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1 | // Copyright John Maddock 2017. |
2 | // Copyright Paul A. Bristow 2016, 2017, 2018. | |
3 | // Copyright Nicholas Thompson 2018 | |
4 | ||
5 | // Distributed under the Boost Software License, Version 1.0. | |
6 | // (See accompanying file LICENSE_1_0.txt or | |
7 | // copy at http ://www.boost.org/LICENSE_1_0.txt). | |
8 | ||
9 | #ifndef BOOST_MATH_SF_LAMBERT_W_HPP | |
10 | #define BOOST_MATH_SF_LAMBERT_W_HPP | |
11 | ||
12 | #ifdef _MSC_VER | |
13 | #pragma warning(disable : 4127) | |
14 | #endif | |
15 | ||
16 | /* | |
17 | Implementation of an algorithm for the Lambert W0 and W-1 real-only functions. | |
18 | ||
19 | This code is based in part on the algorithm by | |
20 | Toshio Fukushima, | |
21 | "Precise and fast computation of Lambert W-functions without transcendental function evaluations", | |
22 | J.Comp.Appl.Math. 244 (2013) 77-89, | |
23 | and on a C/C++ version by Darko Veberic, darko.veberic@ijs.si | |
24 | based on the Fukushima algorithm and Toshio Fukushima's FORTRAN version of his algorithm. | |
25 | ||
26 | First derivative of Lambert_w is derived from | |
27 | Princeton Companion to Applied Mathematics, 'The Lambert-W function', Section 1.3: Series and Generating Functions. | |
28 | ||
29 | */ | |
30 | ||
31 | /* | |
32 | TODO revise this list of macros. | |
33 | Some macros that will show some (or much) diagnostic values if #defined. | |
34 | //[boost_math_instrument_lambert_w_macros | |
35 | ||
36 | // #define-able macros | |
37 | BOOST_MATH_INSTRUMENT_LAMBERT_W_HALLEY // Halley refinement diagnostics. | |
38 | BOOST_MATH_INSTRUMENT_LAMBERT_W_PRECISION // Precision. | |
39 | BOOST_MATH_INSTRUMENT_LAMBERT_WM1 // W1 branch diagnostics. | |
40 | BOOST_MATH_INSTRUMENT_LAMBERT_WM1_HALLEY // Halley refinement diagnostics only for W-1 branch. | |
41 | BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY // K > 64, z > -1.0264389699511303e-26 | |
42 | BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP // Show results from W-1 lookup table. | |
43 | BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER // Schroeder refinement diagnostics. | |
44 | BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS // Number of terms used for near-singularity series. | |
45 | BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES // Show evaluation of series near branch singularity. | |
46 | BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
47 | BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS // Show evaluation of series for small z. | |
48 | //] [/boost_math_instrument_lambert_w_macros] | |
49 | */ | |
50 | ||
51 | #include <boost/math/policies/error_handling.hpp> | |
52 | #include <boost/math/policies/policy.hpp> | |
53 | #include <boost/math/tools/promotion.hpp> | |
54 | #include <boost/math/special_functions/fpclassify.hpp> | |
55 | #include <boost/math/special_functions/log1p.hpp> // for log (1 + x) | |
56 | #include <boost/math/constants/constants.hpp> // For exp_minus_one == 3.67879441171442321595523770161460867e-01. | |
57 | #include <boost/math/special_functions/pow.hpp> // powers with compile time exponent, used in arbitrary precision code. | |
58 | #include <boost/math/tools/series.hpp> // series functor. | |
59 | //#include <boost/math/tools/polynomial.hpp> // polynomial. | |
60 | #include <boost/math/tools/rational.hpp> // evaluate_polynomial. | |
92f5a8d4 TL |
61 | #include <boost/math/tools/precision.hpp> // boost::math::tools::max_value(). |
62 | #include <boost/math/tools/big_constant.hpp> | |
f67539c2 | 63 | #include <boost/math/tools/cxx03_warn.hpp> |
92f5a8d4 | 64 | |
1e59de90 TL |
65 | #ifndef BOOST_MATH_STANDALONE |
66 | #include <boost/lexical_cast.hpp> | |
67 | #endif | |
68 | ||
92f5a8d4 TL |
69 | #include <limits> |
70 | #include <cmath> | |
71 | #include <limits> | |
72 | #include <exception> | |
1e59de90 TL |
73 | #include <type_traits> |
74 | #include <cstdint> | |
92f5a8d4 TL |
75 | |
76 | // Needed for testing and diagnostics only. | |
77 | #include <iostream> | |
78 | #include <typeinfo> | |
79 | #include <boost/math/special_functions/next.hpp> // For float_distance. | |
80 | ||
1e59de90 | 81 | using lookup_t = double; // Type for lookup table (double or float, or even long double?) |
92f5a8d4 TL |
82 | |
83 | //#include "J:\Cpp\Misc\lambert_w_lookup_table_generator\lambert_w_lookup_table.ipp" | |
84 | // #include "lambert_w_lookup_table.ipp" // Boost.Math version. | |
85 | #include <boost/math/special_functions/detail/lambert_w_lookup_table.ipp> | |
86 | ||
87 | #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128) | |
88 | // | |
89 | // This is the only way we can avoid | |
90 | // warning: non-standard suffix on floating constant [-Wpedantic] | |
91 | // when building with -Wall -pedantic. Neither __extension__ | |
f67539c2 | 92 | // nor #pragma diagnostic ignored work :( |
92f5a8d4 TL |
93 | // |
94 | #pragma GCC system_header | |
95 | #endif | |
96 | ||
97 | namespace boost { | |
98 | namespace math { | |
99 | namespace lambert_w_detail { | |
100 | ||
101 | //! \brief Applies a single Halley step to make a better estimate of Lambert W. | |
102 | //! \details Used the simplified formulae obtained from | |
103 | //! http://www.wolframalpha.com/input/?i=%5B2(z+exp(z)-w)+d%2Fdx+(z+exp(z)-w)%5D+%2F+%5B2+(d%2Fdx+(z+exp(z)-w))%5E2+-+(z+exp(z)-w)+d%5E2%2Fdx%5E2+(z+exp(z)-w)%5D | |
104 | //! [2(z exp(z)-w) d/dx (z exp(z)-w)] / [2 (d/dx (z exp(z)-w))^2 - (z exp(z)-w) d^2/dx^2 (z exp(z)-w)] | |
105 | ||
106 | //! \tparam T floating-point (or fixed-point) type. | |
107 | //! \param w_est Lambert W estimate. | |
108 | //! \param z Argument z for Lambert_w function. | |
109 | //! \returns New estimate of Lambert W, hopefully improved. | |
110 | //! | |
1e59de90 | 111 | template <typename T> |
92f5a8d4 TL |
112 | inline T lambert_w_halley_step(T w_est, const T z) |
113 | { | |
114 | BOOST_MATH_STD_USING | |
115 | T e = exp(w_est); | |
116 | w_est -= 2 * (w_est + 1) * (e * w_est - z) / (z * (w_est + 2) + e * (w_est * (w_est + 2) + 2)); | |
117 | return w_est; | |
1e59de90 | 118 | } // template <typename T> lambert_w_halley_step(T w_est, T z) |
92f5a8d4 TL |
119 | |
120 | //! \brief Halley iterate to refine Lambert_w estimate, | |
121 | //! taking at least one Halley_step. | |
122 | //! Repeat Halley steps until the *last step* had fewer than half the digits wrong, | |
123 | //! the step we've just taken should have been sufficient to have completed the iteration. | |
124 | ||
125 | //! \tparam T floating-point (or fixed-point) type. | |
126 | //! \param z Argument z for Lambert_w function. | |
127 | //! \param w_est Lambert w estimate. | |
1e59de90 TL |
128 | template <typename T> |
129 | inline T lambert_w_halley_iterate(T w_est, const T z) | |
92f5a8d4 TL |
130 | { |
131 | BOOST_MATH_STD_USING | |
132 | static const T max_diff = boost::math::tools::root_epsilon<T>() * fabs(w_est); | |
133 | ||
134 | T w_new = lambert_w_halley_step(w_est, z); | |
135 | T diff = fabs(w_est - w_new); | |
136 | while (diff > max_diff) | |
137 | { | |
138 | w_est = w_new; | |
139 | w_new = lambert_w_halley_step(w_est, z); | |
140 | diff = fabs(w_est - w_new); | |
141 | } | |
142 | return w_new; | |
1e59de90 | 143 | } // template <typename T> lambert_w_halley_iterate(T w_est, T z) |
92f5a8d4 TL |
144 | |
145 | // Two Halley function versions that either | |
1e59de90 | 146 | // single step (if std::false_type) or iterate (if std::true_type). |
92f5a8d4 | 147 | // Selected at compile-time using parameter 3. |
1e59de90 TL |
148 | template <typename T> |
149 | inline T lambert_w_maybe_halley_iterate(T z, T w, std::false_type const&) | |
92f5a8d4 TL |
150 | { |
151 | return lambert_w_halley_step(z, w); // Single step. | |
152 | } | |
153 | ||
1e59de90 TL |
154 | template <typename T> |
155 | inline T lambert_w_maybe_halley_iterate(T z, T w, std::true_type const&) | |
92f5a8d4 TL |
156 | { |
157 | return lambert_w_halley_iterate(z, w); // Iterate steps. | |
158 | } | |
159 | ||
160 | //! maybe_reduce_to_double function, | |
161 | //! Two versions that have a compile-time option to | |
f67539c2 | 162 | //! reduce argument z to double precision (if true_type). |
92f5a8d4 TL |
163 | //! Version is selected at compile-time using parameter 2. |
164 | ||
1e59de90 TL |
165 | template <typename T> |
166 | inline double maybe_reduce_to_double(const T& z, const std::true_type&) | |
92f5a8d4 TL |
167 | { |
168 | return static_cast<double>(z); // Reduce to double precision. | |
169 | } | |
170 | ||
1e59de90 TL |
171 | template <typename T> |
172 | inline T maybe_reduce_to_double(const T& z, const std::false_type&) | |
92f5a8d4 TL |
173 | { // Don't reduce to double. |
174 | return z; | |
175 | } | |
176 | ||
1e59de90 TL |
177 | template <typename T> |
178 | inline double must_reduce_to_double(const T& z, const std::true_type&) | |
92f5a8d4 TL |
179 | { |
180 | return static_cast<double>(z); // Reduce to double precision. | |
181 | } | |
182 | ||
1e59de90 TL |
183 | template <typename T> |
184 | inline double must_reduce_to_double(const T& z, const std::false_type&) | |
92f5a8d4 | 185 | { // try a lexical_cast and hope for the best: |
1e59de90 | 186 | #ifndef BOOST_MATH_STANDALONE |
92f5a8d4 | 187 | return boost::lexical_cast<double>(z); |
1e59de90 TL |
188 | #else |
189 | static_assert(sizeof(T) == 0, "Unsupported in standalone mode: don't know how to cast your number type to a double."); | |
190 | return 0.0; | |
191 | #endif | |
92f5a8d4 TL |
192 | } |
193 | ||
194 | //! \brief Schroeder method, fifth-order update formula, | |
195 | //! \details See T. Fukushima page 80-81, and | |
196 | //! A. Householder, The Numerical Treatment of a Single Nonlinear Equation, | |
197 | //! McGraw-Hill, New York, 1970, section 4.4. | |
198 | //! Fukushima algorithm switches to @c schroeder_update after pre-computed bisections, | |
199 | //! chosen to ensure that the result will be achieve the +/- 10 epsilon target. | |
200 | //! \param w Lambert w estimate from bisection or series. | |
201 | //! \param y bracketing value from bisection. | |
202 | //! \returns Refined estimate of Lambert w. | |
203 | ||
204 | // Schroeder refinement, called unless NOT required by precision policy. | |
205 | template<typename T> | |
1e59de90 | 206 | inline T schroeder_update(const T w, const T y) |
92f5a8d4 TL |
207 | { |
208 | // Compute derivatives using 5th order Schroeder refinement. | |
209 | // Since this is the final step, it will always use the highest precision type T. | |
210 | // Example of Call: | |
211 | // result = schroeder_update(w, y); | |
212 | //where | |
213 | // w is estimate of Lambert W (from bisection or series). | |
214 | // y is z * e^-w. | |
215 | ||
216 | BOOST_MATH_STD_USING // Aid argument dependent lookup of abs. | |
217 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER | |
218 | std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); | |
219 | using boost::math::float_distance; | |
220 | T fd = float_distance<T>(w, y); | |
221 | std::cout << "Schroder "; | |
222 | if (abs(fd) < 214748000.) | |
223 | { | |
224 | std::cout << " Distance = "<< static_cast<int>(fd); | |
225 | } | |
226 | else | |
227 | { | |
228 | std::cout << "Difference w - y = " << (w - y) << "."; | |
229 | } | |
230 | std::cout << std::endl; | |
231 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER | |
232 | // Fukushima equation 18, page 6. | |
233 | const T f0 = w - y; // f0 = w - y. | |
234 | const T f1 = 1 + y; // f1 = df/dW | |
235 | const T f00 = f0 * f0; | |
236 | const T f11 = f1 * f1; | |
237 | const T f0y = f0 * y; | |
238 | const T result = | |
239 | w - 4 * f0 * (6 * f1 * (f11 + f0y) + f00 * y) / | |
240 | (f11 * (24 * f11 + 36 * f0y) + | |
241 | f00 * (6 * y * y + 8 * f1 * y + f0y)); // Fukushima Page 81, equation 21 from equation 20. | |
242 | ||
243 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER | |
244 | std::cout << "Schroeder refined " << w << " " << y << ", difference " << w-y << ", change " << w - result << ", to result " << result << std::endl; | |
245 | std::cout.precision(saved_precision); // Restore. | |
246 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER | |
247 | ||
248 | return result; | |
249 | } // template<typename T = double> T schroeder_update(const T w, const T y) | |
250 | ||
251 | //! \brief Series expansion used near the singularity/branch point z = -exp(-1) = -3.6787944. | |
252 | //! Wolfram InverseSeries[Series[sqrt[2(p Exp[1 + p] + 1)], {p,-1, 20}]] | |
253 | //! Wolfram command used to obtain 40 series terms at 50 decimal digit precision was | |
254 | //! N[InverseSeries[Series[Sqrt[2(p Exp[1 + p] + 1)], { p,-1,40 }]], 50] | |
255 | //! -1+p-p^2/3+(11 p^3)/72-(43 p^4)/540+(769 p^5)/17280-(221 p^6)/8505+(680863 p^7)/43545600 ... | |
256 | //! Decimal values of specifications for built-in floating-point types below | |
257 | //! are at least 21 digits precision == max_digits10 for long double. | |
258 | //! Longer decimal digits strings are rationals evaluated using Wolfram. | |
259 | ||
260 | template<typename T> | |
261 | T lambert_w_singularity_series(const T p) | |
262 | { | |
263 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES | |
264 | std::size_t saved_precision = std::cout.precision(3); | |
265 | std::cout << "Singularity_series Lambert_w p argument = " << p << std::endl; | |
266 | std::cout | |
267 | //<< "Argument Type = " << typeid(T).name() | |
268 | //<< ", max_digits10 = " << std::numeric_limits<T>::max_digits10 | |
269 | //<< ", epsilon = " << std::numeric_limits<T>::epsilon() | |
270 | << std::endl; | |
271 | std::cout.precision(saved_precision); | |
272 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES | |
273 | ||
274 | static const T q[] = | |
275 | { | |
276 | -static_cast<T>(1), // j0 | |
277 | +T(1), // j1 | |
278 | -T(1) / 3, // 1/3 j2 | |
279 | +T(11) / 72, // 0.152777777777777778, // 11/72 j3 | |
280 | -T(43) / 540, // 0.0796296296296296296, // 43/540 j4 | |
281 | +T(769) / 17280, // 0.0445023148148148148, j5 | |
282 | -T(221) / 8505, // 0.0259847148736037625, j6 | |
283 | //+T(0.0156356325323339212L), // j7 | |
284 | //+T(0.015635632532333921222810111699000587889476778365667L), // j7 from Wolfram N[680863/43545600, 50] | |
285 | +T(680863uLL) / 43545600uLL, // +0.0156356325323339212, j7 | |
286 | //-T(0.00961689202429943171L), // j8 | |
287 | -T(1963uLL) / 204120uLL, // 0.00961689202429943171, j8 | |
288 | //-T(0.0096168920242994317068391142465216539290613364687439L), // j8 from Wolfram N[1963/204120, 50] | |
289 | +T(226287557uLL) / 37623398400uLL, // 0.00601454325295611786, j9 | |
290 | -T(5776369uLL) / 1515591000uLL, // 0.00381129803489199923, j10 | |
291 | //+T(0.00244087799114398267L), j11 0.0024408779911439826658968585286437530215699919795550 | |
292 | +T(169709463197uLL) / 69528040243200uLL, // j11 | |
293 | // -T(0.00157693034468678425L), // j12 -0.0015769303446867842539234095399314115973161850314723 | |
294 | -T(1118511313uLL) / 709296588000uLL, // j12 | |
295 | +T(667874164916771uLL) / 650782456676352000uLL, // j13 | |
296 | //+T(0.00102626332050760715L), // j13 0.0010262633205076071544375481533906861056468041465973 | |
297 | -T(500525573uLL) / 744761417400uLL, // j14 | |
298 | // -T(0.000672061631156136204L), j14 | |
299 | //+T(1003663334225097487uLL) / 234281684403486720000uLL, // j15 0.00044247306181462090993020760858473726479232802068800 error C2177: constant too big | |
300 | //+T(0.000442473061814620910L, // j15 | |
301 | BOOST_MATH_BIG_CONSTANT(T, 64, +0.000442473061814620910), // j15 | |
302 | // -T(0.000292677224729627445L), // j16 | |
303 | BOOST_MATH_BIG_CONSTANT(T, 64, -0.000292677224729627445), // j16 | |
304 | //+T(0.000194387276054539318L), // j17 | |
305 | BOOST_MATH_BIG_CONSTANT(T, 64, 0.000194387276054539318), // j17 | |
306 | //-T(0.000129574266852748819L), // j18 | |
307 | BOOST_MATH_BIG_CONSTANT(T, 64, -0.000129574266852748819), // j18 | |
308 | //+T(0.0000866503580520812717L), // j19 N[+1150497127780071399782389/13277465363600276402995200000, 50] 0.000086650358052081271660451590462390293190597827783288 | |
309 | BOOST_MATH_BIG_CONSTANT(T, 64, +0.0000866503580520812717), // j19 | |
310 | //-T(0.0000581136075044138168L) // j20 N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913 | |
311 | // -T(2853534237182741069uLL) / 49102686267859224000000uLL // j20 // error C2177: constant too big, | |
312 | // so must use BOOST_MATH_BIG_CONSTANT(T, ) format in hope of using suffix Q for quad or decimal digits string for others. | |
313 | //-T(0.000058113607504413816772205464778828177256611844221913L), // j20 N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913 | |
314 | BOOST_MATH_BIG_CONSTANT(T, 113, -0.000058113607504413816772205464778828177256611844221913) // j20 - last used by Fukushima | |
315 | // More terms don't seem to give any improvement (worse in fact) and are not use for many z values. | |
316 | //BOOST_MATH_BIG_CONSTANT(T, +0.000039076684867439051635395583044527492132109160553593), // j21 | |
317 | //BOOST_MATH_BIG_CONSTANT(T, -0.000026338064747231098738584082718649443078703982217219), // j22 | |
318 | //BOOST_MATH_BIG_CONSTANT(T, +0.000017790345805079585400736282075184540383274460464169), // j23 | |
319 | //BOOST_MATH_BIG_CONSTANT(T, -0.000012040352739559976942274116578992585158113153190354), // j24 | |
320 | //BOOST_MATH_BIG_CONSTANT(T, +8.1635319824966121713827512573558687050675701559448E-6), // j25 | |
321 | //BOOST_MATH_BIG_CONSTANT(T, -5.5442032085673591366657251660804575198155559225316E-6) // j26 | |
322 | // -T(5.5442032085673591366657251660804575198155559225316E-6L) // j26 | |
323 | // 21 to 26 Added for long double. | |
324 | }; // static const T q[] | |
325 | ||
326 | /* | |
327 | // Temporary copy of original double values for comparison; these are reproduced well. | |
328 | static const T q[] = | |
329 | { | |
330 | -1L, // j0 | |
331 | +1L, // j1 | |
332 | -0.333333333333333333L, // 1/3 j2 | |
333 | +0.152777777777777778L, // 11/72 j3 | |
334 | -0.0796296296296296296L, // 43/540 | |
335 | +0.0445023148148148148L, | |
336 | -0.0259847148736037625L, | |
337 | +0.0156356325323339212L, | |
338 | -0.00961689202429943171L, | |
339 | +0.00601454325295611786L, | |
340 | -0.00381129803489199923L, | |
341 | +0.00244087799114398267L, | |
342 | -0.00157693034468678425L, | |
343 | +0.00102626332050760715L, | |
344 | -0.000672061631156136204L, | |
345 | +0.000442473061814620910L, | |
346 | -0.000292677224729627445L, | |
347 | +0.000194387276054539318L, | |
348 | -0.000129574266852748819L, | |
349 | +0.0000866503580520812717L, | |
350 | -0.0000581136075044138168L // j20 | |
351 | }; | |
352 | */ | |
353 | ||
354 | // Decide how many series terms to use, increasing as z approaches the singularity, | |
355 | // balancing run-time versus computational noise from round-off. | |
356 | // In practice, we truncate the series expansion at a certain order. | |
357 | // If the order is too large, not only does the amount of computation increase, | |
358 | // but also the round-off errors accumulate. | |
359 | // See Fukushima equation 35, page 85 for logic of choice of number of series terms. | |
360 | ||
361 | BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs. | |
362 | ||
363 | const T absp = abs(p); | |
364 | ||
365 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS | |
366 | { | |
367 | int terms = 20; // Default to using all terms. | |
f67539c2 | 368 | if (absp < 0.01159) |
92f5a8d4 TL |
369 | { // Very near singularity. |
370 | terms = 6; | |
371 | } | |
372 | else if (absp < 0.0766) | |
373 | { // Near singularity. | |
374 | terms = 10; | |
375 | } | |
376 | std::streamsize saved_precision = std::cout.precision(3); | |
377 | std::cout << "abs(p) = " << absp << ", terms = " << terms << std::endl; | |
378 | std::cout.precision(saved_precision); | |
379 | } | |
380 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS | |
381 | ||
382 | if (absp < 0.01159) | |
383 | { // Only 6 near-singularity series terms are useful. | |
384 | return | |
385 | -1 + | |
386 | p * (1 + | |
387 | p * (q[2] + | |
388 | p * (q[3] + | |
389 | p * (q[4] + | |
390 | p * (q[5] + | |
391 | p * q[6] | |
392 | ))))); | |
393 | } | |
394 | else if (absp < 0.0766) // Use 10 near-singularity series terms. | |
395 | { // Use 10 near-singularity series terms. | |
396 | return | |
397 | -1 + | |
398 | p * (1 + | |
399 | p * (q[2] + | |
400 | p * (q[3] + | |
401 | p * (q[4] + | |
402 | p * (q[5] + | |
403 | p * (q[6] + | |
404 | p * (q[7] + | |
405 | p * (q[8] + | |
406 | p * (q[9] + | |
407 | p * q[10] | |
408 | ))))))))); | |
409 | } | |
410 | else | |
411 | { // Use all 20 near-singularity series terms. | |
412 | return | |
413 | -1 + | |
414 | p * (1 + | |
415 | p * (q[2] + | |
416 | p * (q[3] + | |
417 | p * (q[4] + | |
418 | p * (q[5] + | |
419 | p * (q[6] + | |
420 | p * (q[7] + | |
421 | p * (q[8] + | |
422 | p * (q[9] + | |
423 | p * (q[10] + | |
424 | p * (q[11] + | |
425 | p * (q[12] + | |
426 | p * (q[13] + | |
427 | p * (q[14] + | |
428 | p * (q[15] + | |
429 | p * (q[16] + | |
430 | p * (q[17] + | |
431 | p * (q[18] + | |
432 | p * (q[19] + | |
433 | p * q[20] // Last Fukushima term. | |
434 | ))))))))))))))))))); | |
435 | // + // more terms for more precise T: long double ... | |
436 | //// but makes almost no difference, so don't use more terms? | |
437 | // p*q[21] + | |
438 | // p*q[22] + | |
439 | // p*q[23] + | |
440 | // p*q[24] + | |
441 | // p*q[25] | |
442 | // ))))))))))))))))))); | |
443 | } | |
444 | } // template<typename T = double> T lambert_w_singularity_series(const T p) | |
445 | ||
446 | ||
447 | ///////////////////////////////////////////////////////////////////////////////////////////// | |
448 | ||
449 | //! \brief Series expansion used near zero (abs(z) < 0.05). | |
450 | //! \details | |
451 | //! Coefficients of the inverted series expansion of the Lambert W function around z = 0. | |
452 | //! Tosio Fukushima always uses all 17 terms of a Taylor series computed using Wolfram with | |
453 | //! InverseSeries[Series[z Exp[z],{z,0,17}]] | |
454 | //! Tosio Fukushima / Journal of Computational and Applied Mathematics 244 (2013) page 86. | |
455 | ||
456 | //! Decimal values of specifications for built-in floating-point types below | |
457 | //! are 21 digits precision == max_digits10 for long double. | |
458 | //! Care! Some coefficients might overflow some fixed_point types. | |
459 | ||
460 | //! This version is intended to allow use by user-defined types | |
461 | //! like Boost.Multiprecision quad and cpp_dec_float types. | |
462 | //! The three specializations below for built-in float, double | |
463 | //! (and perhaps long double) will be chosen in preference for these types. | |
464 | ||
465 | //! This version uses rationals computed by Wolfram as far as possible, | |
466 | //! limited by maximum size of uLL integers. | |
467 | //! For higher term, uses decimal digit strings computed by Wolfram up to the maximum possible using uLL rationals, | |
468 | //! and then higher coefficients are computed as necessary using function lambert_w0_small_z_series_term | |
469 | //! until the precision required by the policy is achieved. | |
470 | //! InverseSeries[Series[z Exp[z],{z,0,34}]] also computed. | |
471 | ||
472 | // Series evaluation for LambertW(z) as z -> 0. | |
473 | // See http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/ | |
474 | // http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/MainEq1.L.gif | |
475 | ||
476 | //! \brief lambert_w0_small_z uses a tag_type to select a variant depending on the size of the type. | |
477 | //! The Lambert W is computed by lambert_w0_small_z for small z. | |
478 | //! The cutoff for z smallness determined by Tosio Fukushima by trial and error is (abs(z) < 0.05), | |
479 | //! but the optimum might be a function of the size of the type of z. | |
480 | ||
481 | //! \details | |
482 | //! The tag_type selection is based on the value @c std::numeric_limits<T>::max_digits10. | |
483 | //! This allows distinguishing between long double types that commonly vary between 64 and 80-bits, | |
484 | //! and also compilers that have a float type using 64 bits and/or long double using 128-bits. | |
485 | //! It assumes that max_digits10 is defined correctly or this might fail to make the correct selection. | |
486 | //! causing very small differences in computing lambert_w that would be very difficult to detect and diagnose. | |
487 | //! Cannot switch on @c std::numeric_limits<>::max() because comparison values may overflow the compiler limit. | |
488 | //! Cannot switch on @c std::numeric_limits<long double>::max_exponent10() | |
489 | //! because both 80 and 128 bit floating-point implementations use 11 bits for the exponent. | |
490 | //! So must rely on @c std::numeric_limits<long double>::max_digits10. | |
491 | ||
492 | //! Specialization of float zero series expansion used for small z (abs(z) < 0.05). | |
493 | //! Specializations of lambert_w0_small_z for built-in types. | |
494 | //! These specializations should be chosen in preference to T version. | |
495 | //! For example: lambert_w0_small_z(0.001F) should use the float version. | |
496 | //! (Parameter Policy is not used by built-in types when all terms are used during an inline computation, | |
497 | //! but for the tag_type selection to work, they all must include Policy in their signature. | |
498 | ||
499 | // Forward declaration of variants of lambert_w0_small_z. | |
1e59de90 TL |
500 | template <typename T, typename Policy> |
501 | T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 0> const&); // for float (32-bit) type. | |
92f5a8d4 | 502 | |
1e59de90 TL |
503 | template <typename T, typename Policy> |
504 | T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 1> const&); // for double (64-bit) type. | |
92f5a8d4 | 505 | |
1e59de90 TL |
506 | template <typename T, typename Policy> |
507 | T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 2> const&); // for long double (double extended 80-bit) type. | |
92f5a8d4 | 508 | |
1e59de90 TL |
509 | template <typename T, typename Policy> |
510 | T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 3> const&); // for long double (128-bit) type. | |
92f5a8d4 | 511 | |
1e59de90 TL |
512 | template <typename T, typename Policy> |
513 | T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 4> const&); // for float128 quadmath Q type. | |
92f5a8d4 | 514 | |
1e59de90 TL |
515 | template <typename T, typename Policy> |
516 | T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 5> const&); // Generic multiprecision T. | |
92f5a8d4 | 517 | // Set tag_type depending on max_digits10. |
1e59de90 | 518 | template <typename T, typename Policy> |
92f5a8d4 TL |
519 | T lambert_w0_small_z(T x, const Policy& pol) |
520 | { //std::numeric_limits<T>::max_digits10 == 36 ? 3 : // 128-bit long double. | |
1e59de90 | 521 | using tag_type = std::integral_constant<int, |
92f5a8d4 TL |
522 | std::numeric_limits<T>::is_specialized == 0 ? 5 : |
523 | #ifndef BOOST_NO_CXX11_NUMERIC_LIMITS | |
524 | std::numeric_limits<T>::max_digits10 <= 9 ? 0 : // for float 32-bit. | |
525 | std::numeric_limits<T>::max_digits10 <= 17 ? 1 : // for double 64-bit. | |
526 | std::numeric_limits<T>::max_digits10 <= 22 ? 2 : // for 80-bit double extended. | |
527 | std::numeric_limits<T>::max_digits10 < 37 ? 4 // for both 128-bit long double (3) and 128-bit quad suffix Q type (4). | |
528 | #else | |
529 | std::numeric_limits<T>::radix != 2 ? 5 : | |
530 | std::numeric_limits<T>::digits <= 24 ? 0 : // for float 32-bit. | |
531 | std::numeric_limits<T>::digits <= 53 ? 1 : // for double 64-bit. | |
532 | std::numeric_limits<T>::digits <= 64 ? 2 : // for 80-bit double extended. | |
533 | std::numeric_limits<T>::digits <= 113 ? 4 // for both 128-bit long double (3) and 128-bit quad suffix Q type (4). | |
534 | #endif | |
1e59de90 | 535 | : 5>; // All Generic multiprecision types. |
92f5a8d4 TL |
536 | // std::cout << "\ntag type = " << tag_type << std::endl; // error C2275: 'tag_type': illegal use of this type as an expression. |
537 | return lambert_w0_small_z(x, pol, tag_type()); | |
1e59de90 | 538 | } // template <typename T> T lambert_w0_small_z(T x) |
92f5a8d4 TL |
539 | |
540 | //! Specialization of float (32-bit) series expansion used for small z (abs(z) < 0.05). | |
541 | // Only 9 Coefficients are computed to 21 decimal digits precision, ample for 32-bit float used by most platforms. | |
542 | // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction | |
543 | // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], | |
544 | // as proposed by Tosio Fukushima and implemented by Darko Veberic. | |
545 | ||
1e59de90 TL |
546 | template <typename T, typename Policy> |
547 | T lambert_w0_small_z(T z, const Policy&, std::integral_constant<int, 0> const&) | |
92f5a8d4 TL |
548 | { |
549 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
550 | std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. | |
551 | std::cout << "\ntag_type 0 float lambert_w0_small_z called with z = " << z << " using " << 9 << " terms of precision " | |
552 | << std::numeric_limits<float>::max_digits10 << " decimal digits. " << std::endl; | |
553 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
554 | T result = | |
555 | z * (1 - // j1 z^1 term = 1 | |
556 | z * (1 - // j2 z^2 term = -1 | |
557 | z * (static_cast<float>(3uLL) / 2uLL - // 3/2 // j3 z^3 term = 1.5. | |
558 | z * (2.6666666666666666667F - // 8/3 // j4 | |
559 | z * (5.2083333333333333333F - // -125/24 // j5 | |
560 | z * (10.8F - // j6 | |
561 | z * (23.343055555555555556F - // j7 | |
562 | z * (52.012698412698412698F - // j8 | |
563 | z * 118.62522321428571429F)))))))); // j9 | |
564 | ||
565 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
566 | std::cout << "return w = " << result << std::endl; | |
567 | std::cout.precision(prec); // Restore. | |
568 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
569 | ||
570 | return result; | |
1e59de90 | 571 | } // template <typename T> T lambert_w0_small_z(T x, std::integral_constant<int, 0> const&) |
92f5a8d4 TL |
572 | |
573 | //! Specialization of double (64-bit double) series expansion used for small z (abs(z) < 0.05). | |
574 | // 17 Coefficients are computed to 21 decimal digits precision suitable for 64-bit double used by most platforms. | |
575 | // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction | |
576 | // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], as proposed by Tosio Fukushima and implemented by Veberic. | |
577 | ||
1e59de90 TL |
578 | template <typename T, typename Policy> |
579 | T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 1> const&) | |
92f5a8d4 TL |
580 | { |
581 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
582 | std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. | |
583 | std::cout << "\ntag_type 1 double lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision, " | |
584 | << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl; | |
585 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
586 | T result = | |
587 | z * (1. - // j1 z^1 | |
588 | z * (1. - // j2 z^2 | |
589 | z * (1.5 - // 3/2 // j3 z^3 | |
590 | z * (2.6666666666666666667 - // 8/3 // j4 | |
591 | z * (5.2083333333333333333 - // -125/24 // j5 | |
592 | z * (10.8 - // j6 | |
593 | z * (23.343055555555555556 - // j7 | |
594 | z * (52.012698412698412698 - // j8 | |
595 | z * (118.62522321428571429 - // j9 | |
596 | z * (275.57319223985890653 - // j10 | |
597 | z * (649.78717234347442681 - // j11 | |
598 | z * (1551.1605194805194805 - // j12 | |
599 | z * (3741.4497029592385495 - // j13 | |
600 | z * (9104.5002411580189358 - // j14 | |
601 | z * (22324.308512706601434 - // j15 | |
602 | z * (55103.621972903835338 - // j16 | |
603 | z * 136808.86090394293563)))))))))))))))); // j17 z^17 | |
604 | ||
605 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
606 | std::cout << "return w = " << result << std::endl; | |
607 | std::cout.precision(prec); // Restore. | |
608 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
609 | ||
610 | return result; | |
1e59de90 | 611 | } // T lambert_w0_small_z(const T z, std::integral_constant<int, 1> const&) |
92f5a8d4 TL |
612 | |
613 | //! Specialization of long double (80-bit double extended) series expansion used for small z (abs(z) < 0.05). | |
614 | // 21 Coefficients are computed to 21 decimal digits precision suitable for 80-bit long double used by some | |
615 | // platforms including GCC and Clang when generating for Intel X86 floating-point processors with 80-bit operations enabled (the default). | |
616 | // (This is NOT used by Microsoft Visual Studio where double and long always both use only 64-bit type. | |
617 | // Nor used for 128-bit float128.) | |
1e59de90 TL |
618 | template <typename T, typename Policy> |
619 | T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 2> const&) | |
92f5a8d4 TL |
620 | { |
621 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
622 | std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. | |
623 | std::cout << "\ntag_type 2 long double (80-bit double extended) lambert_w0_small_z called with z = " << z << " using " << 21 << " terms of precision, " | |
624 | << std::numeric_limits<long double>::max_digits10 << " decimal digits. " << std::endl; | |
625 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
626 | // T result = | |
627 | // z * (1.L - // j1 z^1 | |
628 | // z * (1.L - // j2 z^2 | |
629 | // z * (1.5L - // 3/2 // j3 | |
630 | // z * (2.6666666666666666667L - // 8/3 // j4 | |
631 | // z * (5.2083333333333333333L - // -125/24 // j5 | |
632 | // z * (10.800000000000000000L - // j6 | |
633 | // z * (23.343055555555555556L - // j7 | |
634 | // z * (52.012698412698412698L - // j8 | |
635 | // z * (118.62522321428571429L - // j9 | |
636 | // z * (275.57319223985890653L - // j10 | |
637 | // z * (649.78717234347442681L - // j11 | |
638 | // z * (1551.1605194805194805L - // j12 | |
639 | // z * (3741.4497029592385495L - // j13 | |
640 | // z * (9104.5002411580189358L - // j14 | |
641 | // z * (22324.308512706601434L - // j15 | |
642 | // z * (55103.621972903835338L - // j16 | |
643 | // z * (136808.86090394293563L - // j17 z^17 last term used by Fukushima double. | |
644 | // z * (341422.050665838363317L - // z^18 | |
645 | // z * (855992.9659966075514633L - // z^19 | |
646 | // z * (2.154990206091088289321e6L - // z^20 | |
647 | // z * 5.4455529223144624316423e6L // z^21 | |
648 | // )))))))))))))))))))); | |
649 | // | |
650 | ||
651 | T result = | |
652 | z * (1.L - // z j1 | |
653 | z * (1.L - // z^2 | |
654 | z * (1.500000000000000000000000000000000L - // z^3 | |
655 | z * (2.666666666666666666666666666666666L - // z ^ 4 | |
656 | z * (5.208333333333333333333333333333333L - // z ^ 5 | |
657 | z * (10.80000000000000000000000000000000L - // z ^ 6 | |
658 | z * (23.34305555555555555555555555555555L - // z ^ 7 | |
659 | z * (52.01269841269841269841269841269841L - // z ^ 8 | |
660 | z * (118.6252232142857142857142857142857L - // z ^ 9 | |
661 | z * (275.5731922398589065255731922398589L - // z ^ 10 | |
662 | z * (649.7871723434744268077601410934744L - // z ^ 11 | |
663 | z * (1551.160519480519480519480519480519L - // z ^ 12 | |
664 | z * (3741.449702959238549516327294105071L - //z ^ 13 | |
665 | z * (9104.500241158018935796713574491352L - // z ^ 14 | |
666 | z * (22324.308512706601434280005708577137L - // z ^ 15 | |
667 | z * (55103.621972903835337697771560205422L - // z ^ 16 | |
668 | z * (136808.86090394293563342215789305736L - // z ^ 17 | |
669 | z * (341422.05066583836331735491399356945L - // z^18 | |
670 | z * (855992.9659966075514633630250633224L - // z^19 | |
671 | z * (2.154990206091088289321708745358647e6L // z^20 distance -5 without term 20 | |
672 | )))))))))))))))))))); | |
673 | ||
674 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
675 | std::cout << "return w = " << result << std::endl; | |
676 | std::cout.precision(precision); // Restore. | |
677 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
678 | return result; | |
1e59de90 | 679 | } // long double lambert_w0_small_z(const T z, std::integral_constant<int, 1> const&) |
92f5a8d4 TL |
680 | |
681 | //! Specialization of 128-bit long double series expansion used for small z (abs(z) < 0.05). | |
682 | // 34 Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction | |
683 | // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], | |
684 | // and are suffixed by L as they are assumed of type long double. | |
685 | // (This is NOT used for 128-bit quad boost::multiprecision::float128 type which required a suffix Q | |
686 | // nor multiprecision type cpp_bin_float_quad that can only be initialised at full precision of the type | |
687 | // constructed with a decimal digit string like "2.6666666666666666666666666666666666666666666666667".) | |
688 | ||
1e59de90 TL |
689 | template <typename T, typename Policy> |
690 | T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 3> const&) | |
92f5a8d4 TL |
691 | { |
692 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
693 | std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. | |
694 | std::cout << "\ntag_type 3 long double (128-bit) lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision, " | |
695 | << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl; | |
696 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
697 | T result = | |
698 | z * (1.L - // j1 | |
699 | z * (1.L - // j2 | |
700 | z * (1.5L - // 3/2 // j3 | |
701 | z * (2.6666666666666666666666666666666666L - // 8/3 // j4 | |
702 | z * (5.2052083333333333333333333333333333L - // -125/24 // j5 | |
703 | z * (10.800000000000000000000000000000000L - // j6 | |
704 | z * (23.343055555555555555555555555555555L - // j7 | |
705 | z * (52.0126984126984126984126984126984126L - // j8 | |
706 | z * (118.625223214285714285714285714285714L - // j9 | |
707 | z * (275.57319223985890652557319223985890L - // * z ^ 10 - // j10 | |
708 | z * (649.78717234347442680776014109347442680776014109347L - // j11 | |
709 | z * (1551.1605194805194805194805194805194805194805194805L - // j12 | |
710 | z * (3741.4497029592385495163272941050718828496606274384L - // j13 | |
711 | z * (9104.5002411580189357967135744913522691300469078247L - // j14 | |
712 | z * (22324.308512706601434280005708577137148565719994291L - // j15 | |
713 | z * (55103.621972903835337697771560205422639285073147507L - // j16 | |
714 | z * 136808.86090394293563342215789305736395683485630576L // j17 | |
715 | )))))))))))))))); | |
716 | ||
717 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
718 | std::cout << "return w = " << result << std::endl; | |
719 | std::cout.precision(precision); // Restore. | |
720 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
721 | return result; | |
1e59de90 | 722 | } // T lambert_w0_small_z(const T z, std::integral_constant<int, 3> const&) |
92f5a8d4 TL |
723 | |
724 | //! Specialization of 128-bit quad series expansion used for small z (abs(z) < 0.05). | |
725 | // 34 Taylor series coefficients used were computed by Wolfram to 50 decimal digits using instruction | |
726 | // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], | |
727 | // and are suffixed by Q as they are assumed of type quad. | |
728 | // This could be used for 128-bit quad (which requires a suffix Q for full precision). | |
729 | // But experiments with GCC 7.2.0 show that while this gives full 128-bit precision | |
730 | // when the -f-ext-numeric-literals option is in force and the libquadmath library available, | |
731 | // over the range -0.049 to +0.049, | |
732 | // it is slightly slower than getting a double approximation followed by a single Halley step. | |
733 | ||
734 | #ifdef BOOST_HAS_FLOAT128 | |
1e59de90 TL |
735 | template <typename T, typename Policy> |
736 | T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 4> const&) | |
92f5a8d4 TL |
737 | { |
738 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
739 | std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. | |
740 | std::cout << "\ntag_type 4 128-bit quad float128 lambert_w0_small_z called with z = " << z << " using " << 34 << " terms of precision, " | |
741 | << std::numeric_limits<float128>::max_digits10 << " max decimal digits." << std::endl; | |
742 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
743 | T result = | |
744 | z * (1.Q - // z j1 | |
745 | z * (1.Q - // z^2 | |
746 | z * (1.500000000000000000000000000000000Q - // z^3 | |
747 | z * (2.666666666666666666666666666666666Q - // z ^ 4 | |
748 | z * (5.208333333333333333333333333333333Q - // z ^ 5 | |
749 | z * (10.80000000000000000000000000000000Q - // z ^ 6 | |
750 | z * (23.34305555555555555555555555555555Q - // z ^ 7 | |
751 | z * (52.01269841269841269841269841269841Q - // z ^ 8 | |
752 | z * (118.6252232142857142857142857142857Q - // z ^ 9 | |
753 | z * (275.5731922398589065255731922398589Q - // z ^ 10 | |
754 | z * (649.7871723434744268077601410934744Q - // z ^ 11 | |
755 | z * (1551.160519480519480519480519480519Q - // z ^ 12 | |
756 | z * (3741.449702959238549516327294105071Q - //z ^ 13 | |
757 | z * (9104.500241158018935796713574491352Q - // z ^ 14 | |
758 | z * (22324.308512706601434280005708577137Q - // z ^ 15 | |
759 | z * (55103.621972903835337697771560205422Q - // z ^ 16 | |
760 | z * (136808.86090394293563342215789305736Q - // z ^ 17 | |
761 | z * (341422.05066583836331735491399356945Q - // z^18 | |
762 | z * (855992.9659966075514633630250633224Q - // z^19 | |
763 | z * (2.154990206091088289321708745358647e6Q - // 20 | |
764 | z * (5.445552922314462431642316420035073e6Q - // 21 | |
765 | z * (1.380733000216662949061923813184508e7Q - // 22 | |
766 | z * (3.511704498513923292853869855945334e7Q - // 23 | |
767 | z * (8.956800256102797693072819557780090e7Q - // 24 | |
768 | z * (2.290416846187949813964782641734774e8Q - // 25 | |
769 | z * (5.871035041171798492020292225245235e8Q - // 26 | |
770 | z * (1.508256053857792919641317138812957e9Q - // 27 | |
771 | z * (3.882630161293188940385873468413841e9Q - // 28 | |
772 | z * (1.001394313665482968013913601565723e10Q - // 29 | |
773 | z * (2.587356736265760638992878359024929e10Q - // 30 | |
774 | z * (6.696209709358073856946120522333454e10Q - // 31 | |
775 | z * (1.735711659599198077777078238043644e11Q - // 32 | |
776 | z * (4.505680465642353886756098108484670e11Q - // 33 | |
777 | z * (1.171223178256487391904047636564823e12Q //z^34 | |
778 | )))))))))))))))))))))))))))))))))); | |
779 | ||
780 | ||
781 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
782 | std::cout << "return w = " << result << std::endl; | |
783 | std::cout.precision(precision); // Restore. | |
784 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
785 | ||
786 | return result; | |
1e59de90 | 787 | } // T lambert_w0_small_z(const T z, std::integral_constant<int, 4> const&) float128 |
92f5a8d4 TL |
788 | |
789 | #else | |
790 | ||
1e59de90 TL |
791 | template <typename T, typename Policy> |
792 | inline T lambert_w0_small_z(const T z, const Policy& pol, std::integral_constant<int, 4> const&) | |
92f5a8d4 | 793 | { |
1e59de90 | 794 | return lambert_w0_small_z(z, pol, std::integral_constant<int, 5>()); |
92f5a8d4 TL |
795 | } |
796 | ||
797 | #endif // BOOST_HAS_FLOAT128 | |
798 | ||
799 | //! Series functor to compute series term using pow and factorial. | |
800 | //! \details Functor is called after evaluating polynomial with the coefficients as rationals below. | |
1e59de90 | 801 | template <typename T> |
92f5a8d4 TL |
802 | struct lambert_w0_small_z_series_term |
803 | { | |
1e59de90 | 804 | using result_type = T; |
92f5a8d4 TL |
805 | //! \param _z Lambert W argument z. |
806 | //! \param -term -pow<18>(z) / 6402373705728000uLL | |
807 | //! \param _k number of terms == initially 18 | |
808 | ||
809 | // Note *after* evaluating N terms, its internal state has k = N and term = (-1)^N z^N. | |
810 | ||
811 | lambert_w0_small_z_series_term(T _z, T _term, int _k) | |
812 | : k(_k), z(_z), term(_term) { } | |
813 | ||
814 | T operator()() | |
815 | { // Called by sum_series until needs precision set by factor (policy::get_epsilon). | |
816 | using std::pow; | |
817 | ++k; | |
818 | term *= -z / k; | |
819 | //T t = pow(z, k) * pow(T(k), -1 + k) / factorial<T>(k); // (z^k * k(k-1)^k) / k! | |
820 | T result = term * pow(T(k), -1 + k); // term * k^(k-1) | |
821 | // std::cout << " k = " << k << ", term = " << term << ", result = " << result << std::endl; | |
822 | return result; // | |
823 | } | |
824 | private: | |
825 | int k; | |
826 | T z; | |
827 | T term; | |
1e59de90 | 828 | }; // template <typename T> struct lambert_w0_small_z_series_term |
92f5a8d4 TL |
829 | |
830 | //! Generic variant for T a User-defined types like Boost.Multiprecision. | |
1e59de90 TL |
831 | template <typename T, typename Policy> |
832 | inline T lambert_w0_small_z(T z, const Policy& pol, std::integral_constant<int, 5> const&) | |
92f5a8d4 TL |
833 | { |
834 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
835 | std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. | |
836 | std::cout << "Generic lambert_w0_small_z called with z = " << z << " using as many terms needed for precision." << std::endl; | |
837 | std::cout << "Argument z is of type " << typeid(T).name() << std::endl; | |
838 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
839 | ||
840 | // First several terms of the series are tabulated and evaluated as a polynomial: | |
841 | // this will save us a bunch of expensive calls to pow. | |
842 | // Then our series functor is initialized "as if" it had already reached term 18, | |
843 | // enough evaluation of built-in 64-bit double and float (and 80-bit long double?) types. | |
844 | ||
845 | // Coefficients should be stored such that the coefficients for the x^i terms are in poly[i]. | |
846 | static const T coeff[] = | |
847 | { | |
848 | 0, // z^0 Care: zeroth term needed by tools::evaluate_polynomial, but not in the Wolfram equation, so indexes are one different! | |
849 | 1, // z^1 term. | |
850 | -1, // z^2 term | |
851 | static_cast<T>(3uLL) / 2uLL, // z^3 term. | |
852 | -static_cast<T>(8uLL) / 3uLL, // z^4 | |
853 | static_cast<T>(125uLL) / 24uLL, // z^5 | |
854 | -static_cast<T>(54uLL) / 5uLL, // z^6 | |
855 | static_cast<T>(16807uLL) / 720uLL, // z^7 | |
856 | -static_cast<T>(16384uLL) / 315uLL, // z^8 | |
857 | static_cast<T>(531441uLL) / 4480uLL, // z^9 | |
858 | -static_cast<T>(156250uLL) / 567uLL, // z^10 | |
859 | static_cast<T>(2357947691uLL) / 3628800uLL, // z^11 | |
860 | -static_cast<T>(2985984uLL) / 1925uLL, // z^12 | |
861 | static_cast<T>(1792160394037uLL) / 479001600uLL, // z^13 | |
862 | -static_cast<T>(7909306972uLL) / 868725uLL, // z^14 | |
863 | static_cast<T>(320361328125uLL) / 14350336uLL, // z^15 | |
864 | -static_cast<T>(35184372088832uLL) / 638512875uLL, // z^16 | |
865 | static_cast<T>(2862423051509815793uLL) / 20922789888000uLL, // z^17 term | |
866 | -static_cast<T>(5083731656658uLL) / 14889875uLL, | |
867 | // z^18 term. = 136808.86090394293563342215789305735851647769682393 | |
868 | ||
869 | // z^18 is biggest that can be computed as rational using the largest possible uLL integers, | |
870 | // so higher terms cannot be potentially compiler-computed as uLL rationals. | |
871 | // Wolfram (5083731656658 z ^ 18) / 14889875 or | |
872 | // -341422.05066583836331735491399356945575432970390954 z^18 | |
873 | ||
874 | // See note below calling the functor to compute another term, | |
875 | // sufficient for 80-bit long double precision. | |
876 | // Wolfram -341422.05066583836331735491399356945575432970390954 z^19 term. | |
877 | // (5480386857784802185939 z^19)/6402373705728000 | |
878 | // But now this variant is not used to compute long double | |
879 | // as specializations are provided above. | |
880 | }; // static const T coeff[] | |
881 | ||
882 | /* | |
883 | Table of 19 computed coefficients: | |
884 | ||
885 | #0 0 | |
886 | #1 1 | |
887 | #2 -1 | |
888 | #3 1.5 | |
889 | #4 -2.6666666666666666666666666666666665382713370408509 | |
890 | #5 5.2083333333333333333333333333333330765426740817019 | |
891 | #6 -10.800000000000000000000000000000000616297582203915 | |
892 | #7 23.343055555555555555555555555555555076212991619177 | |
893 | #8 -52.012698412698412698412698412698412659282693193402 | |
894 | #9 118.62522321428571428571428571428571146835390992496 | |
895 | #10 -275.57319223985890652557319223985891400375196748314 | |
896 | #11 649.7871723434744268077601410934743969785223845882 | |
897 | #12 -1551.1605194805194805194805194805194947599566007429 | |
898 | #13 3741.4497029592385495163272941050719510009019331763 | |
899 | #14 -9104.5002411580189357967135744913524243896052869184 | |
900 | #15 22324.308512706601434280005708577137322392070452582 | |
901 | #16 -55103.621972903835337697771560205423203318720697224 | |
902 | #17 136808.86090394293563342215789305735851647769682393 | |
903 | 136808.86090394293563342215789305735851647769682393 == Exactly same as Wolfram computed value. | |
904 | #18 -341422.05066583836331735491399356947486381600607416 | |
905 | 341422.05066583836331735491399356945575432970390954 z^19 Wolfram value differs at 36 decimal digit, as expected. | |
906 | */ | |
907 | ||
908 | using boost::math::policies::get_epsilon; // for type T. | |
909 | using boost::math::tools::sum_series; | |
910 | using boost::math::tools::evaluate_polynomial; | |
911 | // http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/roots/rational.html | |
912 | ||
913 | // std::streamsize prec = std::cout.precision(std::numeric_limits <T>::max_digits10); | |
914 | ||
915 | T result = evaluate_polynomial(coeff, z); | |
1e59de90 | 916 | // template <std::size_t N, typename T, typename V> |
92f5a8d4 TL |
917 | // V evaluate_polynomial(const T(&poly)[N], const V& val); |
918 | // Size of coeff found from N | |
919 | //std::cout << "evaluate_polynomial(coeff, z); == " << result << std::endl; | |
920 | //std::cout << "result = " << result << std::endl; | |
921 | // It's an artefact of the way I wrote the functor: *after* evaluating N | |
922 | // terms, its internal state has k = N and term = (-1)^N z^N. So after | |
923 | // evaluating 18 terms, we initialize the functor to the term we've just | |
924 | // evaluated, and then when it's called, it increments itself to the next term. | |
925 | // So 18!is 6402373705728000, which is where that comes from. | |
926 | ||
927 | // The 19th coefficient of the polynomial is actually, 19 ^ 18 / 19!= | |
928 | // 104127350297911241532841 / 121645100408832000 which after removing GCDs | |
929 | // reduces down to Wolfram rational 5480386857784802185939 / 6402373705728000. | |
930 | // Wolfram z^19 term +(5480386857784802185939 z^19) /6402373705728000 | |
931 | // +855992.96599660755146336302506332246623424823099755 z^19 | |
932 | ||
933 | //! Evaluate Functor. | |
934 | lambert_w0_small_z_series_term<T> s(z, -pow<18>(z) / 6402373705728000uLL, 18); | |
935 | ||
936 | // Temporary to list the coefficients. | |
937 | //std::cout << " Table of coefficients" << std::endl; | |
938 | //std::streamsize saved_precision = std::cout.precision(50); | |
939 | //for (size_t i = 0; i != 19; i++) | |
940 | //{ | |
941 | // std::cout << "#" << i << " " << coeff[i] << std::endl; | |
942 | //} | |
943 | //std::cout.precision(saved_precision); | |
944 | ||
1e59de90 | 945 | std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); // Max iterations from policy. |
92f5a8d4 TL |
946 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES |
947 | std::cout << "max iter from policy = " << max_iter << std::endl; | |
948 | // // max iter from policy = 1000000 is default. | |
949 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES | |
950 | ||
951 | result = sum_series(s, get_epsilon<T, Policy>(), max_iter, result); | |
952 | // result == evaluate_polynomial. | |
1e59de90 | 953 | //sum_series(Functor& func, int bits, std::uintmax_t& max_terms, const U& init_value) |
92f5a8d4 TL |
954 | // std::cout << "sum_series(s, get_epsilon<T, Policy>(), max_iter, result); = " << result << std::endl; |
955 | ||
956 | //T epsilon = get_epsilon<T, Policy>(); | |
f67539c2 TL |
957 | //std::cout << "epsilon from policy = " << epsilon << std::endl; |
958 | // epsilon from policy = 1.93e-34 for T == quad | |
92f5a8d4 TL |
959 | // 5.35e-51 for t = cpp_bin_float_50 |
960 | ||
961 | // std::cout << " get eps = " << get_epsilon<T, Policy>() << std::endl; // quad eps = 1.93e-34, bin_float_50 eps = 5.35e-51 | |
962 | policies::check_series_iterations<T>("boost::math::lambert_w0_small_z<%1%>(%1%)", max_iter, pol); | |
963 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS | |
964 | std::cout << "z = " << z << " needed " << max_iter << " iterations." << std::endl; | |
965 | std::cout.precision(prec); // Restore. | |
966 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS | |
967 | return result; | |
1e59de90 | 968 | } // template <typename T, typename Policy> inline T lambert_w0_small_z_series(T z, const Policy& pol) |
92f5a8d4 TL |
969 | |
970 | // Approximate lambert_w0 (used for z values that are outside range of lookup table or rational functions) | |
971 | // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162. | |
972 | template <typename T> | |
1e59de90 | 973 | inline T lambert_w0_approx(T z) |
92f5a8d4 TL |
974 | { |
975 | BOOST_MATH_STD_USING | |
976 | T lz = log(z); | |
977 | T llz = log(lz); | |
978 | T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162. | |
979 | return w; | |
980 | // std::cout << "w max " << max_w << std::endl; // double 703.227 | |
981 | } | |
982 | ||
983 | ////////////////////////////////////////////////////////////////////////////////////////// | |
984 | ||
985 | //! \brief Lambert_w0 implementations for float, double and higher precisions. | |
986 | //! 3rd parameter used to select which version is used. | |
987 | ||
988 | //! /details Rational polynomials are provided for several range of argument z. | |
989 | //! For very small values of z, and for z very near the branch singularity at -e^-1 (~= -0.367879), | |
990 | //! two other series functions are used. | |
991 | ||
992 | //! float precision polynomials are used for 32-bit (usually float) precision (for speed) | |
993 | //! double precision polynomials are used for 64-bit (usually double) precision. | |
994 | //! For higher precisions, a 64-bit double approximation is computed first, | |
f67539c2 | 995 | //! and then refined using Halley iterations. |
92f5a8d4 | 996 | |
1e59de90 | 997 | template <typename T> |
20effc67 | 998 | inline T do_get_near_singularity_param(T z) |
92f5a8d4 TL |
999 | { |
1000 | BOOST_MATH_STD_USING | |
1001 | const T p2 = 2 * (boost::math::constants::e<T>() * z + 1); | |
1002 | const T p = sqrt(p2); | |
1003 | return p; | |
1004 | } | |
1e59de90 | 1005 | template <typename T, typename Policy> |
20effc67 | 1006 | inline T get_near_singularity_param(T z, const Policy) |
92f5a8d4 | 1007 | { |
1e59de90 | 1008 | using value_type = typename policies::evaluation<T, Policy>::type; |
20effc67 | 1009 | return static_cast<T>(do_get_near_singularity_param(static_cast<value_type>(z))); |
92f5a8d4 TL |
1010 | } |
1011 | ||
1012 | // Forward declarations: | |
1013 | ||
1e59de90 TL |
1014 | //template <typename T, typename Policy> T lambert_w0_small_z(T z, const Policy& pol); |
1015 | //template <typename T, typename Policy> | |
1016 | //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 0>&); // 32 bit usually float. | |
1017 | //template <typename T, typename Policy> | |
1018 | //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 1>&); // 64 bit usually double. | |
1019 | //template <typename T, typename Policy> | |
1020 | //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 2>&); // 80-bit long double. | |
92f5a8d4 | 1021 | |
1e59de90 | 1022 | template <typename T> |
92f5a8d4 TL |
1023 | T lambert_w_positive_rational_float(T z) |
1024 | { | |
1025 | BOOST_MATH_STD_USING | |
1026 | if (z < 2) | |
1027 | { | |
1028 | if (z < 0.5) | |
1029 | { // 0.05 < z < 0.5 | |
1030 | // Maximum Deviation Found: 2.993e-08 | |
1031 | // Expected Error Term : 2.993e-08 | |
1032 | // Maximum Relative Change in Control Points : 7.555e-04 Y offset : -8.196592331e-01 | |
1033 | static const T Y = 8.196592331e-01f; | |
1034 | static const T P[] = { | |
1035 | 1.803388345e-01f, | |
1036 | -4.820256838e-01f, | |
1037 | -1.068349741e+00f, | |
1038 | -3.506624319e-02f, | |
1039 | }; | |
1040 | static const T Q[] = { | |
1041 | 1.000000000e+00f, | |
1042 | 2.871703469e+00f, | |
1043 | 1.690949264e+00f, | |
1044 | }; | |
1045 | return z * (Y + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z)); | |
1046 | } | |
1047 | else | |
1048 | { // 0.5 < z < 2 | |
1049 | // Max error in interpolated form: 1.018e-08 | |
1050 | static const T Y = 5.503368378e-01f; | |
1051 | static const T P[] = { | |
1052 | 4.493332766e-01f, | |
1053 | 2.543432707e-01f, | |
1054 | -4.808788799e-01f, | |
1055 | -1.244425316e-01f, | |
1056 | }; | |
1057 | static const T Q[] = { | |
1058 | 1.000000000e+00f, | |
1059 | 2.780661241e+00f, | |
1060 | 1.830840318e+00f, | |
1061 | 2.407221031e-01f, | |
1062 | }; | |
1063 | return z * (Y + boost::math::tools::evaluate_rational(P, Q, z)); | |
1064 | } | |
1065 | } | |
1066 | else if (z < 6) | |
1067 | { | |
1068 | // 2 < z < 6 | |
1069 | // Max error in interpolated form: 2.944e-08 | |
1070 | static const T Y = 1.162393570e+00f; | |
1071 | static const T P[] = { | |
1072 | -1.144183394e+00f, | |
1073 | -4.712732855e-01f, | |
1074 | 1.563162512e-01f, | |
1075 | 1.434010911e-02f, | |
1076 | }; | |
1077 | static const T Q[] = { | |
1078 | 1.000000000e+00f, | |
1079 | 1.192626340e+00f, | |
1080 | 2.295580708e-01f, | |
1081 | 5.477869455e-03f, | |
1082 | }; | |
1083 | return Y + boost::math::tools::evaluate_rational(P, Q, z); | |
1084 | } | |
1085 | else if (z < 18) | |
1086 | { | |
1087 | // 6 < z < 18 | |
1088 | // Max error in interpolated form: 5.893e-08 | |
1089 | static const T Y = 1.809371948e+00f; | |
1090 | static const T P[] = { | |
1091 | -1.689291769e+00f, | |
1092 | -3.337812742e-01f, | |
1093 | 3.151434873e-02f, | |
1094 | 1.134178734e-03f, | |
1095 | }; | |
1096 | static const T Q[] = { | |
1097 | 1.000000000e+00f, | |
1098 | 5.716915685e-01f, | |
1099 | 4.489521292e-02f, | |
1100 | 4.076716763e-04f, | |
1101 | }; | |
1102 | return Y + boost::math::tools::evaluate_rational(P, Q, z); | |
1103 | } | |
1104 | else if (z < 9897.12905874) // 2.8 < log(z) < 9.2 | |
1105 | { | |
1106 | // Max error in interpolated form: 1.771e-08 | |
1107 | static const T Y = -1.402973175e+00f; | |
1108 | static const T P[] = { | |
1109 | 1.966174312e+00f, | |
1110 | 2.350864728e-01f, | |
1111 | -5.098074353e-02f, | |
1112 | -1.054818339e-02f, | |
1113 | }; | |
1114 | static const T Q[] = { | |
1115 | 1.000000000e+00f, | |
1116 | 4.388208264e-01f, | |
1117 | 8.316639634e-02f, | |
1118 | 3.397187918e-03f, | |
1119 | -1.321489743e-05f, | |
1120 | }; | |
1121 | T log_w = log(z); | |
1122 | return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w); | |
1123 | } | |
1124 | else if (z < 7.896296e+13) // 9.2 < log(z) <= 32 | |
1125 | { | |
1126 | // Max error in interpolated form: 5.821e-08 | |
1127 | static const T Y = -2.735729218e+00f; | |
1128 | static const T P[] = { | |
1129 | 3.424903470e+00f, | |
1130 | 7.525631787e-02f, | |
1131 | -1.427309584e-02f, | |
1132 | -1.435974178e-05f, | |
1133 | }; | |
1134 | static const T Q[] = { | |
1135 | 1.000000000e+00f, | |
1136 | 2.514005579e-01f, | |
1137 | 6.118994652e-03f, | |
1138 | -1.357889535e-05f, | |
1139 | 7.312865624e-08f, | |
1140 | }; | |
1141 | T log_w = log(z); | |
1142 | return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w); | |
1143 | } | |
1144 | else // 32 < log(z) < 100 | |
1145 | { | |
1146 | // Max error in interpolated form: 1.491e-08 | |
1147 | static const T Y = -4.012863159e+00f; | |
1148 | static const T P[] = { | |
1149 | 4.431629226e+00f, | |
1150 | 2.756690487e-01f, | |
1151 | -2.992956930e-03f, | |
1152 | -4.912259384e-05f, | |
1153 | }; | |
1154 | static const T Q[] = { | |
1155 | 1.000000000e+00f, | |
1156 | 2.015434591e-01f, | |
1157 | 4.949426142e-03f, | |
1158 | 1.609659944e-05f, | |
1159 | -5.111523436e-09f, | |
1160 | }; | |
1161 | T log_w = log(z); | |
1162 | return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w); | |
1163 | } | |
1164 | } | |
1165 | ||
1e59de90 | 1166 | template <typename T, typename Policy> |
92f5a8d4 TL |
1167 | T lambert_w_negative_rational_float(T z, const Policy& pol) |
1168 | { | |
1169 | BOOST_MATH_STD_USING | |
1170 | if (z > -0.27) | |
1171 | { | |
1172 | if (z < -0.051) | |
1173 | { | |
1174 | // -0.27 < z < -0.051 | |
1175 | // Max error in interpolated form: 5.080e-08 | |
1176 | static const T Y = 1.255809784e+00f; | |
1177 | static const T P[] = { | |
1178 | -2.558083412e-01f, | |
1179 | -2.306524098e+00f, | |
1180 | -5.630887033e+00f, | |
1181 | -3.803974556e+00f, | |
1182 | }; | |
1183 | static const T Q[] = { | |
1184 | 1.000000000e+00f, | |
1185 | 5.107680783e+00f, | |
1186 | 7.914062868e+00f, | |
1187 | 3.501498501e+00f, | |
1188 | }; | |
1189 | return z * (Y + boost::math::tools::evaluate_rational(P, Q, z)); | |
1190 | } | |
1191 | else | |
1192 | { | |
1193 | // Very small z so use a series function. | |
1194 | return lambert_w0_small_z(z, pol); | |
1195 | } | |
1196 | } | |
1197 | else if (z > -0.3578794411714423215955237701) | |
1198 | { // Very close to branch singularity. | |
1199 | // Max error in interpolated form: 5.269e-08 | |
1200 | static const T Y = 1.220928431e-01f; | |
1201 | static const T P[] = { | |
1202 | -1.221787446e-01f, | |
1203 | -6.816155875e+00f, | |
1204 | 7.144582035e+01f, | |
1205 | 1.128444390e+03f, | |
1206 | }; | |
1207 | static const T Q[] = { | |
1208 | 1.000000000e+00f, | |
1209 | 6.480326790e+01f, | |
1210 | 1.869145243e+02f, | |
1211 | -1.361804274e+03f, | |
1212 | 1.117826726e+03f, | |
1213 | }; | |
1214 | T d = z + 0.367879441171442321595523770161460867445811f; | |
1215 | return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d)); | |
1216 | } | |
1217 | else | |
1218 | { | |
1219 | // z is very close (within 0.01) of the singularity at e^-1. | |
20effc67 | 1220 | return lambert_w_singularity_series(get_near_singularity_param(z, pol)); |
92f5a8d4 TL |
1221 | } |
1222 | } | |
1223 | ||
1224 | //! Lambert_w0 @b 'float' implementation, selected when T is 32-bit precision. | |
1e59de90 TL |
1225 | template <typename T, typename Policy> |
1226 | inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 1>&) | |
92f5a8d4 TL |
1227 | { |
1228 | static const char* function = "boost::math::lambert_w0<%1%>"; // For error messages. | |
1229 | BOOST_MATH_STD_USING // Aid ADL of std functions. | |
1230 | ||
1231 | if ((boost::math::isnan)(z)) | |
1232 | { | |
1233 | return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol); | |
1234 | } | |
1235 | if ((boost::math::isinf)(z)) | |
1236 | { | |
1237 | return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol); | |
1238 | } | |
1239 | ||
1240 | if (z >= 0.05) // Fukushima switch point. | |
1241 | // if (z >= 0.045) // 34 terms makes 128-bit 'exact' below 0.045. | |
1242 | { // Normal ranges using several rational polynomials. | |
1243 | return lambert_w_positive_rational_float(z); | |
1244 | } | |
1245 | else if (z <= -0.3678794411714423215955237701614608674458111310f) | |
1246 | { | |
1247 | if (z < -0.3678794411714423215955237701614608674458111310f) | |
1248 | return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol); | |
1249 | return -1; | |
1250 | } | |
1251 | else // z < 0.05 | |
1252 | { | |
1253 | return lambert_w_negative_rational_float(z, pol); | |
1254 | } | |
1e59de90 | 1255 | } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 1>&) for 32-bit usually float. |
92f5a8d4 | 1256 | |
1e59de90 | 1257 | template <typename T> |
92f5a8d4 TL |
1258 | T lambert_w_positive_rational_double(T z) |
1259 | { | |
1260 | BOOST_MATH_STD_USING | |
1261 | if (z < 2) | |
1262 | { | |
1263 | if (z < 0.5) | |
1264 | { | |
1265 | // Max error in interpolated form: 2.255e-17 | |
1266 | static const T offset = 8.19659233093261719e-01; | |
1267 | static const T P[] = { | |
1268 | 1.80340766906685177e-01, | |
1269 | 3.28178241493119307e-01, | |
1270 | -2.19153620687139706e+00, | |
1271 | -7.24750929074563990e+00, | |
1272 | -7.28395876262524204e+00, | |
1273 | -2.57417169492512916e+00, | |
1274 | -2.31606948888704503e-01 | |
1275 | }; | |
1276 | static const T Q[] = { | |
1277 | 1.00000000000000000e+00, | |
1278 | 7.36482529307436604e+00, | |
1279 | 2.03686007856430677e+01, | |
1280 | 2.62864592096657307e+01, | |
1281 | 1.59742041380858333e+01, | |
1282 | 4.03760534788374589e+00, | |
1283 | 2.91327346750475362e-01 | |
1284 | }; | |
1285 | return z * (offset + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z)); | |
1286 | } | |
1287 | else | |
1288 | { | |
1289 | // Max error in interpolated form: 3.806e-18 | |
1290 | static const T offset = 5.50335884094238281e-01; | |
1291 | static const T P[] = { | |
1292 | 4.49664083944098322e-01, | |
1293 | 1.90417666196776909e+00, | |
1294 | 1.99951368798255994e+00, | |
1295 | -6.91217310299270265e-01, | |
1296 | -1.88533935998617058e+00, | |
1297 | -7.96743968047750836e-01, | |
1298 | -1.02891726031055254e-01, | |
1299 | -3.09156013592636568e-03 | |
1300 | }; | |
1301 | static const T Q[] = { | |
1302 | 1.00000000000000000e+00, | |
1303 | 6.45854489419584014e+00, | |
1304 | 1.54739232422116048e+01, | |
1305 | 1.72606164253337843e+01, | |
1306 | 9.29427055609544096e+00, | |
1307 | 2.29040824649748117e+00, | |
1308 | 2.21610620995418981e-01, | |
1309 | 5.70597669908194213e-03 | |
1310 | }; | |
1311 | return z * (offset + boost::math::tools::evaluate_rational(P, Q, z)); | |
1312 | } | |
1313 | } | |
1314 | else if (z < 6) | |
1315 | { | |
1316 | // 2 < z < 6 | |
1317 | // Max error in interpolated form: 1.216e-17 | |
1318 | static const T Y = 1.16239356994628906e+00; | |
1319 | static const T P[] = { | |
1320 | -1.16230494982099475e+00, | |
1321 | -3.38528144432561136e+00, | |
1322 | -2.55653717293161565e+00, | |
1323 | -3.06755172989214189e-01, | |
1324 | 1.73149743765268289e-01, | |
1325 | 3.76906042860014206e-02, | |
1326 | 1.84552217624706666e-03, | |
1327 | 1.69434126904822116e-05, | |
1328 | }; | |
1329 | static const T Q[] = { | |
1330 | 1.00000000000000000e+00, | |
1331 | 3.77187616711220819e+00, | |
1332 | 4.58799960260143701e+00, | |
1333 | 2.24101228462292447e+00, | |
1334 | 4.54794195426212385e-01, | |
1335 | 3.60761772095963982e-02, | |
1336 | 9.25176499518388571e-04, | |
1337 | 4.43611344705509378e-06, | |
1338 | }; | |
1339 | return Y + boost::math::tools::evaluate_rational(P, Q, z); | |
1340 | } | |
1341 | else if (z < 18) | |
1342 | { | |
1343 | // 6 < z < 18 | |
1344 | // Max error in interpolated form: 1.985e-19 | |
1345 | static const T offset = 1.80937194824218750e+00; | |
1346 | static const T P[] = | |
1347 | { | |
1348 | -1.80690935424793635e+00, | |
1349 | -3.66995929380314602e+00, | |
1350 | -1.93842957940149781e+00, | |
1351 | -2.94269984375794040e-01, | |
1352 | 1.81224710627677778e-03, | |
1353 | 2.48166798603547447e-03, | |
1354 | 1.15806592415397245e-04, | |
1355 | 1.43105573216815533e-06, | |
1356 | 3.47281483428369604e-09 | |
1357 | }; | |
1358 | static const T Q[] = { | |
1359 | 1.00000000000000000e+00, | |
1360 | 2.57319080723908597e+00, | |
1361 | 1.96724528442680658e+00, | |
1362 | 5.84501352882650722e-01, | |
1363 | 7.37152837939206240e-02, | |
1364 | 3.97368430940416778e-03, | |
1365 | 8.54941838187085088e-05, | |
1366 | 6.05713225608426678e-07, | |
1367 | 8.17517283816615732e-10 | |
1368 | }; | |
1369 | return offset + boost::math::tools::evaluate_rational(P, Q, z); | |
1370 | } | |
1371 | else if (z < 9897.12905874) // 2.8 < log(z) < 9.2 | |
1372 | { | |
1373 | // Max error in interpolated form: 1.195e-18 | |
1374 | static const T Y = -1.40297317504882812e+00; | |
1375 | static const T P[] = { | |
1376 | 1.97011826279311924e+00, | |
1377 | 1.05639945701546704e+00, | |
1378 | 3.33434529073196304e-01, | |
1379 | 3.34619153200386816e-02, | |
1380 | -5.36238353781326675e-03, | |
1381 | -2.43901294871308604e-03, | |
1382 | -2.13762095619085404e-04, | |
1383 | -4.85531936495542274e-06, | |
1384 | -2.02473518491905386e-08, | |
1385 | }; | |
1386 | static const T Q[] = { | |
1387 | 1.00000000000000000e+00, | |
1388 | 8.60107275833921618e-01, | |
1389 | 4.10420467985504373e-01, | |
1390 | 1.18444884081994841e-01, | |
1391 | 2.16966505556021046e-02, | |
1392 | 2.24529766630769097e-03, | |
1393 | 9.82045090226437614e-05, | |
1394 | 1.36363515125489502e-06, | |
1395 | 3.44200749053237945e-09, | |
1396 | }; | |
1397 | T log_w = log(z); | |
1398 | return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w); | |
1399 | } | |
1400 | else if (z < 7.896296e+13) // 9.2 < log(z) <= 32 | |
1401 | { | |
1402 | // Max error in interpolated form: 6.529e-18 | |
1403 | static const T Y = -2.73572921752929688e+00; | |
1404 | static const T P[] = { | |
1405 | 3.30547638424076217e+00, | |
1406 | 1.64050071277550167e+00, | |
1407 | 4.57149576470736039e-01, | |
1408 | 4.03821227745424840e-02, | |
1409 | -4.99664976882514362e-04, | |
1410 | -1.28527893803052956e-04, | |
1411 | -2.95470325373338738e-06, | |
1412 | -1.76662025550202762e-08, | |
1413 | -1.98721972463709290e-11, | |
1414 | }; | |
1415 | static const T Q[] = { | |
1416 | 1.00000000000000000e+00, | |
1417 | 6.91472559412458759e-01, | |
1418 | 2.48154578891676774e-01, | |
1419 | 4.60893578284335263e-02, | |
1420 | 3.60207838982301946e-03, | |
1421 | 1.13001153242430471e-04, | |
1422 | 1.33690948263488455e-06, | |
1423 | 4.97253225968548872e-09, | |
1424 | 3.39460723731970550e-12, | |
1425 | }; | |
1426 | T log_w = log(z); | |
1427 | return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w); | |
1428 | } | |
1429 | else if (z < 2.6881171e+43) // 32 < log(z) < 100 | |
1430 | { | |
1431 | // Max error in interpolated form: 2.015e-18 | |
1432 | static const T Y = -4.01286315917968750e+00; | |
1433 | static const T P[] = { | |
1434 | 5.07714858354309672e+00, | |
1435 | -3.32994414518701458e+00, | |
1436 | -8.61170416909864451e-01, | |
1437 | -4.01139705309486142e-02, | |
1438 | -1.85374201771834585e-04, | |
1439 | 1.08824145844270666e-05, | |
1440 | 1.17216905810452396e-07, | |
1441 | 2.97998248101385990e-10, | |
1442 | 1.42294856434176682e-13, | |
1443 | }; | |
1444 | static const T Q[] = { | |
1445 | 1.00000000000000000e+00, | |
1446 | -4.85840770639861485e-01, | |
1447 | -3.18714850604827580e-01, | |
1448 | -3.20966129264610534e-02, | |
1449 | -1.06276178044267895e-03, | |
1450 | -1.33597828642644955e-05, | |
1451 | -6.27900905346219472e-08, | |
1452 | -9.35271498075378319e-11, | |
1453 | -2.60648331090076845e-14, | |
1454 | }; | |
1455 | T log_w = log(z); | |
1456 | return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w); | |
1457 | } | |
1458 | else // 100 < log(z) < 710 | |
1459 | { | |
1460 | // Max error in interpolated form: 5.277e-18 | |
1461 | static const T Y = -5.70115661621093750e+00; | |
1462 | static const T P[] = { | |
1463 | 6.42275660145116698e+00, | |
1464 | 1.33047964073367945e+00, | |
1465 | 6.72008923401652816e-02, | |
1466 | 1.16444069958125895e-03, | |
1467 | 7.06966760237470501e-06, | |
1468 | 5.48974896149039165e-09, | |
1469 | -7.00379652018853621e-11, | |
1470 | -1.89247635913659556e-13, | |
1471 | -1.55898770790170598e-16, | |
1472 | -4.06109208815303157e-20, | |
1473 | -2.21552699006496737e-24, | |
1474 | }; | |
1475 | static const T Q[] = { | |
1476 | 1.00000000000000000e+00, | |
1477 | 3.34498588416632854e-01, | |
1478 | 2.51519862456384983e-02, | |
1479 | 6.81223810622416254e-04, | |
1480 | 7.94450897106903537e-06, | |
1481 | 4.30675039872881342e-08, | |
1482 | 1.10667669458467617e-10, | |
1483 | 1.31012240694192289e-13, | |
1484 | 6.53282047177727125e-17, | |
1485 | 1.11775518708172009e-20, | |
1486 | 3.78250395617836059e-25, | |
1487 | }; | |
1488 | T log_w = log(z); | |
1489 | return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w); | |
1490 | } | |
1491 | } | |
1492 | ||
1e59de90 | 1493 | template <typename T, typename Policy> |
92f5a8d4 TL |
1494 | T lambert_w_negative_rational_double(T z, const Policy& pol) |
1495 | { | |
1496 | BOOST_MATH_STD_USING | |
1497 | if (z > -0.1) | |
1498 | { | |
1499 | if (z < -0.051) | |
1500 | { | |
1501 | // -0.1 < z < -0.051 | |
1502 | // Maximum Deviation Found: 4.402e-22 | |
1503 | // Expected Error Term : 4.240e-22 | |
1504 | // Maximum Relative Change in Control Points : 4.115e-03 | |
1505 | static const T Y = 1.08633995056152344e+00; | |
1506 | static const T P[] = { | |
1507 | -8.63399505615014331e-02, | |
1508 | -1.64303871814816464e+00, | |
1509 | -7.71247913918273738e+00, | |
1510 | -1.41014495545382454e+01, | |
1511 | -1.02269079949257616e+01, | |
1512 | -2.17236002836306691e+00, | |
1513 | }; | |
1514 | static const T Q[] = { | |
1515 | 1.00000000000000000e+00, | |
1516 | 7.44775406945739243e+00, | |
1517 | 2.04392643087266541e+01, | |
1518 | 2.51001961077774193e+01, | |
1519 | 1.31256080849023319e+01, | |
1520 | 2.11640324843601588e+00, | |
1521 | }; | |
1522 | return z * (Y + boost::math::tools::evaluate_rational(P, Q, z)); | |
1523 | } | |
1524 | else | |
1525 | { | |
1526 | // Very small z > 0.051: | |
1527 | return lambert_w0_small_z(z, pol); | |
1528 | } | |
1529 | } | |
1530 | else if (z > -0.2) | |
1531 | { | |
1532 | // -0.2 < z < -0.1 | |
1533 | // Maximum Deviation Found: 2.898e-20 | |
1534 | // Expected Error Term : 2.873e-20 | |
1535 | // Maximum Relative Change in Control Points : 3.779e-04 | |
1536 | static const T Y = 1.20359611511230469e+00; | |
1537 | static const T P[] = { | |
1538 | -2.03596115108465635e-01, | |
1539 | -2.95029082937201859e+00, | |
1540 | -1.54287922188671648e+01, | |
1541 | -3.81185809571116965e+01, | |
1542 | -4.66384358235575985e+01, | |
1543 | -2.59282069989642468e+01, | |
1544 | -4.70140451266553279e+00, | |
1545 | }; | |
1546 | static const T Q[] = { | |
1547 | 1.00000000000000000e+00, | |
1548 | 9.57921436074599929e+00, | |
1549 | 3.60988119290234377e+01, | |
1550 | 6.73977699505546007e+01, | |
1551 | 6.41104992068148823e+01, | |
1552 | 2.82060127225153607e+01, | |
1553 | 4.10677610657724330e+00, | |
1554 | }; | |
1555 | return z * (Y + boost::math::tools::evaluate_rational(P, Q, z)); | |
1556 | } | |
1557 | else if (z > -0.3178794411714423215955237) | |
1558 | { | |
1559 | // Max error in interpolated form: 6.996e-18 | |
1560 | static const T Y = 3.49680423736572266e-01; | |
1561 | static const T P[] = { | |
1562 | -3.49729841718749014e-01, | |
1563 | -6.28207407760709028e+01, | |
1564 | -2.57226178029669171e+03, | |
1565 | -2.50271008623093747e+04, | |
1566 | 1.11949239154711388e+05, | |
1567 | 1.85684566607844318e+06, | |
1568 | 4.80802490427638643e+06, | |
1569 | 2.76624752134636406e+06, | |
1570 | }; | |
1571 | static const T Q[] = { | |
1572 | 1.00000000000000000e+00, | |
1573 | 1.82717661215113000e+02, | |
1574 | 8.00121119810280100e+03, | |
1575 | 1.06073266717010129e+05, | |
1576 | 3.22848993926057721e+05, | |
1577 | -8.05684814514171256e+05, | |
1578 | -2.59223192927265737e+06, | |
1579 | -5.61719645211570871e+05, | |
1580 | 6.27765369292636844e+04, | |
1581 | }; | |
1582 | T d = z + 0.367879441171442321595523770161460867445811; | |
1583 | return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d)); | |
1584 | } | |
1585 | else if (z > -0.3578794411714423215955237701) | |
1586 | { | |
1587 | // Max error in interpolated form: 1.404e-17 | |
1588 | static const T Y = 5.00126481056213379e-02; | |
1589 | static const T P[] = { | |
1590 | -5.00173570682372162e-02, | |
1591 | -4.44242461870072044e+01, | |
1592 | -9.51185533619946042e+03, | |
1593 | -5.88605699015429386e+05, | |
1594 | -1.90760843597427751e+06, | |
1595 | 5.79797663818311404e+08, | |
1596 | 1.11383352508459134e+10, | |
1597 | 5.67791253678716467e+10, | |
1598 | 6.32694500716584572e+10, | |
1599 | }; | |
1600 | static const T Q[] = { | |
1601 | 1.00000000000000000e+00, | |
1602 | 9.08910517489981551e+02, | |
1603 | 2.10170163753340133e+05, | |
1604 | 1.67858612416470327e+07, | |
1605 | 4.90435561733227953e+08, | |
1606 | 4.54978142622939917e+09, | |
1607 | 2.87716585708739168e+09, | |
1608 | -4.59414247951143131e+10, | |
1609 | -1.72845216404874299e+10, | |
1610 | }; | |
1611 | T d = z + 0.36787944117144232159552377016146086744581113103176804; | |
1612 | return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d)); | |
1613 | } | |
1614 | else | |
1615 | { // z is very close (within 0.01) of the singularity at -e^-1, | |
1616 | // so use a series expansion from R. M. Corless et al. | |
1617 | const T p2 = 2 * (boost::math::constants::e<T>() * z + 1); | |
1618 | const T p = sqrt(p2); | |
1619 | return lambert_w_detail::lambert_w_singularity_series(p); | |
1620 | } | |
1621 | } | |
1622 | ||
1623 | //! Lambert_w0 @b 'double' implementation, selected when T is 64-bit precision. | |
1e59de90 TL |
1624 | template <typename T, typename Policy> |
1625 | inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 2>&) | |
92f5a8d4 TL |
1626 | { |
1627 | static const char* function = "boost::math::lambert_w0<%1%>"; | |
1628 | BOOST_MATH_STD_USING // Aid ADL of std functions. | |
1629 | ||
1630 | // Detect unusual case of 32-bit double with a wider/64-bit long double | |
1e59de90 | 1631 | static_assert(std::numeric_limits<double>::digits >= 53, |
92f5a8d4 TL |
1632 | "Our double precision coefficients will be truncated, " |
1633 | "please file a bug report with details of your platform's floating point types " | |
1634 | "- or possibly edit the coefficients to have " | |
1635 | "an appropriate size-suffix for 64-bit floats on your platform - L?"); | |
1636 | ||
1637 | if ((boost::math::isnan)(z)) | |
1638 | { | |
1639 | return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol); | |
1640 | } | |
1641 | if ((boost::math::isinf)(z)) | |
1642 | { | |
1643 | return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol); | |
1644 | } | |
1645 | ||
1646 | if (z >= 0.05) | |
1647 | { | |
1648 | return lambert_w_positive_rational_double(z); | |
1649 | } | |
1650 | else if (z <= -0.36787944117144232159552377016146086744581113103176804) // Precision is max_digits10(cpp_bin_float_50). | |
1651 | { | |
1652 | if (z < -0.36787944117144232159552377016146086744581113103176804) | |
1653 | { | |
1654 | return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol); | |
1655 | } | |
1656 | return -1; | |
1657 | } | |
1658 | else | |
1659 | { | |
1660 | return lambert_w_negative_rational_double(z, pol); | |
1661 | } | |
1e59de90 | 1662 | } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 2>&) 64-bit precision, usually double. |
92f5a8d4 TL |
1663 | |
1664 | //! lambert_W0 implementation for extended precision types including | |
1665 | //! long double (80-bit and 128-bit), ??? | |
1666 | //! quad float128, Boost.Multiprecision types like cpp_bin_float_quad, cpp_bin_float_50... | |
1667 | ||
1e59de90 TL |
1668 | template <typename T, typename Policy> |
1669 | inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 0>&) | |
92f5a8d4 TL |
1670 | { |
1671 | static const char* function = "boost::math::lambert_w0<%1%>"; | |
1672 | BOOST_MATH_STD_USING // Aid ADL of std functions. | |
1673 | ||
1674 | // Filter out special cases first: | |
1675 | if ((boost::math::isnan)(z)) | |
1676 | { | |
1677 | return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol); | |
1678 | } | |
1679 | if (fabs(z) <= 0.05f) | |
1680 | { | |
1681 | // Very small z: | |
1682 | return lambert_w0_small_z(z, pol); | |
1683 | } | |
1684 | if (z > (std::numeric_limits<double>::max)()) | |
1685 | { | |
1686 | if ((boost::math::isinf)(z)) | |
1687 | { | |
1688 | return policies::raise_overflow_error<T>(function, 0, pol); | |
1689 | // Or might return infinity if available else max_value, | |
1690 | // but other Boost.Math special functions raise overflow. | |
1691 | } | |
1692 | // z is larger than the largest double, so cannot use the polynomial to get an approximation, | |
1693 | // so use the asymptotic approximation and Halley iterate: | |
1694 | ||
1695 | T w = lambert_w0_approx(z); // Make an inline function as also used elsewhere. | |
1696 | //T lz = log(z); | |
1697 | //T llz = log(lz); | |
1698 | //T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162. | |
1699 | return lambert_w_halley_iterate(w, z); | |
1700 | } | |
1701 | if (z < -0.3578794411714423215955237701) | |
1702 | { // Very close to branch point so rational polynomials are not usable. | |
1703 | if (z <= -boost::math::constants::exp_minus_one<T>()) | |
1704 | { | |
1705 | if (z == -boost::math::constants::exp_minus_one<T>()) | |
1706 | { // Exactly at the branch point singularity. | |
1707 | return -1; | |
1708 | } | |
1709 | return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol); | |
1710 | } | |
1711 | // z is very close (within 0.01) of the branch singularity at -e^-1 | |
1712 | // so use a series approximation proposed by Corless et al. | |
1713 | const T p2 = 2 * (boost::math::constants::e<T>() * z + 1); | |
1714 | const T p = sqrt(p2); | |
1715 | T w = lambert_w_detail::lambert_w_singularity_series(p); | |
1716 | return lambert_w_halley_iterate(w, z); | |
1717 | } | |
1718 | ||
1719 | // Phew! If we get here we are in the normal range of the function, | |
1720 | // so get a double precision approximation first, then iterate to full precision of T. | |
1721 | // We define a tag_type that is: | |
f67539c2 TL |
1722 | // true_type if there are so many digits precision wanted that iteration is necessary. |
1723 | // false_type if a single Halley step is sufficient. | |
92f5a8d4 | 1724 | |
1e59de90 TL |
1725 | using precision_type = typename policies::precision<T, Policy>::type; |
1726 | using tag_type = std::integral_constant<bool, | |
92f5a8d4 TL |
1727 | (precision_type::value == 0) || (precision_type::value > 113) ? |
1728 | true // Unknown at compile-time, variable/arbitrary, or more than float128 or cpp_bin_quad 128-bit precision. | |
1729 | : false // float, double, float128, cpp_bin_quad 128-bit, so single Halley step. | |
1e59de90 | 1730 | >; |
92f5a8d4 TL |
1731 | |
1732 | // For speed, we also cast z to type double when that is possible | |
1e59de90 TL |
1733 | // if (std::is_constructible<double, T>() == true). |
1734 | T w = lambert_w0_imp(maybe_reduce_to_double(z, std::is_constructible<double, T>()), pol, std::integral_constant<int, 2>()); | |
92f5a8d4 TL |
1735 | |
1736 | return lambert_w_maybe_halley_iterate(w, z, tag_type()); | |
1737 | ||
1e59de90 | 1738 | } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 0>&) all extended precision types. |
92f5a8d4 TL |
1739 | |
1740 | // Lambert w-1 implementation | |
1741 | // ============================================================================================== | |
1742 | ||
1743 | //! Lambert W for W-1 branch, -max(z) < z <= -1/e. | |
1744 | // TODO is -max(z) allowed? | |
1e59de90 | 1745 | template<typename T, typename Policy> |
92f5a8d4 TL |
1746 | T lambert_wm1_imp(const T z, const Policy& pol) |
1747 | { | |
1748 | // Catch providing an integer value as parameter x to lambert_w, for example, lambert_w(1). | |
1749 | // Need to ensure it is a floating-point type (of the desired type, float 1.F, double 1., or long double 1.L), | |
1750 | // or static_casted integer, for example: static_cast<float>(1) or static_cast<cpp_dec_float_50>(1). | |
1751 | // Want to allow fixed_point types too, so do not just test for floating-point. | |
1752 | // Integral types should be promoted to double by user Lambert w functions. | |
1753 | // If integral type provided to user function lambert_w0 or lambert_wm1, | |
1754 | // then should already have been promoted to double. | |
1e59de90 | 1755 | static_assert(!std::is_integral<T>::value, |
92f5a8d4 TL |
1756 | "Must be floating-point or fixed type (not integer type), for example: lambert_wm1(1.), not lambert_wm1(1)!"); |
1757 | ||
1758 | BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs. | |
1759 | ||
1760 | const char* function = "boost::math::lambert_wm1<RealType>(<RealType>)"; // Used for error messages. | |
1761 | ||
1762 | // Check for edge and corner cases first: | |
1763 | if ((boost::math::isnan)(z)) | |
1764 | { | |
1765 | return policies::raise_domain_error(function, | |
1766 | "Argument z is NaN!", | |
1767 | z, pol); | |
1768 | } // isnan | |
1769 | ||
1770 | if ((boost::math::isinf)(z)) | |
1771 | { | |
1772 | return policies::raise_domain_error(function, | |
1773 | "Argument z is infinite!", | |
1774 | z, pol); | |
1775 | } // isinf | |
1776 | ||
1777 | if (z == static_cast<T>(0)) | |
1778 | { // z is exactly zero so return -std::numeric_limits<T>::infinity(); | |
1779 | if (std::numeric_limits<T>::has_infinity) | |
1780 | { | |
1781 | return -std::numeric_limits<T>::infinity(); | |
1782 | } | |
1783 | else | |
1784 | { | |
1785 | return -tools::max_value<T>(); | |
1786 | } | |
1787 | } | |
1788 | if (std::numeric_limits<T>::has_denorm) | |
1789 | { // All real types except arbitrary precision. | |
1790 | if (!(boost::math::isnormal)(z)) | |
1791 | { // Almost zero - might also just return infinity like z == 0 or max_value? | |
1792 | return policies::raise_overflow_error(function, | |
1793 | "Argument z = %1% is denormalized! (must be z > (std::numeric_limits<RealType>::min)() or z == 0)", | |
1794 | z, pol); | |
1795 | } | |
1796 | } | |
1797 | ||
1798 | if (z > static_cast<T>(0)) | |
1799 | { // | |
1800 | return policies::raise_domain_error(function, | |
1801 | "Argument z = %1% is out of range (z <= 0) for Lambert W-1 branch! (Try Lambert W0 branch?)", | |
1802 | z, pol); | |
1803 | } | |
1804 | if (z > -boost::math::tools::min_value<T>()) | |
1805 | { // z is denormalized, so cannot be computed. | |
1806 | // -std::numeric_limits<T>::min() is smallest for type T, | |
1807 | // for example, for double: lambert_wm1(-2.2250738585072014e-308) = -714.96865723796634 | |
1808 | return policies::raise_overflow_error(function, | |
1809 | "Argument z = %1% is too small (z < -std::numeric_limits<T>::min so denormalized) for Lambert W-1 branch!", | |
1810 | z, pol); | |
1811 | } | |
1812 | if (z == -boost::math::constants::exp_minus_one<T>()) // == singularity/branch point z = -exp(-1) = -3.6787944. | |
1813 | { // At singularity, so return exactly -1. | |
1814 | return -static_cast<T>(1); | |
1815 | } | |
1816 | // z is too negative for the W-1 (or W0) branch. | |
1817 | if (z < -boost::math::constants::exp_minus_one<T>()) // > singularity/branch point z = -exp(-1) = -3.6787944. | |
1818 | { | |
1819 | return policies::raise_domain_error(function, | |
1820 | "Argument z = %1% is out of range (z < -exp(-1) = -3.6787944... <= 0) for Lambert W-1 (or W0) branch!", | |
1821 | z, pol); | |
1822 | } | |
1823 | if (z < static_cast<T>(-0.35)) | |
1824 | { // Close to singularity/branch point z = -0.3678794411714423215955237701614608727 but on W-1 branch. | |
1825 | const T p2 = 2 * (boost::math::constants::e<T>() * z + 1); | |
1826 | if (p2 == 0) | |
1827 | { // At the singularity at branch point. | |
1828 | return -1; | |
1829 | } | |
1830 | if (p2 > 0) | |
1831 | { | |
1832 | T w_series = lambert_w_singularity_series(T(-sqrt(p2))); | |
1833 | if (boost::math::tools::digits<T>() > 53) | |
1834 | { // Multiprecision, so try a Halley refinement. | |
1835 | w_series = lambert_w_detail::lambert_w_halley_iterate(w_series, z); | |
1836 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN | |
1837 | std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); | |
1838 | std::cout << "Lambert W-1 Halley updated to " << w_series << std::endl; | |
1839 | std::cout.precision(saved_precision); | |
1840 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN | |
1841 | } | |
1842 | return w_series; | |
1843 | } | |
1844 | // Should not get here. | |
1845 | return policies::raise_domain_error(function, | |
1846 | "Argument z = %1% is out of range for Lambert W-1 branch. (Should not get here - please report!)", | |
1847 | z, pol); | |
1848 | } // if (z < -0.35) | |
1849 | ||
1850 | using lambert_w_lookup::wm1es; | |
1851 | using lambert_w_lookup::wm1zs; | |
1852 | using lambert_w_lookup::noof_wm1zs; // size == 64 | |
1853 | ||
1854 | // std::cout <<" Wm1zs[63] (== G[64]) = " << " " << wm1zs[63] << std::endl; // Wm1zs[63] (== G[64]) = -1.0264389699511283e-26 | |
1855 | // Check that z argument value is not smaller than lookup_table G[64] | |
1856 | // std::cout << "(z > wm1zs[63]) = " << std::boolalpha << (z > wm1zs[63]) << std::endl; | |
1857 | ||
1858 | if (z >= wm1zs[63]) // wm1zs[63] = -1.0264389699511282259046957018510946438e-26L W = 64.00000000000000000 | |
1859 | { // z >= -1.0264389699511303e-26 (but z != 0 and z >= std::numeric_limits<T>::min() and so NOT denormalized). | |
1860 | ||
1861 | // Some info on Lambert W-1 values for extreme values of z. | |
1862 | // std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); | |
1863 | // std::cout << "-std::numeric_limits<float>::min() = " << -(std::numeric_limits<float>::min)() << std::endl; | |
1864 | // std::cout << "-std::numeric_limits<double>::min() = " << -(std::numeric_limits<double>::min)() << std::endl; | |
1865 | // -std::numeric_limits<float>::min() = -1.1754943508222875e-38 | |
1866 | // -std::numeric_limits<double>::min() = -2.2250738585072014e-308 | |
1867 | // N[productlog(-1, -1.1754943508222875 * 10^-38 ), 50] = -91.856775324595479509567756730093823993834155027858 | |
1868 | // N[productlog(-1, -2.2250738585072014e-308 * 10^-308 ), 50] = -1424.8544521230553853558132180518404363617968042942 | |
1869 | // N[productlog(-1, -1.4325445274604020119111357113179868158* 10^-27), 37] = -65.99999999999999999999999999999999955 | |
1870 | ||
1871 | // R.M.Corless, G.H.Gonnet, D.E.G.Hare, D.J.Jeffrey, and D.E.Knuth, | |
1872 | // On the Lambert W function, Adv.Comput.Math., vol. 5, pp. 329, 1996. | |
1873 | // Francois Chapeau-Blondeau and Abdelilah Monir | |
1874 | // Numerical Evaluation of the Lambert W Function | |
1875 | // IEEE Transactions On Signal Processing, VOL. 50, NO. 9, Sep 2002 | |
1876 | // https://pdfs.semanticscholar.org/7a5a/76a9369586dd0dd34dda156d8f2779d1fd59.pdf | |
1877 | // Estimate Lambert W using ln(-z) ... | |
1878 | // This is roughly the power of ten * ln(10) ~= 2.3. n ~= 10^n | |
1879 | // and improve by adding a second term -ln(ln(-z)) | |
1880 | T guess; // bisect lowest possible Gk[=64] (for lookup_t type) | |
1881 | T lz = log(-z); | |
1882 | T llz = log(-lz); | |
1883 | guess = lz - llz + (llz / lz); // Chapeau-Blondeau equation 20, page 2162. | |
1884 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY | |
1885 | std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); | |
1886 | std::cout << "z = " << z << ", guess = " << guess << ", ln(-z) = " << lz << ", ln(-ln(-z) = " << llz << ", llz/lz = " << (llz / lz) << std::endl; | |
1887 | // z = -1.0000000000000001e-30, guess = -73.312782616731482, ln(-z) = -69.077552789821368, ln(-ln(-z) = 4.2352298269101114, llz/lz = -0.061311231447304194 | |
1888 | // z = -9.9999999999999999e-91, guess = -212.56650048504233, ln(-z) = -207.23265836946410, ln(-ln(-z) = 5.3338421155782205, llz/lz = -0.025738424423764311 | |
1889 | // >z = -2.2250738585072014e-308, guess = -714.95942238244606, ln(-z) = -708.39641853226408, ln(-ln(-z) = 6.5630038501819854, llz/lz = -0.0092645920821846622 | |
1890 | int d10 = policies::digits_base10<T, Policy>(); // policy template parameter digits10 | |
1891 | int d2 = policies::digits<T, Policy>(); // digits base 2 from policy. | |
1892 | std::cout << "digits10 = " << d10 << ", digits2 = " << d2 // For example: digits10 = 1, digits2 = 5 | |
1893 | << std::endl; | |
1894 | std::cout.precision(saved_precision); | |
1895 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY | |
1896 | if (policies::digits<T, Policy>() < 12) | |
1897 | { // For the worst case near w = 64, the error in the 'guess' is ~0.008, ratio ~ 0.0001 or 1 in 10,000 digits 10 ~= 4, or digits2 ~= 12. | |
1898 | return guess; | |
1899 | } | |
1900 | T result = lambert_w_detail::lambert_w_halley_iterate(guess, z); | |
1901 | return result; | |
1902 | ||
1903 | // Was Fukushima | |
1904 | // G[k=64] == g[63] == -1.02643897e-26 | |
1905 | //return policies::raise_domain_error(function, | |
1906 | // "Argument z = %1% is too small (< -1.02643897e-26) ! (Should not occur, please report.", | |
1907 | // z, pol); | |
1908 | } // Z too small so use approximation and Halley. | |
1909 | // Else Use a lookup table to find the nearest integer part of Lambert W-1 as starting point for Bisection. | |
1910 | ||
1911 | if (boost::math::tools::digits<T>() > 53) | |
1912 | { // T is more precise than 64-bit double (or long double, or ?), | |
1913 | // so compute an approximate value using only one Schroeder refinement, | |
1914 | // (avoiding any double-precision Halley refinement from policy double_digits2<50> 53 - 3 = 50 | |
1915 | // because are next going to use Halley refinement at full/high precision using this as an approximation). | |
1916 | using boost::math::policies::precision; | |
1917 | using boost::math::policies::digits10; | |
1918 | using boost::math::policies::digits2; | |
1919 | using boost::math::policies::policy; | |
1920 | // Compute a 50-bit precision approximate W0 in a double (no Halley refinement). | |
1e59de90 | 1921 | T double_approx(static_cast<T>(lambert_wm1_imp(must_reduce_to_double(z, std::is_constructible<double, T>()), policy<digits2<50>>()))); |
92f5a8d4 TL |
1922 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN |
1923 | std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); | |
1924 | std::cout << "Lambert_wm1 Argument Type " << typeid(T).name() << " approximation double = " << double_approx << std::endl; | |
1925 | std::cout.precision(saved_precision); | |
1926 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1 | |
1927 | // Perform additional Halley refinement(s) to ensure that | |
1928 | // get a near as possible to correct result (usually +/- one epsilon). | |
1929 | T result = lambert_w_halley_iterate(double_approx, z); | |
1930 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1 | |
1931 | std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); | |
1932 | std::cout << "Result " << typeid(T).name() << " precision Halley refinement = " << result << std::endl; | |
1933 | std::cout.precision(saved_precision); | |
1934 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1 | |
1935 | return result; | |
1936 | } // digits > 53 - higher precision than double. | |
1937 | else // T is double or less precision. | |
1938 | { // Use a lookup table to find the nearest integer part of Lambert W as starting point for Bisection. | |
1939 | using namespace boost::math::lambert_w_detail::lambert_w_lookup; | |
1940 | // Bracketing sequence n = (2, 4, 8, 16, 32, 64) for W-1 branch. (0 is -infinity) | |
1941 | // Since z is probably quite small, start with lowest n (=2). | |
1942 | int n = 2; | |
1943 | if (wm1zs[n - 1] > z) | |
1944 | { | |
1945 | goto bisect; | |
1946 | } | |
1947 | for (int j = 1; j <= 5; ++j) | |
1948 | { | |
1949 | n *= 2; | |
1950 | if (wm1zs[n - 1] > z) | |
1951 | { | |
1952 | goto overshot; | |
1953 | } | |
1954 | } | |
1955 | // else z < g[63] == -1.0264389699511303e-26, so Lambert W-1 integer part > 64. | |
1956 | // This should not now occur (should be caught by test and code above) so should be a logic_error? | |
1957 | return policies::raise_domain_error(function, | |
1958 | "Argument z = %1% is too small (< -1.026439e-26) (logic error - please report!)", | |
1959 | z, pol); | |
1960 | overshot: | |
1961 | { | |
1962 | int nh = n / 2; | |
1963 | for (int j = 1; j <= 5; ++j) | |
1964 | { | |
1965 | nh /= 2; // halve step size. | |
1966 | if (nh <= 0) | |
1967 | { | |
1968 | break; // goto bisect; | |
1969 | } | |
1970 | if (wm1zs[n - nh - 1] > z) | |
1971 | { | |
1972 | n -= nh; | |
1973 | } | |
1974 | } | |
1975 | } | |
1976 | bisect: | |
1977 | --n; | |
1978 | // g[n] now holds lambert W of floor integer n and g[n+1] the ceil part; | |
1979 | // these are used as initial values for bisection. | |
1980 | #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP | |
1981 | std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); | |
1982 | std::cout << "Result lookup W-1(" << z << ") bisection between wm1zs[" << n - 1 << "] = " << wm1zs[n - 1] << " and wm1zs[" << n << "] = " << wm1zs[n] | |
1983 | << ", bisect mean = " << (wm1zs[n - 1] + wm1zs[n]) / 2 << std::endl; | |
1984 | std::cout.precision(saved_precision); | |
1985 | #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP | |
1986 | ||
1987 | // Compute bisections is the number of bisections computed from n, | |
1988 | // such that a single application of the fifth-order Schroeder update formula | |
1989 | // after the bisections is enough to evaluate Lambert W-1 with (near?) 53-bit accuracy. | |
1990 | // Fukushima established these by trial and error? | |
1991 | int bisections = 11; // Assume maximum number of bisections will be needed (most common case). | |
1992 | if (n >= 8) | |
1993 | { | |
1994 | bisections = 8; | |
1995 | } | |
1996 | else if (n >= 3) | |
1997 | { | |
1998 | bisections = 9; | |
1999 | } | |
2000 | else if (n >= 2) | |
2001 | { | |
2002 | bisections = 10; | |
2003 | } | |
2004 | // Bracketing, Fukushima section 2.3, page 82: | |
2005 | // (Avoiding using exponential function for speed). | |
2006 | // Only use @c lookup_t precision, default double, for bisection (again for speed), | |
2007 | // and use later Halley refinement for higher precisions. | |
2008 | using lambert_w_lookup::halves; | |
2009 | using lambert_w_lookup::sqrtwm1s; | |
2010 | ||
1e59de90 | 2011 | using calc_type = typename std::conditional<std::is_constructible<lookup_t, T>::value, lookup_t, T>::type; |
92f5a8d4 TL |
2012 | |
2013 | calc_type w = -static_cast<calc_type>(n); // Equation 25, | |
2014 | calc_type y = static_cast<calc_type>(z * wm1es[n - 1]); // Equation 26, | |
2015 | // Perform the bisections fractional bisections for necessary precision. | |
2016 | for (int j = 0; j < bisections; ++j) | |
2017 | { // Equation 27. | |
2018 | calc_type wj = w - halves[j]; // Subtract 1/2, 1/4, 1/8 ... | |
2019 | calc_type yj = y * sqrtwm1s[j]; // Multiply by sqrt(1/e), ... | |
2020 | if (wj < yj) | |
2021 | { | |
2022 | w = wj; | |
2023 | y = yj; | |
2024 | } | |
2025 | } // for j | |
2026 | return static_cast<T>(schroeder_update(w, y)); // Schroeder 5th order method refinement. | |
2027 | ||
2028 | // else // Perform additional Halley refinement(s) to ensure that | |
2029 | // // get a near as possible to correct result (usually +/- epsilon). | |
2030 | // { | |
2031 | // // result = lambert_w_halley_iterate(result, z); | |
2032 | // result = lambert_w_halley_step(result, z); // Just one Halley step should be enough. | |
2033 | //#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_HALLEY | |
2034 | // std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); | |
2035 | // std::cout << "Halley refinement estimate = " << result << std::endl; | |
2036 | // std::cout.precision(saved_precision); | |
2037 | //#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W1_HALLEY | |
2038 | // return result; // Halley | |
2039 | // } // Schroeder or Schroeder and Halley. | |
2040 | } | |
2041 | } // template<typename T = double> T lambert_wm1_imp(const T z) | |
2042 | } // namespace lambert_w_detail | |
2043 | ||
2044 | ///////////////////////////// User Lambert w functions. ////////////////////////////// | |
2045 | ||
2046 | //! Lambert W0 using User-defined policy. | |
1e59de90 | 2047 | template <typename T, typename Policy> |
92f5a8d4 TL |
2048 | inline |
2049 | typename boost::math::tools::promote_args<T>::type | |
2050 | lambert_w0(T z, const Policy& pol) | |
2051 | { | |
2052 | // Promote integer or expression template arguments to double, | |
2053 | // without doing any other internal promotion like float to double. | |
1e59de90 | 2054 | using result_type = typename tools::promote_args<T>::type; |
92f5a8d4 TL |
2055 | |
2056 | // Work out what precision has been selected, | |
2057 | // based on the Policy and the number type. | |
1e59de90 | 2058 | using precision_type = typename policies::precision<result_type, Policy>::type; |
92f5a8d4 | 2059 | // and then select the correct implementation based on that precision (not the type T): |
1e59de90 | 2060 | using tag_type = std::integral_constant<int, |
92f5a8d4 TL |
2061 | (precision_type::value == 0) || (precision_type::value > 53) ? |
2062 | 0 // either variable precision (0), or greater than 64-bit precision. | |
2063 | : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision. | |
2064 | : 2 // 64-bit (probably double) precision. | |
1e59de90 | 2065 | >; |
92f5a8d4 TL |
2066 | |
2067 | return lambert_w_detail::lambert_w0_imp(result_type(z), pol, tag_type()); // | |
2068 | } // lambert_w0(T z, const Policy& pol) | |
2069 | ||
2070 | //! Lambert W0 using default policy. | |
1e59de90 | 2071 | template <typename T> |
92f5a8d4 TL |
2072 | inline |
2073 | typename tools::promote_args<T>::type | |
2074 | lambert_w0(T z) | |
2075 | { | |
2076 | // Promote integer or expression template arguments to double, | |
2077 | // without doing any other internal promotion like float to double. | |
1e59de90 | 2078 | using result_type = typename tools::promote_args<T>::type; |
92f5a8d4 TL |
2079 | |
2080 | // Work out what precision has been selected, based on the Policy and the number type. | |
2081 | // For the default policy version, we want the *default policy* precision for T. | |
1e59de90 | 2082 | using precision_type = typename policies::precision<result_type, policies::policy<>>::type; |
92f5a8d4 | 2083 | // and then select the correct implementation based on that (not the type T): |
1e59de90 | 2084 | using tag_type = std::integral_constant<int, |
92f5a8d4 TL |
2085 | (precision_type::value == 0) || (precision_type::value > 53) ? |
2086 | 0 // either variable precision (0), or greater than 64-bit precision. | |
2087 | : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision. | |
2088 | : 2 // 64-bit (probably double) precision. | |
1e59de90 | 2089 | >; |
92f5a8d4 TL |
2090 | return lambert_w_detail::lambert_w0_imp(result_type(z), policies::policy<>(), tag_type()); |
2091 | } // lambert_w0(T z) using default policy. | |
2092 | ||
2093 | //! W-1 branch (-max(z) < z <= -1/e). | |
2094 | ||
2095 | //! Lambert W-1 using User-defined policy. | |
1e59de90 | 2096 | template <typename T, typename Policy> |
92f5a8d4 TL |
2097 | inline |
2098 | typename tools::promote_args<T>::type | |
2099 | lambert_wm1(T z, const Policy& pol) | |
2100 | { | |
2101 | // Promote integer or expression template arguments to double, | |
2102 | // without doing any other internal promotion like float to double. | |
1e59de90 | 2103 | using result_type = typename tools::promote_args<T>::type; |
92f5a8d4 TL |
2104 | return lambert_w_detail::lambert_wm1_imp(result_type(z), pol); // |
2105 | } | |
2106 | ||
2107 | //! Lambert W-1 using default policy. | |
1e59de90 | 2108 | template <typename T> |
92f5a8d4 TL |
2109 | inline |
2110 | typename tools::promote_args<T>::type | |
2111 | lambert_wm1(T z) | |
2112 | { | |
1e59de90 | 2113 | using result_type = typename tools::promote_args<T>::type; |
92f5a8d4 TL |
2114 | return lambert_w_detail::lambert_wm1_imp(result_type(z), policies::policy<>()); |
2115 | } // lambert_wm1(T z) | |
2116 | ||
2117 | // First derivative of Lambert W0 and W-1. | |
1e59de90 | 2118 | template <typename T, typename Policy> |
92f5a8d4 TL |
2119 | inline typename tools::promote_args<T>::type |
2120 | lambert_w0_prime(T z, const Policy& pol) | |
2121 | { | |
1e59de90 | 2122 | using result_type = typename tools::promote_args<T>::type; |
92f5a8d4 TL |
2123 | using std::numeric_limits; |
2124 | if (z == 0) | |
2125 | { | |
2126 | return static_cast<result_type>(1); | |
2127 | } | |
2128 | // This is the sensible choice if we regard the Lambert-W function as complex analytic. | |
2129 | // Of course on the real line, it's just undefined. | |
2130 | if (z == - boost::math::constants::exp_minus_one<result_type>()) | |
2131 | { | |
2132 | return numeric_limits<result_type>::has_infinity ? numeric_limits<result_type>::infinity() : boost::math::tools::max_value<result_type>(); | |
2133 | } | |
2134 | // if z < -1/e, we'll let lambert_w0 do the error handling: | |
2135 | result_type w = lambert_w0(result_type(z), pol); | |
2136 | // If w ~ -1, then presumably this can get inaccurate. | |
2137 | // Is there an accurate way to evaluate 1 + W(-1/e + eps)? | |
2138 | // Yes: This is discussed in the Princeton Companion to Applied Mathematics, | |
2139 | // 'The Lambert-W function', Section 1.3: Series and Generating Functions. | |
2140 | // 1 + W(-1/e + x) ~ sqrt(2ex). | |
2141 | // Nick is not convinced this formula is more accurate than the naive one. | |
2142 | // However, for z != -1/e, we never get rounded to w = -1 in any precision I've tested (up to cpp_bin_float_100). | |
2143 | return w / (z * (1 + w)); | |
2144 | } // lambert_w0_prime(T z) | |
2145 | ||
1e59de90 | 2146 | template <typename T> |
92f5a8d4 TL |
2147 | inline typename tools::promote_args<T>::type |
2148 | lambert_w0_prime(T z) | |
2149 | { | |
2150 | return lambert_w0_prime(z, policies::policy<>()); | |
2151 | } | |
2152 | ||
1e59de90 | 2153 | template <typename T, typename Policy> |
92f5a8d4 TL |
2154 | inline typename tools::promote_args<T>::type |
2155 | lambert_wm1_prime(T z, const Policy& pol) | |
2156 | { | |
2157 | using std::numeric_limits; | |
1e59de90 | 2158 | using result_type = typename tools::promote_args<T>::type; |
92f5a8d4 TL |
2159 | //if (z == 0) |
2160 | //{ | |
2161 | // return static_cast<result_type>(1); | |
2162 | //} | |
2163 | //if (z == - boost::math::constants::exp_minus_one<result_type>()) | |
2164 | if (z == 0 || z == - boost::math::constants::exp_minus_one<result_type>()) | |
2165 | { | |
2166 | return numeric_limits<result_type>::has_infinity ? -numeric_limits<result_type>::infinity() : -boost::math::tools::max_value<result_type>(); | |
2167 | } | |
2168 | ||
2169 | result_type w = lambert_wm1(z, pol); | |
2170 | return w/(z*(1+w)); | |
2171 | } // lambert_wm1_prime(T z) | |
2172 | ||
1e59de90 | 2173 | template <typename T> |
92f5a8d4 TL |
2174 | inline typename tools::promote_args<T>::type |
2175 | lambert_wm1_prime(T z) | |
2176 | { | |
2177 | return lambert_wm1_prime(z, policies::policy<>()); | |
2178 | } | |
2179 | ||
2180 | }} //boost::math namespaces | |
2181 | ||
2182 | #endif // #ifdef BOOST_MATH_SF_LAMBERT_W_HPP | |
2183 |