1 | |
2 | /////////////////////////////////////////////////////////////////////////////// |
3 | // Copyright 2013 Nikhar Agrawal |
4 | // Copyright 2013 Christopher Kormanyos |
5 | // Copyright 2014 John Maddock |
6 | // Copyright 2013 Paul Bristow |
7 | // Distributed under the Boost |
8 | // Software License, Version 1.0. (See accompanying file |
9 | // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) |
10 | |
11 | #ifndef _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_ |
12 | #define _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_ |
13 | |
14 | #include <cmath> |
15 | #include <limits> |
16 | #include <boost/cstdint.hpp> |
17 | #include <boost/math/policies/policy.hpp> |
18 | #include <boost/math/special_functions/bernoulli.hpp> |
19 | #include <boost/math/special_functions/trunc.hpp> |
20 | #include <boost/math/special_functions/zeta.hpp> |
21 | #include <boost/math/special_functions/digamma.hpp> |
22 | #include <boost/math/special_functions/sin_pi.hpp> |
23 | #include <boost/math/special_functions/cos_pi.hpp> |
24 | #include <boost/math/special_functions/pow.hpp> |
25 | #include <boost/mpl/if.hpp> |
26 | #include <boost/mpl/int.hpp> |
27 | #include <boost/static_assert.hpp> |
28 | #include <boost/type_traits/is_convertible.hpp> |
29 | |
30 | #ifdef _MSC_VER |
31 | #pragma once |
32 | #pragma warning(push) |
33 | #pragma warning(disable:4702) // Unreachable code (release mode only warning) |
34 | #endif |
35 | |
36 | namespace boost { namespace math { namespace detail{ |
37 | |
38 | template<class T, class Policy> |
39 | T polygamma_atinfinityplus(const int n, const T& x, const Policy& pol, const char* function) // for large values of x such as for x> 400 |
40 | { |
41 | // See http://functions.wolfram.com/GammaBetaErf/PolyGamma2/06/02/0001/ |
42 | BOOST_MATH_STD_USING |
43 | // |
44 | // sum == current value of accumulated sum. |
45 | // term == value of current term to be added to sum. |
46 | // part_term == value of current term excluding the Bernoulli number part |
47 | // |
48 | if(n + x == x) |
49 | { |
50 | // x is crazy large, just concentrate on the first part of the expression and use logs: |
51 | if(n == 1) return 1 / x; |
52 | T nlx = n * log(x); |
53 | if((nlx < tools::log_max_value<T>()) && (n < (int)max_factorial<T>::value)) |
54 | return ((n & 1) ? 1 : -1) * boost::math::factorial<T>(n - 1) * pow(x, -n); |
55 | else |
56 | return ((n & 1) ? 1 : -1) * exp(boost::math::lgamma(T(n), pol) - n * log(x)); |
57 | } |
58 | T term, sum, part_term; |
59 | T x_squared = x * x; |
60 | // |
61 | // Start by setting part_term to: |
62 | // |
63 | // (n-1)! / x^(n+1) |
64 | // |
65 | // which is common to both the first term of the series (with k = 1) |
66 | // and to the leading part. |
67 | // We can then get to the leading term by: |
68 | // |
69 | // part_term * (n + 2 * x) / 2 |
70 | // |
71 | // and to the first term in the series |
72 | // (excluding the Bernoulli number) by: |
73 | // |
74 | // part_term n * (n + 1) / (2x) |
75 | // |
76 | // If either the factorial would overflow, |
77 | // or the power term underflows, this just gets set to 0 and then we |
78 | // know that we have to use logs for the initial terms: |
79 | // |
80 | part_term = ((n > (int)boost::math::max_factorial<T>::value) && (T(n) * n > tools::log_max_value<T>())) |
81 | ? T(0) : static_cast<T>(boost::math::factorial<T>(n - 1, pol) * pow(x, -n - 1)); |
82 | if(part_term == 0) |
83 | { |
84 | // Either n is very large, or the power term underflows, |
85 | // set the initial values of part_term, term and sum via logs: |
86 | part_term = static_cast<T>(boost::math::lgamma(n, pol) - (n + 1) * log(x)); |
87 | sum = exp(part_term + log(n + 2 * x) - boost::math::constants::ln_two<T>()); |
88 | part_term += log(T(n) * (n + 1)) - boost::math::constants::ln_two<T>() - log(x); |
89 | part_term = exp(part_term); |
90 | } |
91 | else |
92 | { |
93 | sum = part_term * (n + 2 * x) / 2; |
94 | part_term *= (T(n) * (n + 1)) / 2; |
95 | part_term /= x; |
96 | } |
97 | // |
98 | // If the leading term is 0, so is the result: |
99 | // |
100 | if(sum == 0) |
101 | return sum; |
102 | |
103 | for(unsigned k = 1;;) |
104 | { |
105 | term = part_term * boost::math::bernoulli_b2n<T>(k, pol); |
106 | sum += term; |
107 | // |
108 | // Normal termination condition: |
109 | // |
110 | if(fabs(term / sum) < tools::epsilon<T>()) |
111 | break; |
112 | // |
113 | // Increment our counter, and move part_term on to the next value: |
114 | // |
115 | ++k; |
116 | part_term *= T(n + 2 * k - 2) * (n - 1 + 2 * k); |
117 | part_term /= (2 * k - 1) * 2 * k; |
118 | part_term /= x_squared; |
119 | // |
120 | // Emergency get out termination condition: |
121 | // |
122 | if(k > policies::get_max_series_iterations<Policy>()) |
123 | { |
124 | return policies::raise_evaluation_error(function, "Series did not converge, closest value was %1%" , sum, pol); |
125 | } |
126 | } |
127 | |
128 | if((n - 1) & 1) |
129 | sum = -sum; |
130 | |
131 | return sum; |
132 | } |
133 | |
134 | template<class T, class Policy> |
135 | T polygamma_attransitionplus(const int n, const T& x, const Policy& pol, const char* function) |
136 | { |
137 | // See: http://functions.wolfram.com/GammaBetaErf/PolyGamma2/16/01/01/0017/ |
138 | |
139 | // Use N = (0.4 * digits) + (4 * n) for target value for x: |
140 | BOOST_MATH_STD_USING |
141 | const int d4d = static_cast<int>(0.4F * policies::digits_base10<T, Policy>()); |
142 | const int N = d4d + (4 * n); |
143 | const int m = n; |
144 | const int iter = N - itrunc(x); |
145 | |
146 | if(iter > (int)policies::get_max_series_iterations<Policy>()) |
147 | return policies::raise_evaluation_error<T>(function, ("Exceeded maximum series evaluations evaluating at n = " + boost::lexical_cast<std::string>(arg: n) + " and x = %1%" ).c_str(), x, pol); |
148 | |
149 | const int minus_m_minus_one = -m - 1; |
150 | |
151 | T z(x); |
152 | T sum0(0); |
153 | T z_plus_k_pow_minus_m_minus_one(0); |
154 | |
155 | // Forward recursion to larger x, need to check for overflow first though: |
156 | if(log(z + iter) * minus_m_minus_one > -tools::log_max_value<T>()) |
157 | { |
158 | for(int k = 1; k <= iter; ++k) |
159 | { |
160 | z_plus_k_pow_minus_m_minus_one = pow(z, minus_m_minus_one); |
161 | sum0 += z_plus_k_pow_minus_m_minus_one; |
162 | z += 1; |
163 | } |
164 | sum0 *= boost::math::factorial<T>(n); |
165 | } |
166 | else |
167 | { |
168 | for(int k = 1; k <= iter; ++k) |
169 | { |
170 | T log_term = log(z) * minus_m_minus_one + boost::math::lgamma(T(n + 1), pol); |
171 | sum0 += exp(log_term); |
172 | z += 1; |
173 | } |
174 | } |
175 | if((n - 1) & 1) |
176 | sum0 = -sum0; |
177 | |
178 | return sum0 + polygamma_atinfinityplus(n, z, pol, function); |
179 | } |
180 | |
181 | template <class T, class Policy> |
182 | T polygamma_nearzero(int n, T x, const Policy& pol, const char* function) |
183 | { |
184 | BOOST_MATH_STD_USING |
185 | // |
186 | // If we take this expansion for polygamma: http://functions.wolfram.com/06.15.06.0003.02 |
187 | // and substitute in this expression for polygamma(n, 1): http://functions.wolfram.com/06.15.03.0009.01 |
188 | // we get an alternating series for polygamma when x is small in terms of zeta functions of |
189 | // integer arguments (which are easy to evaluate, at least when the integer is even). |
190 | // |
191 | // In order to avoid spurious overflow, save the n! term for later, and rescale at the end: |
192 | // |
193 | T scale = boost::math::factorial<T>(n, pol); |
194 | // |
195 | // "factorial_part" contains everything except the zeta function |
196 | // evaluations in each term: |
197 | // |
198 | T factorial_part = 1; |
199 | // |
200 | // "prefix" is what we'll be adding the accumulated sum to, it will |
201 | // be n! / z^(n+1), but since we're scaling by n! it's just |
202 | // 1 / z^(n+1) for now: |
203 | // |
204 | T prefix = pow(x, n + 1); |
205 | if(prefix == 0) |
206 | return boost::math::policies::raise_overflow_error<T>(function, 0, pol); |
207 | prefix = 1 / prefix; |
208 | // |
209 | // First term in the series is necessarily < zeta(2) < 2, so |
210 | // ignore the sum if it will have no effect on the result anyway: |
211 | // |
212 | if(prefix > 2 / policies::get_epsilon<T, Policy>()) |
213 | return ((n & 1) ? 1 : -1) * |
214 | (tools::max_value<T>() / prefix < scale ? policies::raise_overflow_error<T>(function, 0, pol) : prefix * scale); |
215 | // |
216 | // As this is an alternating series we could accelerate it using |
217 | // "Convergence Acceleration of Alternating Series", |
218 | // Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Experimental Mathematics, 1999. |
219 | // In practice however, it appears not to make any difference to the number of terms |
220 | // required except in some edge cases which are filtered out anyway before we get here. |
221 | // |
222 | T sum = prefix; |
223 | for(unsigned k = 0;;) |
224 | { |
225 | // Get the k'th term: |
226 | T term = factorial_part * boost::math::zeta(T(k + n + 1), pol); |
227 | sum += term; |
228 | // Termination condition: |
229 | if(fabs(term) < fabs(sum * boost::math::policies::get_epsilon<T, Policy>())) |
230 | break; |
231 | // |
232 | // Move on k and factorial_part: |
233 | // |
234 | ++k; |
235 | factorial_part *= (-x * (n + k)) / k; |
236 | // |
237 | // Last chance exit: |
238 | // |
239 | if(k > policies::get_max_series_iterations<Policy>()) |
240 | return policies::raise_evaluation_error<T>(function, "Series did not converge, best value is %1%" , sum, pol); |
241 | } |
242 | // |
243 | // We need to multiply by the scale, at each stage checking for overflow: |
244 | // |
245 | if(boost::math::tools::max_value<T>() / scale < sum) |
246 | return boost::math::policies::raise_overflow_error<T>(function, 0, pol); |
247 | sum *= scale; |
248 | return n & 1 ? sum : T(-sum); |
249 | } |
250 | |
251 | // |
252 | // Helper function which figures out which slot our coefficient is in |
253 | // given an angle multiplier for the cosine term of power: |
254 | // |
255 | template <class Table> |
256 | typename Table::value_type::reference dereference_table(Table& table, unsigned row, unsigned power) |
257 | { |
258 | return table[row][power / 2]; |
259 | } |
260 | |
261 | |
262 | |
263 | template <class T, class Policy> |
264 | T poly_cot_pi(int n, T x, T xc, const Policy& pol, const char* function) |
265 | { |
266 | BOOST_MATH_STD_USING |
267 | // Return n'th derivative of cot(pi*x) at x, these are simply |
268 | // tabulated for up to n = 9, beyond that it is possible to |
269 | // calculate coefficients as follows: |
270 | // |
271 | // The general form of each derivative is: |
272 | // |
273 | // pi^n * SUM{k=0, n} C[k,n] * cos^k(pi * x) * csc^(n+1)(pi * x) |
274 | // |
275 | // With constant C[0,1] = -1 and all other C[k,n] = 0; |
276 | // Then for each k < n+1: |
277 | // C[k-1, n+1] -= k * C[k, n]; |
278 | // C[k+1, n+1] += (k-n-1) * C[k, n]; |
279 | // |
280 | // Note that there are many different ways of representing this derivative thanks to |
281 | // the many trigonometric identies available. In particular, the sum of powers of |
282 | // cosines could be replaced by a sum of cosine multiple angles, and indeed if you |
283 | // plug the derivative into Mathematica this is the form it will give. The two |
284 | // forms are related via the Chebeshev polynomials of the first kind and |
285 | // T_n(cos(x)) = cos(n x). The polynomial form has the great advantage that |
286 | // all the cosine terms are zero at half integer arguments - right where this |
287 | // function has it's minimum - thus avoiding cancellation error in this region. |
288 | // |
289 | // And finally, since every other term in the polynomials is zero, we can save |
290 | // space by only storing the non-zero terms. This greatly complexifies |
291 | // subscripting the tables in the calculation, but halves the storage space |
292 | // (and complexity for that matter). |
293 | // |
294 | T s = fabs(x) < fabs(xc) ? boost::math::sin_pi(x, pol) : boost::math::sin_pi(xc, pol); |
295 | T c = boost::math::cos_pi(x, pol); |
296 | switch(n) |
297 | { |
298 | case 1: |
299 | return -constants::pi<T, Policy>() / (s * s); |
300 | case 2: |
301 | { |
302 | return 2 * constants::pi<T, Policy>() * constants::pi<T, Policy>() * c / boost::math::pow<3>(s, pol); |
303 | } |
304 | case 3: |
305 | { |
306 | int P[] = { -2, -4 }; |
307 | return boost::math::pow<3>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<4>(s, pol); |
308 | } |
309 | case 4: |
310 | { |
311 | int P[] = { 16, 8 }; |
312 | return boost::math::pow<4>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<5>(s, pol); |
313 | } |
314 | case 5: |
315 | { |
316 | int P[] = { -16, -88, -16 }; |
317 | return boost::math::pow<5>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<6>(s, pol); |
318 | } |
319 | case 6: |
320 | { |
321 | int P[] = { 272, 416, 32 }; |
322 | return boost::math::pow<6>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<7>(s, pol); |
323 | } |
324 | case 7: |
325 | { |
326 | int P[] = { -272, -2880, -1824, -64 }; |
327 | return boost::math::pow<7>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<8>(s, pol); |
328 | } |
329 | case 8: |
330 | { |
331 | int P[] = { 7936, 24576, 7680, 128 }; |
332 | return boost::math::pow<8>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<9>(s, pol); |
333 | } |
334 | case 9: |
335 | { |
336 | int P[] = { -7936, -137216, -185856, -31616, -256 }; |
337 | return boost::math::pow<9>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<10>(s, pol); |
338 | } |
339 | case 10: |
340 | { |
341 | int P[] = { 353792, 1841152, 1304832, 128512, 512 }; |
342 | return boost::math::pow<10>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<11>(s, pol); |
343 | } |
344 | case 11: |
345 | { |
346 | int P[] = { -353792, -9061376, -21253376, -8728576, -518656, -1024}; |
347 | return boost::math::pow<11>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<12>(s, pol); |
348 | } |
349 | case 12: |
350 | { |
351 | int P[] = { 22368256, 175627264, 222398464, 56520704, 2084864, 2048 }; |
352 | return boost::math::pow<12>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<13>(s, pol); |
353 | } |
354 | #ifndef BOOST_NO_LONG_LONG |
355 | case 13: |
356 | { |
357 | long long P[] = { -22368256LL, -795300864LL, -2868264960LL, -2174832640LL, -357888000LL, -8361984LL, -4096 }; |
358 | return boost::math::pow<13>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<14>(s, pol); |
359 | } |
360 | case 14: |
361 | { |
362 | long long P[] = { 1903757312LL, 21016670208LL, 41731645440LL, 20261765120LL, 2230947840LL, 33497088LL, 8192 }; |
363 | return boost::math::pow<14>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<15>(s, pol); |
364 | } |
365 | case 15: |
366 | { |
367 | long long P[] = { -1903757312LL, -89702612992LL, -460858269696LL, -559148810240LL, -182172651520LL, -13754155008LL, -134094848LL, -16384 }; |
368 | return boost::math::pow<15>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<16>(s, pol); |
369 | } |
370 | case 16: |
371 | { |
372 | long long P[] = { 209865342976LL, 3099269660672LL, 8885192097792LL, 7048869314560LL, 1594922762240LL, 84134068224LL, 536608768LL, 32768 }; |
373 | return boost::math::pow<16>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<17>(s, pol); |
374 | } |
375 | case 17: |
376 | { |
377 | long long P[] = { -209865342976LL, -12655654469632LL, -87815735738368LL, -155964390375424LL, -84842998005760LL, -13684856848384LL, -511780323328LL, -2146926592LL, -65536 }; |
378 | return boost::math::pow<17>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<18>(s, pol); |
379 | } |
380 | case 18: |
381 | { |
382 | long long P[] = { 29088885112832LL, 553753414467584LL, 2165206642589696LL, 2550316668551168LL, 985278548541440LL, 115620218667008LL, 3100738912256LL, 8588754944LL, 131072 }; |
383 | return boost::math::pow<18>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<19>(s, pol); |
384 | } |
385 | case 19: |
386 | { |
387 | long long P[] = { -29088885112832LL, -2184860175433728LL, -19686087844429824LL, -48165109676113920LL, -39471306959486976LL, -11124607890751488LL, -965271355195392LL, -18733264797696LL, -34357248000LL, -262144 }; |
388 | return boost::math::pow<19>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<20>(s, pol); |
389 | } |
390 | case 20: |
391 | { |
392 | long long P[] = { 4951498053124096LL, 118071834535526400LL, 603968063567560704LL, 990081991141490688LL, 584901762421358592LL, 122829335169859584LL, 7984436548730880LL, 112949304754176LL, 137433710592LL, 524288 }; |
393 | return boost::math::pow<20>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<21>(s, pol); |
394 | } |
395 | #endif |
396 | } |
397 | |
398 | // |
399 | // We'll have to compute the coefficients up to n, |
400 | // complexity is O(n^2) which we don't worry about for now |
401 | // as the values are computed once and then cached. |
402 | // However, if the final evaluation would have too many |
403 | // terms just bail out right away: |
404 | // |
405 | if((unsigned)n / 2u > policies::get_max_series_iterations<Policy>()) |
406 | return policies::raise_evaluation_error<T>(function, "The value of n is so large that we're unable to compute the result in reasonable time, best guess is %1%" , 0, pol); |
407 | #ifdef BOOST_HAS_THREADS |
408 | static boost::detail::lightweight_mutex m; |
409 | boost::detail::lightweight_mutex::scoped_lock l(m); |
410 | #endif |
411 | static int digits = tools::digits<T>(); |
412 | static std::vector<std::vector<T> > table(1, std::vector<T>(1, T(-1))); |
413 | |
414 | int current_digits = tools::digits<T>(); |
415 | |
416 | if(digits != current_digits) |
417 | { |
418 | // Oh my... our precision has changed! |
419 | table = std::vector<std::vector<T> >(1, std::vector<T>(1, T(-1))); |
420 | digits = current_digits; |
421 | } |
422 | |
423 | int index = n - 1; |
424 | |
425 | if(index >= (int)table.size()) |
426 | { |
427 | for(int i = (int)table.size() - 1; i < index; ++i) |
428 | { |
429 | int offset = i & 1; // 1 if the first cos power is 0, otherwise 0. |
430 | int sin_order = i + 2; // order of the sin term |
431 | int max_cos_order = sin_order - 1; // largest order of the polynomial of cos terms |
432 | int max_columns = (max_cos_order - offset) / 2; // How many entries there are in the current row. |
433 | int next_offset = offset ? 0 : 1; |
434 | int next_max_columns = (max_cos_order + 1 - next_offset) / 2; // How many entries there will be in the next row |
435 | table.push_back(std::vector<T>(next_max_columns + 1, T(0))); |
436 | |
437 | for(int column = 0; column <= max_columns; ++column) |
438 | { |
439 | int cos_order = 2 * column + offset; // order of the cosine term in entry "column" |
440 | BOOST_ASSERT(column < (int)table[i].size()); |
441 | BOOST_ASSERT((cos_order + 1) / 2 < (int)table[i + 1].size()); |
442 | table[i + 1][(cos_order + 1) / 2] += ((cos_order - sin_order) * table[i][column]) / (sin_order - 1); |
443 | if(cos_order) |
444 | table[i + 1][(cos_order - 1) / 2] += (-cos_order * table[i][column]) / (sin_order - 1); |
445 | } |
446 | } |
447 | |
448 | } |
449 | T sum = boost::math::tools::evaluate_even_polynomial(&table[index][0], c, table[index].size()); |
450 | if(index & 1) |
451 | sum *= c; // First coefficient is order 1, and really an odd polynomial. |
452 | if(sum == 0) |
453 | return sum; |
454 | // |
455 | // The remaining terms are computed using logs since the powers and factorials |
456 | // get real large real quick: |
457 | // |
458 | T power_terms = n * log(boost::math::constants::pi<T>()); |
459 | if(s == 0) |
460 | return sum * boost::math::policies::raise_overflow_error<T>(function, 0, pol); |
461 | power_terms -= log(fabs(s)) * (n + 1); |
462 | power_terms += boost::math::lgamma(T(n)); |
463 | power_terms += log(fabs(sum)); |
464 | |
465 | if(power_terms > boost::math::tools::log_max_value<T>()) |
466 | return sum * boost::math::policies::raise_overflow_error<T>(function, 0, pol); |
467 | |
468 | return exp(power_terms) * ((s < 0) && ((n + 1) & 1) ? -1 : 1) * boost::math::sign(sum); |
469 | } |
470 | |
471 | template <class T, class Policy> |
472 | struct polygamma_initializer |
473 | { |
474 | struct init |
475 | { |
476 | init() |
477 | { |
478 | // Forces initialization of our table of coefficients and mutex: |
479 | boost::math::polygamma(30, T(-2.5f), Policy()); |
480 | } |
481 | void force_instantiate()const{} |
482 | }; |
483 | static const init initializer; |
484 | static void force_instantiate() |
485 | { |
486 | initializer.force_instantiate(); |
487 | } |
488 | }; |
489 | |
490 | template <class T, class Policy> |
491 | const typename polygamma_initializer<T, Policy>::init polygamma_initializer<T, Policy>::initializer; |
492 | |
493 | template<class T, class Policy> |
494 | inline T polygamma_imp(const int n, T x, const Policy &pol) |
495 | { |
496 | BOOST_MATH_STD_USING |
497 | static const char* function = "boost::math::polygamma<%1%>(int, %1%)" ; |
498 | polygamma_initializer<T, Policy>::initializer.force_instantiate(); |
499 | if(n < 0) |
500 | return policies::raise_domain_error<T>(function, "Order must be >= 0, but got %1%" , static_cast<T>(n), pol); |
501 | if(x < 0) |
502 | { |
503 | if(floor(x) == x) |
504 | { |
505 | // |
506 | // Result is infinity if x is odd, and a pole error if x is even. |
507 | // |
508 | if(lltrunc(x) & 1) |
509 | return policies::raise_overflow_error<T>(function, 0, pol); |
510 | else |
511 | return policies::raise_pole_error<T>(function, "Evaluation at negative integer %1%" , x, pol); |
512 | } |
513 | T z = 1 - x; |
514 | T result = polygamma_imp(n, z, pol) + constants::pi<T, Policy>() * poly_cot_pi(n, z, x, pol, function); |
515 | return n & 1 ? T(-result) : result; |
516 | } |
517 | // |
518 | // Limit for use of small-x-series is chosen |
519 | // so that the series doesn't go too divergent |
520 | // in the first few terms. Ordinarily this |
521 | // would mean setting the limit to ~ 1 / n, |
522 | // but we can tolerate a small amount of divergence: |
523 | // |
524 | T small_x_limit = (std::min)(T(T(5) / n), T(0.25f)); |
525 | if(x < small_x_limit) |
526 | { |
527 | return polygamma_nearzero(n, x, pol, function); |
528 | } |
529 | else if(x > 0.4F * policies::digits_base10<T, Policy>() + 4.0f * n) |
530 | { |
531 | return polygamma_atinfinityplus(n, x, pol, function); |
532 | } |
533 | else if(x == 1) |
534 | { |
535 | return (n & 1 ? 1 : -1) * boost::math::factorial<T>(n, pol) * boost::math::zeta(T(n + 1), pol); |
536 | } |
537 | else if(x == 0.5f) |
538 | { |
539 | T result = (n & 1 ? 1 : -1) * boost::math::factorial<T>(n, pol) * boost::math::zeta(T(n + 1), pol); |
540 | if(fabs(result) >= ldexp(tools::max_value<T>(), -n - 1)) |
541 | return boost::math::sign(result) * policies::raise_overflow_error<T>(function, 0, pol); |
542 | result *= ldexp(T(1), n + 1) - 1; |
543 | return result; |
544 | } |
545 | else |
546 | { |
547 | return polygamma_attransitionplus(n, x, pol, function); |
548 | } |
549 | } |
550 | |
551 | } } } // namespace boost::math::detail |
552 | |
553 | #ifdef _MSC_VER |
554 | #pragma warning(pop) |
555 | #endif |
556 | |
557 | #endif // _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_ |
558 | |
559 | |