2 ///////////////////////////////////////////////////////////////////////////////
3 // Copyright 2018 John Maddock
4 // Distributed under the Boost
5 // Software License, Version 1.0. (See accompanying file
6 // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
8 #ifndef BOOST_HYPERGEOMETRIC_1F1_LARGE_ABZ_HPP_
9 #define BOOST_HYPERGEOMETRIC_1F1_LARGE_ABZ_HPP_
11 #include <boost/math/special_functions/detail/hypergeometric_1F1_bessel.hpp>
12 #include <boost/math/special_functions/detail/hypergeometric_series.hpp>
13 #include <boost/math/special_functions/gamma.hpp>
14 #include <boost/math/special_functions/trunc.hpp>
16 namespace boost { namespace math { namespace detail {
19 inline bool is_negative_integer(const T& x)
22 return (x <= 0) && (floor(x) == x);
26 template <class T, class Policy>
27 struct hypergeometric_1F1_igamma_series
29 enum{ cache_size = 64 };
31 typedef T result_type;
32 hypergeometric_1F1_igamma_series(const T& alpha, const T& delta, const T& x, const Policy& pol)
33 : delta_poch(-delta), alpha_poch(alpha), x(x), k(0), cache_offset(0), pol(pol)
36 T log_term = log(x) * -alpha;
37 log_scaling = lltrunc(log_term - 3 - boost::math::tools::log_min_value<T>() / 50);
38 term = exp(log_term - log_scaling);
43 if (k - cache_offset >= cache_size)
45 cache_offset += cache_size;
48 T result = term * gamma_cache[k - cache_offset];
49 term *= delta_poch * alpha_poch / (++k * x);
56 typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
58 gamma_cache[cache_size - 1] = boost::math::gamma_p(alpha_poch + ((int)cache_size - 1), x, pol);
59 for (int i = cache_size - 1; i > 0; --i)
61 gamma_cache[i - 1] = gamma_cache[i] >= 1 ? T(1) : T(gamma_cache[i] + regularised_gamma_prefix(T(alpha_poch + (i - 1)), x, pol, lanczos_type()) / (alpha_poch + (i - 1)));
64 T delta_poch, alpha_poch, x, term;
65 T gamma_cache[cache_size];
67 long long log_scaling;
72 template <class T, class Policy>
73 T hypergeometric_1F1_igamma(const T& a, const T& b, const T& x, const T& b_minus_a, const Policy& pol, long long& log_scaling)
78 // special case: M(a,a,z) == exp(z)
79 long long scale = lltrunc(x, pol);
81 return exp(x - scale);
83 hypergeometric_1F1_igamma_series<T, Policy> s(b_minus_a, a - 1, x, pol);
84 log_scaling += s.log_scaling;
85 std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>();
86 T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter);
87 boost::math::policies::check_series_iterations<T>("boost::math::tgamma<%1%>(%1%,%1%)", max_iter, pol);
88 T log_prefix = x + boost::math::lgamma(b, pol) - boost::math::lgamma(a, pol);
89 long long scale = lltrunc(log_prefix);
91 return result * exp(log_prefix - scale);
94 template <class T, class Policy>
95 T hypergeometric_1F1_shift_on_a(T h, const T& a_local, const T& b_local, const T& x, int a_shift, const Policy& pol, long long& log_scaling)
98 T a = a_local + a_shift;
101 else if (a_shift > 0)
104 // Forward recursion on a is stable as long as 2a-b+z > 0.
105 // If 2a-b+z < 0 then backwards recursion is stable even though
106 // the function may be strictly increasing with a. Potentially
107 // we may need to split the recurrence in 2 sections - one using
108 // forward recursion, and one backwards.
110 // We will get the next seed value from the ratio
111 // on b as that's stable and quick to compute.
114 T crossover_a = (b_local - x) / 2;
115 int crossover_shift = itrunc(crossover_a - a_local);
117 if (crossover_shift > 1)
120 // Forwards recursion will start off unstable, but may switch to the stable direction later.
121 // Start in the middle and go in both directions:
123 if (crossover_shift > a_shift)
124 crossover_shift = a_shift;
125 crossover_a = a_local + crossover_shift;
126 boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef(crossover_a, b_local, x);
127 std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>();
128 T b_ratio = boost::math::tools::function_ratio_from_backwards_recurrence(b_coef, boost::math::policies::get_epsilon<T, Policy>(), max_iter);
129 boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_large_abz<%1%>(%1%,%1%,%1%)", max_iter, pol);
131 // Convert to a ratio:
132 // (1+a-b)M(a, b, z) - aM(a+1, b, z) + (b-1)M(a, b-1, z) = 0
134 // hence: M(a+1,b,z) = ((1+a-b) / a) M(a,b,z) + ((b-1) / a) M(a,b,z)/b_ratio
137 T second = ((1 + crossover_a - b_local) / crossover_a) + ((b_local - 1) / crossover_a) / b_ratio;
139 // Recurse down to a_local, compare values and re-normalise first and second:
141 boost::math::detail::hypergeometric_1F1_recurrence_a_coefficients<T> a_coef(crossover_a, b_local, x);
142 long long backwards_scale = 0;
143 T comparitor = boost::math::tools::apply_recurrence_relation_backward(a_coef, crossover_shift, second, first, &backwards_scale);
144 log_scaling -= backwards_scale;
145 if ((h < 1) && (tools::max_value<T>() * h > comparitor))
148 long long scale = lltrunc(log(h), pol) + 1;
150 log_scaling += scale;
154 second /= comparitor;
156 // Now we can recurse forwards for the rest of the range:
158 if (crossover_shift < a_shift)
160 boost::math::detail::hypergeometric_1F1_recurrence_a_coefficients<T> a_coef_2(crossover_a + 1, b_local, x);
161 h = boost::math::tools::apply_recurrence_relation_forward(a_coef_2, a_shift - crossover_shift - 1, first, second, &log_scaling);
169 // Regular case where forwards iteration is stable right from the start:
171 boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef(a_local, b_local, x);
172 std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>();
173 T b_ratio = boost::math::tools::function_ratio_from_backwards_recurrence(b_coef, boost::math::policies::get_epsilon<T, Policy>(), max_iter);
174 boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_large_abz<%1%>(%1%,%1%,%1%)", max_iter, pol);
176 // Convert to a ratio:
177 // (1+a-b)M(a, b, z) - aM(a+1, b, z) + (b-1)M(a, b-1, z) = 0
179 // hence: M(a+1,b,z) = ((1+a-b) / a) M(a,b,z) + ((b-1) / a) M(a,b,z)/b_ratio
181 T second = ((1 + a_local - b_local) / a_local) * h + ((b_local - 1) / a_local) * h / b_ratio;
182 boost::math::detail::hypergeometric_1F1_recurrence_a_coefficients<T> a_coef(a_local + 1, b_local, x);
183 h = boost::math::tools::apply_recurrence_relation_forward(a_coef, --a_shift, h, second, &log_scaling);
189 // We've calculated h for a larger value of a than we want, and need to recurse down.
190 // However, only forward iteration is stable, so calculate the ratio, compare values,
191 // and normalise. Note that we calculate the ratio on b and convert to a since the
192 // direction is the minimal solution for N->+INF.
194 // IMPORTANT: this is only currently called for a > b and therefore forwards iteration
195 // is the only stable direction as we will only iterate down until a ~ b, but we
196 // will check this with an assert:
198 BOOST_MATH_ASSERT(2 * a - b_local + x > 0);
199 boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef(a, b_local, x);
200 std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>();
201 T b_ratio = boost::math::tools::function_ratio_from_backwards_recurrence(b_coef, boost::math::policies::get_epsilon<T, Policy>(), max_iter);
202 boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_large_abz<%1%>(%1%,%1%,%1%)", max_iter, pol);
204 // Convert to a ratio:
205 // (1+a-b)M(a, b, z) - aM(a+1, b, z) + (b-1)M(a, b-1, z) = 0
207 // hence: M(a+1,b,z) = (1+a-b) / a M(a,b,z) + (b-1) / a M(a,b,z)/ (M(a,b,z)/M(a,b-1,z))
209 T first = 1; // arbitrary value;
210 T second = ((1 + a - b_local) / a) + ((b_local - 1) / a) * (1 / b_ratio);
216 boost::math::detail::hypergeometric_1F1_recurrence_a_coefficients<T> a_coef(a + 1, b_local, x);
217 T comparitor = boost::math::tools::apply_recurrence_relation_forward(a_coef, -(a_shift + 1), first, second);
218 if (boost::math::tools::min_value<T>() * comparitor > h)
220 // Ooops, need to rescale h:
221 long long rescale = lltrunc(log(fabs(h)));
222 T scale = exp(T(-rescale));
224 log_scaling += rescale;
232 template <class T, class Policy>
233 T hypergeometric_1F1_shift_on_b(T h, const T& a, const T& b_local, const T& x, int b_shift, const Policy& pol, long long& log_scaling)
237 T b = b_local + b_shift;
240 else if (b_shift > 0)
243 // We get here for b_shift > 0 when b > z. We can't use forward recursion on b - it's unstable,
244 // so grab the ratio and work backwards to b - b_shift and normalise.
246 boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef(a, b, x);
247 std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>();
249 T first = 1; // arbitrary value;
250 T second = 1 / boost::math::tools::function_ratio_from_backwards_recurrence(b_coef, boost::math::policies::get_epsilon<T, Policy>(), max_iter);
251 boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_large_abz<%1%>(%1%,%1%,%1%)", max_iter, pol);
257 // Reset coefficients and recurse:
259 boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef_2(a, b - 1, x);
260 long long local_scale = 0;
261 T comparitor = boost::math::tools::apply_recurrence_relation_backward(b_coef_2, --b_shift, first, second, &local_scale);
262 log_scaling -= local_scale;
263 if (boost::math::tools::min_value<T>() * comparitor > h)
265 // Ooops, need to rescale h:
266 long long rescale = lltrunc(log(fabs(h)));
267 T scale = exp(T(-rescale));
269 log_scaling += rescale;
279 // recurrence is trivial for a == b and method of ratios fails as the c-term goes to zero:
280 second = -b_local * (1 - b_local - x) * h / (b_local * (b_local - 1));
284 BOOST_MATH_ASSERT(!is_negative_integer(b - a));
285 boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef(a, b_local, x);
286 std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>();
287 second = h / boost::math::tools::function_ratio_from_backwards_recurrence(b_coef, boost::math::policies::get_epsilon<T, Policy>(), max_iter);
288 boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_large_abz<%1%>(%1%,%1%,%1%)", max_iter, pol);
294 boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef_2(a, b_local - 1, x);
295 h = boost::math::tools::apply_recurrence_relation_backward(b_coef_2, -(++b_shift), h, second, &log_scaling);
302 template <class T, class Policy>
303 T hypergeometric_1F1_large_igamma(const T& a, const T& b, const T& x, const T& b_minus_a, const Policy& pol, long long& log_scaling)
307 // We need a < b < z in order to ensure there's at least a chance of convergence,
308 // we can use recurrence relations to shift forwards on a+b or just a to achieve this,
309 // for decent accuracy, try to keep 2b - 1 < a < 2b < z
311 int b_shift = b * 2 < x ? 0 : itrunc(b - x / 2);
312 int a_shift = a > b - b_shift ? -itrunc(b - b_shift - a - 1) : -itrunc(b - b_shift - a);
316 // Might as well do all the shifting on b as scale a downwards:
321 T a_local = a - a_shift;
322 T b_local = b - b_shift;
323 T b_minus_a_local = (a_shift == 0) && (b_shift == 0) ? b_minus_a : b_local - a_local;
324 long long local_scaling = 0;
325 T h = hypergeometric_1F1_igamma(a_local, b_local, x, b_minus_a_local, pol, local_scaling);
326 log_scaling += local_scaling;
329 // Apply shifts on a and b as required:
331 h = hypergeometric_1F1_shift_on_a(h, a_local, b_local, x, a_shift, pol, log_scaling);
332 h = hypergeometric_1F1_shift_on_b(h, a, b_local, x, b_shift, pol, log_scaling);
337 template <class T, class Policy>
338 T hypergeometric_1F1_large_series(const T& a, const T& b, const T& z, const Policy& pol, long long& log_scaling)
342 // We make a small, and b > z:
344 int a_shift(0), b_shift(0);
347 a_shift = itrunc(a) - 5;
348 b_shift = b < z ? itrunc(b - z - 1) : 0;
351 // If a_shift is trivially small, there's really not much point in losing
352 // accuracy to save a couple of iterations:
356 T a_local = a - a_shift;
357 T b_local = b - b_shift;
358 T h = boost::math::detail::hypergeometric_1F1_generic_series(a_local, b_local, z, pol, log_scaling, "hypergeometric_1F1_large_series<%1%>(a,b,z)");
360 // Apply shifts on a and b as required:
362 if (a_shift && (a_local == 0))
365 // Shifting on a via method of ratios in hypergeometric_1F1_shift_on_a fails when
366 // a_local == 0. However, the value of h calculated was trivial (unity), so
367 // calculate a second 1F1 for a == 1 and recurse as normal:
370 T h2 = boost::math::detail::hypergeometric_1F1_generic_series(T(a_local + 1), b_local, z, pol, scale, "hypergeometric_1F1_large_series<%1%>(a,b,z)");
371 if (scale != log_scaling)
373 h2 *= exp(T(scale - log_scaling));
375 boost::math::detail::hypergeometric_1F1_recurrence_a_coefficients<T> coef(a_local + 1, b_local, z);
376 h = boost::math::tools::apply_recurrence_relation_forward(coef, a_shift - 1, h, h2, &log_scaling);
377 h = hypergeometric_1F1_shift_on_b(h, a, b_local, z, b_shift, pol, log_scaling);
381 h = hypergeometric_1F1_shift_on_a(h, a_local, b_local, z, a_shift, pol, log_scaling);
382 h = hypergeometric_1F1_shift_on_b(h, a, b_local, z, b_shift, pol, log_scaling);
387 template <class T, class Policy>
388 T hypergeometric_1F1_large_13_3_6_series(const T& a, const T& b, const T& z, const Policy& pol, long long& log_scaling)
392 // A&S 13.3.6 is good only when a ~ b, but isn't too fussy on the size of z.
393 // So shift b to match a (b shifting seems to be more stable via method of ratios).
395 int b_shift = itrunc(b - a);
396 T b_local = b - b_shift;
397 T h = boost::math::detail::hypergeometric_1F1_AS_13_3_6(a, b_local, z, T(b_local - a), pol, log_scaling);
398 return hypergeometric_1F1_shift_on_b(h, a, b_local, z, b_shift, pol, log_scaling);
401 template <class T, class Policy>
402 T hypergeometric_1F1_large_abz(const T& a, const T& b, const T& z, const Policy& pol, long long& log_scaling)
406 // This is the selection logic to pick the "best" method.
407 // We have a,b,z >> 0 and need to compute the approximate cost of each method
408 // and then select whichever wins out.
413 method_shifted_series,
418 // Cost of direct series, is the approx number of terms required until we hit convergence:
420 T current_cost = (sqrt(16 * z * (3 * a + z) + 9 * b * b - 24 * b * z) - 3 * b + 4 * z) / 6;
421 method current_method = method_series;
423 // Cost of shifted series, is the number of recurrences required to move to a zone where
424 // the series is convergent right from the start.
425 // Note that recurrence relations fail for very small b, and too many recurrences on a
426 // will completely destroy all our digits.
427 // Also note that the method fails when b-a is a negative integer unless b is already
428 // larger than z and thus does not need shifting.
430 T cost = a + ((b < z) ? T(z - b) : T(0));
431 if((b > 1) && (cost < current_cost) && ((b > z) || !is_negative_integer(b-a)))
433 current_method = method_shifted_series;
437 // Cost for gamma function method is the number of recurrences required to move it
438 // into a convergent zone, note that recurrence relations fail for very small b.
439 // Also add on a fudge factor to account for the fact that this method is both
440 // more expensive to compute (requires gamma functions), and less accurate than the
443 T b_shift = fabs(b * 2 < z ? T(0) : T(b - z / 2));
444 T a_shift = fabs(a > b - b_shift ? T(-(b - b_shift - a - 1)) : T(-(b - b_shift - a)));
445 cost = 1000 + b_shift + a_shift;
446 if((b > 1) && (cost <= current_cost))
448 current_method = method_gamma;
452 // Cost for bessel approximation is the number of recurrences required to make a ~ b,
453 // Note that recurrence relations fail for very small b. We also have issue with large
454 // z: either overflow/numeric instability or else the series goes divergent. We seem to be
455 // OK for z smaller than log_max_value<Quad> though, maybe we can stretch a little further
456 // but that's not clear...
457 // Also need to add on a fudge factor to the cost to account for the fact that we need
458 // to calculate the Bessel functions, this is not quite as high as the gamma function
459 // method above as this is generally more accurate and so preferred if the methods are close:
461 cost = 50 + fabs(b - a);
462 if((b > 1) && (cost <= current_cost) && (z < tools::log_max_value<T>()) && (z < 11356) && (b - a != 0.5f))
464 current_method = method_bessel;
468 switch (current_method)
471 return detail::hypergeometric_1F1_generic_series(a, b, z, pol, log_scaling, "hypergeometric_1f1_large_abz<%1%>(a,b,z)");
472 case method_shifted_series:
473 return detail::hypergeometric_1F1_large_series(a, b, z, pol, log_scaling);
475 return detail::hypergeometric_1F1_large_igamma(a, b, z, T(b - a), pol, log_scaling);
477 return detail::hypergeometric_1F1_large_13_3_6_series(a, b, z, pol, log_scaling);
479 return 0; // We don't get here!
484 #endif // BOOST_HYPERGEOMETRIC_1F1_LARGE_ABZ_HPP_