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)
11 #ifndef _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_
12 #define _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_
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>
33 #pragma warning(disable:4702) // Unreachable code (release mode only warning)
36 namespace boost { namespace math { namespace detail{
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
41 // See http://functions.wolfram.com/GammaBetaErf/PolyGamma2/06/02/0001/
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
50 // x is crazy large, just concentrate on the first part of the expression and use logs:
51 if(n == 1) return 1 / 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);
56 return ((n & 1) ? 1 : -1) * exp(boost::math::lgamma(T(n), pol) - n * log(x));
58 T term, sum, part_term;
61 // Start by setting part_term to:
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:
69 // part_term * (n + 2 * x) / 2
71 // and to the first term in the series
72 // (excluding the Bernoulli number) by:
74 // part_term n * (n + 1) / (2x)
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:
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));
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);
93 sum = part_term * (n + 2 * x) / 2;
94 part_term *= (T(n) * (n + 1)) / 2;
98 // If the leading term is 0, so is the result:
103 for(unsigned k = 1;;)
105 term = part_term * boost::math::bernoulli_b2n<T>(k, pol);
108 // Normal termination condition:
110 if(fabs(term / sum) < tools::epsilon<T>())
113 // Increment our counter, and move part_term on to the next value:
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;
120 // Emergency get out termination condition:
122 if(k > policies::get_max_series_iterations<Policy>())
124 return policies::raise_evaluation_error(function, "Series did not converge, closest value was %1%", sum, pol);
134 template<class T, class Policy>
135 T polygamma_attransitionplus(const int n, const T& x, const Policy& pol, const char* function)
137 // See: http://functions.wolfram.com/GammaBetaErf/PolyGamma2/16/01/01/0017/
139 // Use N = (0.4 * digits) + (4 * n) for target value for x:
141 const int d4d = static_cast<int>(0.4F * policies::digits_base10<T, Policy>());
142 const int N = d4d + (4 * n);
144 const int iter = N - itrunc(x);
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>(n) + " and x = %1%").c_str(), x, pol);
149 const int minus_m_minus_one = -m - 1;
153 T z_plus_k_pow_minus_m_minus_one(0);
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>())
158 for(int k = 1; k <= iter; ++k)
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;
164 sum0 *= boost::math::factorial<T>(n);
168 for(int k = 1; k <= iter; ++k)
170 T log_term = log(z) * minus_m_minus_one + boost::math::lgamma(T(n + 1), pol);
171 sum0 += exp(log_term);
178 return sum0 + polygamma_atinfinityplus(n, z, pol, function);
181 template <class T, class Policy>
182 T polygamma_nearzero(int n, T x, const Policy& pol, const char* function)
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).
191 // In order to avoid spurious overflow, save the n! term for later, and rescale at the end:
193 T scale = boost::math::factorial<T>(n, pol);
195 // "factorial_part" contains everything except the zeta function
196 // evaluations in each term:
198 T factorial_part = 1;
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:
204 T prefix = pow(x, n + 1);
206 return boost::math::policies::raise_overflow_error<T>(function, 0, pol);
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:
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);
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.
223 for(unsigned k = 0;;)
225 // Get the k'th term:
226 T term = factorial_part * boost::math::zeta(T(k + n + 1), pol);
228 // Termination condition:
229 if(fabs(term) < fabs(sum * boost::math::policies::get_epsilon<T, Policy>()))
232 // Move on k and factorial_part:
235 factorial_part *= (-x * (n + k)) / k;
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);
243 // We need to multiply by the scale, at each stage checking for oveflow:
245 if(boost::math::tools::max_value<T>() / scale < sum)
246 return boost::math::policies::raise_overflow_error<T>(function, 0, pol);
248 return n & 1 ? sum : T(-sum);
252 // Helper function which figures out which slot our coefficient is in
253 // given an angle multiplier for the cosine term of power:
255 template <class Table>
256 typename Table::value_type::reference dereference_table(Table& table, unsigned row, unsigned power)
258 return table[row][power / 2];
263 template <class T, class Policy>
264 T poly_cot_pi(int n, T x, T xc, const Policy& pol, const char* function)
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:
271 // The general form of each derivative is:
273 // pi^n * SUM{k=0, n} C[k,n] * cos^k(pi * x) * csc^(n+1)(pi * x)
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];
280 // Note that there are many different ways of representing this derivative thanks to
281 // the many trigomonetric 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 minumum - thus avoiding cancellation error in this region.
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).
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);
299 return -constants::pi<T, Policy>() / (s * s);
302 return 2 * constants::pi<T, Policy>() * constants::pi<T, Policy>() * c / boost::math::pow<3>(s, pol);
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);
312 return boost::math::pow<4>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<5>(s, pol);
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);
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);
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);
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);
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);
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);
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);
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);
354 #ifndef BOOST_NO_LONG_LONG
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);
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);
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);
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);
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);
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);
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);
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);
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:
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);
411 static int digits = tools::digits<T>();
412 static std::vector<std::vector<T> > table(1, std::vector<T>(1, T(-1)));
414 int current_digits = tools::digits<T>();
416 if(digits != current_digits)
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;
425 if(index >= (int)table.size())
427 for(int i = (int)table.size() - 1; i < index; ++i)
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)));
437 for(int column = 0; column <= max_columns; ++column)
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);
444 table[i + 1][(cos_order - 1) / 2] += (-cos_order * table[i][column]) / (sin_order - 1);
449 T sum = boost::math::tools::evaluate_even_polynomial(&table[index][0], c, table[index].size());
451 sum *= c; // First coeffient is order 1, and really an odd polynomial.
455 // The remaining terms are computed using logs since the powers and factorials
456 // get real large real quick:
458 T power_terms = n * log(boost::math::constants::pi<T>());
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));
465 if(power_terms > boost::math::tools::log_max_value<T>())
466 return sum * boost::math::policies::raise_overflow_error<T>(function, 0, pol);
468 return exp(power_terms) * ((s < 0) && ((n + 1) & 1) ? -1 : 1) * boost::math::sign(sum);
471 template <class T, class Policy>
472 struct polygamma_initializer
478 // Forces initialization of our table of coefficients and mutex:
479 boost::math::polygamma(30, T(-2.5f), Policy());
481 void force_instantiate()const{}
483 static const init initializer;
484 static void force_instantiate()
486 initializer.force_instantiate();
490 template <class T, class Policy>
491 const typename polygamma_initializer<T, Policy>::init polygamma_initializer<T, Policy>::initializer;
493 template<class T, class Policy>
494 inline T polygamma_imp(const int n, T x, const Policy &pol)
497 static const char* function = "boost::math::polygamma<%1%>(int, %1%)";
498 polygamma_initializer<T, Policy>::initializer.force_instantiate();
500 return policies::raise_domain_error<T>(function, "Order must be >= 0, but got %1%", static_cast<T>(n), pol);
506 // Result is infinity if x is odd, and a pole error if x is even.
509 return policies::raise_overflow_error<T>(function, 0, pol);
511 return policies::raise_pole_error<T>(function, "Evaluation at negative integer %1%", x, pol);
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;
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:
524 T small_x_limit = (std::min)(T(T(5) / n), T(0.25f));
525 if(x < small_x_limit)
527 return polygamma_nearzero(n, x, pol, function);
529 else if(x > 0.4F * policies::digits_base10<T, Policy>() + 4.0f * n)
531 return polygamma_atinfinityplus(n, x, pol, function);
535 return (n & 1 ? 1 : -1) * boost::math::factorial<T>(n, pol) * boost::math::zeta(T(n + 1), pol);
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;
547 return polygamma_attransitionplus(n, x, pol, function);
551 } } } // namespace boost::math::detail
557 #endif // _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_