |
348 | T a = 4 * (u - u * u); //1 - 4 * (u - 0.5f) * (u - 0.5f); | |
349 | T b = pow(a, T(1) / 5); | |
350 | T p = 10 * (1 + c10 * (1 / b - 1)); | |
351 | T p0; | |
352 | do{ | |
353 | T p6 = p * p; | |
354 | p6 *= p6 * p6; | |
355 | p0 = p; | |
356 | // Next term given by Eq 43: | |
357 | p = (8 * p) / 9 + (218750 * (21875 + 4 * p * (625 + p * (75 + 2 * p * (5 + p))))) / | |
358 | (9 * (-68359375 + 8 * p * (-2343750 + p * (-546875 - 175000 * p + 8 * a * p6)))); | |
359 | }while(fabs((p - p0) / p) > tolerance); | |
360 | // | |
361 | // Use Eq 45 to extract the result: | |
362 | // | |
363 | p = sqrt(p - df); | |
364 | result = (u - 0.5f) < 0 ? -p : p; | |
365 | break; | |
366 | } | |
367 | #endif | |
368 | default: | |
369 | goto calculate_real; | |
370 | } | |
371 | } | |
372 | else | |
373 | { | |
374 | calculate_real: | |
375 | if(df > 0x10000000) | |
376 | { | |
377 | result = -boost::math::erfc_inv(2 * u, pol) * constants::root_two<T>(); | |
378 | if((pexact) && (df >= 1e20)) | |
379 | *pexact = true; | |
380 | } | |
381 | else if(df < 3) | |
382 | { | |
383 | // | |
384 | // Use a roughly linear scheme to choose between Shaw's | |
385 | // tail series and body series: | |
386 | // | |
387 | T crossover = 0.2742f - df * 0.0242143f; | |
388 | if(u > crossover) | |
389 | { | |
390 | result = boost::math::detail::inverse_students_t_body_series(df, u, pol); | |
391 | } | |
392 | else | |
393 | { | |
394 | result = boost::math::detail::inverse_students_t_tail_series(df, u, pol); | |
395 | } | |
396 | } | |
397 | else | |
398 | { | |
399 | // | |
400 | // Use Hill's method except in the exteme tails | |
401 | // where we use Shaw's tail series. | |
402 | // The crossover point is roughly exponential in -df: | |
403 | // | |
404 | T crossover = ldexp(1.0f, iround(T(df / -0.654f), typename policies::normalise<Policy, policies::rounding_error<policies::ignore_error> >::type())); | |
405 | if(u > crossover) | |
406 | { | |
407 | result = boost::math::detail::inverse_students_t_hill(df, u, pol); | |
408 | } | |
409 | else | |
410 | { | |
411 | result = boost::math::detail::inverse_students_t_tail_series(df, u, pol); | |
412 | } | |
413 | } | |
414 | } | |
415 | return invert ? (T)-result : result; | |
416 | } | |
417 | ||
418 | template <class T, class Policy> | |
419 | inline T find_ibeta_inv_from_t_dist(T a, T p, T /*q*/, T* py, const Policy& pol) | |
420 | { | |
421 | T u = p / 2; | |
422 | T v = 1 - u; | |
423 | T df = a * 2; | |
424 | T t = boost::math::detail::inverse_students_t(df, u, v, pol); | |
425 | *py = t * t / (df + t * t); | |
426 | return df / (df + t * t); | |
427 | } | |
428 | ||
429 | template <class T, class Policy> | |
430 | inline T fast_students_t_quantile_imp(T df, T p, const Policy& pol, const mpl::false_*) | |
431 | { | |
432 | BOOST_MATH_STD_USING | |
433 | // | |
434 | // Need to use inverse incomplete beta to get | |
435 | // required precision so not so fast: | |
436 | // | |
437 | T probability = (p > 0.5) ? 1 - p : p; | |
438 | T t, x, y(0); | |
439 | x = ibeta_inv(df / 2, T(0.5), 2 * probability, &y, pol); | |
440 | if(df * y > tools::max_value<T>() * x) | |
441 | t = policies::raise_overflow_error<T>("boost::math::students_t_quantile<%1%>(%1%,%1%)", 0, pol); | |
442 | else | |
443 | t = sqrt(df * y / x); | |
444 | // | |
445 | // Figure out sign based on the size of p: | |
446 | // | |
447 | if(p < 0.5) | |
448 | t = -t; | |
449 | return t; | |
450 | } | |
451 | ||
452 | template <class T, class Policy> | |
453 | T fast_students_t_quantile_imp(T df, T p, const Policy& pol, const mpl::true_*) | |
454 | { | |
455 | BOOST_MATH_STD_USING | |
456 | bool invert = false; | |
457 | if((df < 2) && (floor(df) != df)) | |
458 | return boost::math::detail::fast_students_t_quantile_imp(df, p, pol, static_cast<mpl::false_*>(0)); | |
459 | if(p > 0.5) | |
460 | { | |
461 | p = 1 - p; | |
462 | invert = true; | |
463 | } | |
464 | // | |
465 | // Get an estimate of the result: | |
466 | // | |
467 | bool exact; | |
468 | T t = inverse_students_t(df, p, T(1-p), pol, &exact); | |
469 | if((t == 0) || exact) | |
470 | return invert ? -t : t; // can't do better! | |
471 | // | |
472 | // Change variables to inverse incomplete beta: | |
473 | // | |
474 | T t2 = t * t; | |
475 | T xb = df / (df + t2); | |
476 | T y = t2 / (df + t2); | |
477 | T a = df / 2; | |
478 | // | |
479 | // t can be so large that x underflows, | |
480 | // just return our estimate in that case: | |
481 | // | |
482 | if(xb == 0) | |
483 | return t; | |
484 | // | |
485 | // Get incomplete beta and it's derivative: | |
486 | // | |
487 | T f1; | |
488 | T f0 = xb < y ? ibeta_imp(a, constants::half<T>(), xb, pol, false, true, &f1) | |
489 | : ibeta_imp(constants::half<T>(), a, y, pol, true, true, &f1); | |
490 | ||
491 | // Get cdf from incomplete beta result: | |
492 | T p0 = f0 / 2 - p; | |
493 | // Get pdf from derivative: | |
494 | T p1 = f1 * sqrt(y * xb * xb * xb / df); | |
495 | // | |
496 | // Second derivative divided by p1: | |
497 | // | |
498 | // yacas gives: | |
499 | // | |
500 | // In> PrettyForm(Simplify(D(t) (1 + t^2/v) ^ (-(v+1)/2))) | |
501 | // | |
502 | // | | v + 1 | | | |
503 | // | -| ----- + 1 | | | |
504 | // | | 2 | | | |
505 | // -| | 2 | | | |
506 | // | | t | | | |
507 | // | | -- + 1 | | | |
508 | // | ( v + 1 ) * | v | * t | | |
509 | // --------------------------------------------- | |
510 | // v | |
511 | // | |
512 | // Which after some manipulation is: | |
513 | // | |
514 | // -p1 * t * (df + 1) / (t^2 + df) | |
515 | // | |
516 | T p2 = t * (df + 1) / (t * t + df); | |
517 | // Halley step: | |
518 | t = fabs(t); | |
519 | t += p0 / (p1 + p0 * p2 / 2); | |
520 | return !invert ? -t : t; | |
521 | } | |
522 | ||
523 | template <class T, class Policy> | |
524 | inline T fast_students_t_quantile(T df, T p, const Policy& pol) | |
525 | { | |
526 | typedef typename policies::evaluation<T, Policy>::type value_type; | |
527 | typedef typename policies::normalise< | |
528 | Policy, | |
529 | policies::promote_float<false>, | |
530 | policies::promote_double<false>, | |
531 | policies::discrete_quantile<>, | |
532 | policies::assert_undefined<> >::type forwarding_policy; | |
533 | ||
534 | typedef mpl::bool_< | |
535 | (std::numeric_limits<T>::digits <= 53) | |
536 | && | |
537 | (std::numeric_limits<T>::is_specialized) | |
538 | && | |
539 | (std::numeric_limits<T>::radix == 2) | |
540 | > tag_type; | |
541 | return policies::checked_narrowing_cast<T, forwarding_policy>(fast_students_t_quantile_imp(static_cast<value_type>(df), static_cast<value_type>(p), pol, static_cast<tag_type*>(0)), "boost::math::students_t_quantile<%1%>(%1%,%1%,%1%)"); | |
542 | } | |
543 | ||
544 | }}} // namespaces | |
545 | ||
546 | #endif // BOOST_MATH_SF_DETAIL_INV_T_HPP | |
547 | ||
548 | ||
549 |