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.
61 #include <boost/math/tools/precision.hpp> // boost::math::tools::max_value().
62 #include <boost/math/tools/big_constant.hpp>
63 #include <boost/math/tools/cxx03_warn.hpp>
64
65 #include <limits>
66 #include <cmath>
67 #include <limits>
68 #include <exception>
69 #include <type_traits>
70 #include <cstdint>
71
72 // Needed for testing and diagnostics only.
73 #include <iostream>
74 #include <typeinfo>
75 #include <boost/math/special_functions/next.hpp> // For float_distance.
76
77 using lookup_t = double; // Type for lookup table (double or float, or even long double?)
78
79 //#include "J:\Cpp\Misc\lambert_w_lookup_table_generator\lambert_w_lookup_table.ipp"
80 // #include "lambert_w_lookup_table.ipp" // Boost.Math version.
81 #include <boost/math/special_functions/detail/lambert_w_lookup_table.ipp>
82
83 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
84 //
85 // This is the only way we can avoid
86 // warning: non-standard suffix on floating constant [-Wpedantic]
87 // when building with -Wall -pedantic. Neither __extension__
88 // nor #pragma diagnostic ignored work :(
89 //
90 #pragma GCC system_header
91 #endif
92
93 namespace boost {
94 namespace math {
95 namespace lambert_w_detail {
96
97 //! \brief Applies a single Halley step to make a better estimate of Lambert W.
98 //! \details Used the simplified formulae obtained from
99 //! 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
100 //! [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)]
101
102 //! \tparam T floating-point (or fixed-point) type.
103 //! \param w_est Lambert W estimate.
104 //! \param z Argument z for Lambert_w function.
105 //! \returns New estimate of Lambert W, hopefully improved.
106 //!
107 template <typename T>
lambert_w_halley_step(T w_est,const T z)108 inline T lambert_w_halley_step(T w_est, const T z)
109 {
110 BOOST_MATH_STD_USING
111 T e = exp(w_est);
112 w_est -= 2 * (w_est + 1) * (e * w_est - z) / (z * (w_est + 2) + e * (w_est * (w_est + 2) + 2));
113 return w_est;
114 } // template <typename T> lambert_w_halley_step(T w_est, T z)
115
116 //! \brief Halley iterate to refine Lambert_w estimate,
117 //! taking at least one Halley_step.
118 //! Repeat Halley steps until the *last step* had fewer than half the digits wrong,
119 //! the step we've just taken should have been sufficient to have completed the iteration.
120
121 //! \tparam T floating-point (or fixed-point) type.
122 //! \param z Argument z for Lambert_w function.
123 //! \param w_est Lambert w estimate.
124 template <typename T>
lambert_w_halley_iterate(T w_est,const T z)125 inline T lambert_w_halley_iterate(T w_est, const T z)
126 {
127 BOOST_MATH_STD_USING
128 static const T max_diff = boost::math::tools::root_epsilon<T>() * fabs(w_est);
129
130 T w_new = lambert_w_halley_step(w_est, z);
131 T diff = fabs(w_est - w_new);
132 while (diff > max_diff)
133 {
134 w_est = w_new;
135 w_new = lambert_w_halley_step(w_est, z);
136 diff = fabs(w_est - w_new);
137 }
138 return w_new;
139 } // template <typename T> lambert_w_halley_iterate(T w_est, T z)
140
141 // Two Halley function versions that either
142 // single step (if std::false_type) or iterate (if std::true_type).
143 // Selected at compile-time using parameter 3.
144 template <typename T>
lambert_w_maybe_halley_iterate(T z,T w,std::false_type const &)145 inline T lambert_w_maybe_halley_iterate(T z, T w, std::false_type const&)
146 {
147 return lambert_w_halley_step(z, w); // Single step.
148 }
149
150 template <typename T>
lambert_w_maybe_halley_iterate(T z,T w,std::true_type const &)151 inline T lambert_w_maybe_halley_iterate(T z, T w, std::true_type const&)
152 {
153 return lambert_w_halley_iterate(z, w); // Iterate steps.
154 }
155
156 //! maybe_reduce_to_double function,
157 //! Two versions that have a compile-time option to
158 //! reduce argument z to double precision (if true_type).
159 //! Version is selected at compile-time using parameter 2.
160
161 template <typename T>
maybe_reduce_to_double(const T & z,const std::true_type &)162 inline double maybe_reduce_to_double(const T& z, const std::true_type&)
163 {
164 return static_cast<double>(z); // Reduce to double precision.
165 }
166
167 template <typename T>
maybe_reduce_to_double(const T & z,const std::false_type &)168 inline T maybe_reduce_to_double(const T& z, const std::false_type&)
169 { // Don't reduce to double.
170 return z;
171 }
172
173 template <typename T>
must_reduce_to_double(const T & z,const std::true_type &)174 inline double must_reduce_to_double(const T& z, const std::true_type&)
175 {
176 return static_cast<double>(z); // Reduce to double precision.
177 }
178
179 template <typename T>
must_reduce_to_double(const T & z,const std::false_type &)180 inline double must_reduce_to_double(const T& z, const std::false_type&)
181 { // try a lexical_cast and hope for the best:
182 return boost::lexical_cast<double>(z);
183 }
184
185 //! \brief Schroeder method, fifth-order update formula,
186 //! \details See T. Fukushima page 80-81, and
187 //! A. Householder, The Numerical Treatment of a Single Nonlinear Equation,
188 //! McGraw-Hill, New York, 1970, section 4.4.
189 //! Fukushima algorithm switches to @c schroeder_update after pre-computed bisections,
190 //! chosen to ensure that the result will be achieve the +/- 10 epsilon target.
191 //! \param w Lambert w estimate from bisection or series.
192 //! \param y bracketing value from bisection.
193 //! \returns Refined estimate of Lambert w.
194
195 // Schroeder refinement, called unless NOT required by precision policy.
196 template<typename T>
schroeder_update(const T w,const T y)197 inline T schroeder_update(const T w, const T y)
198 {
199 // Compute derivatives using 5th order Schroeder refinement.
200 // Since this is the final step, it will always use the highest precision type T.
201 // Example of Call:
202 // result = schroeder_update(w, y);
203 //where
204 // w is estimate of Lambert W (from bisection or series).
205 // y is z * e^-w.
206
207 BOOST_MATH_STD_USING // Aid argument dependent lookup of abs.
208 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
209 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
210 using boost::math::float_distance;
211 T fd = float_distance<T>(w, y);
212 std::cout << "Schroder ";
213 if (abs(fd) < 214748000.)
214 {
215 std::cout << " Distance = "<< static_cast<int>(fd);
216 }
217 else
218 {
219 std::cout << "Difference w - y = " << (w - y) << ".";
220 }
221 std::cout << std::endl;
222 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
223 // Fukushima equation 18, page 6.
224 const T f0 = w - y; // f0 = w - y.
225 const T f1 = 1 + y; // f1 = df/dW
226 const T f00 = f0 * f0;
227 const T f11 = f1 * f1;
228 const T f0y = f0 * y;
229 const T result =
230 w - 4 * f0 * (6 * f1 * (f11 + f0y) + f00 * y) /
231 (f11 * (24 * f11 + 36 * f0y) +
232 f00 * (6 * y * y + 8 * f1 * y + f0y)); // Fukushima Page 81, equation 21 from equation 20.
233
234 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
235 std::cout << "Schroeder refined " << w << " " << y << ", difference " << w-y << ", change " << w - result << ", to result " << result << std::endl;
236 std::cout.precision(saved_precision); // Restore.
237 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
238
239 return result;
240 } // template<typename T = double> T schroeder_update(const T w, const T y)
241
242 //! \brief Series expansion used near the singularity/branch point z = -exp(-1) = -3.6787944.
243 //! Wolfram InverseSeries[Series[sqrt[2(p Exp[1 + p] + 1)], {p,-1, 20}]]
244 //! Wolfram command used to obtain 40 series terms at 50 decimal digit precision was
245 //! N[InverseSeries[Series[Sqrt[2(p Exp[1 + p] + 1)], { p,-1,40 }]], 50]
246 //! -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 ...
247 //! Decimal values of specifications for built-in floating-point types below
248 //! are at least 21 digits precision == max_digits10 for long double.
249 //! Longer decimal digits strings are rationals evaluated using Wolfram.
250
251 template<typename T>
lambert_w_singularity_series(const T p)252 T lambert_w_singularity_series(const T p)
253 {
254 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES
255 std::size_t saved_precision = std::cout.precision(3);
256 std::cout << "Singularity_series Lambert_w p argument = " << p << std::endl;
257 std::cout
258 //<< "Argument Type = " << typeid(T).name()
259 //<< ", max_digits10 = " << std::numeric_limits<T>::max_digits10
260 //<< ", epsilon = " << std::numeric_limits<T>::epsilon()
261 << std::endl;
262 std::cout.precision(saved_precision);
263 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES
264
265 static const T q[] =
266 {
267 -static_cast<T>(1), // j0
268 +T(1), // j1
269 -T(1) / 3, // 1/3 j2
270 +T(11) / 72, // 0.152777777777777778, // 11/72 j3
271 -T(43) / 540, // 0.0796296296296296296, // 43/540 j4
272 +T(769) / 17280, // 0.0445023148148148148, j5
273 -T(221) / 8505, // 0.0259847148736037625, j6
274 //+T(0.0156356325323339212L), // j7
275 //+T(0.015635632532333921222810111699000587889476778365667L), // j7 from Wolfram N[680863/43545600, 50]
276 +T(680863uLL) / 43545600uLL, // +0.0156356325323339212, j7
277 //-T(0.00961689202429943171L), // j8
278 -T(1963uLL) / 204120uLL, // 0.00961689202429943171, j8
279 //-T(0.0096168920242994317068391142465216539290613364687439L), // j8 from Wolfram N[1963/204120, 50]
280 +T(226287557uLL) / 37623398400uLL, // 0.00601454325295611786, j9
281 -T(5776369uLL) / 1515591000uLL, // 0.00381129803489199923, j10
282 //+T(0.00244087799114398267L), j11 0.0024408779911439826658968585286437530215699919795550
283 +T(169709463197uLL) / 69528040243200uLL, // j11
284 // -T(0.00157693034468678425L), // j12 -0.0015769303446867842539234095399314115973161850314723
285 -T(1118511313uLL) / 709296588000uLL, // j12
286 +T(667874164916771uLL) / 650782456676352000uLL, // j13
287 //+T(0.00102626332050760715L), // j13 0.0010262633205076071544375481533906861056468041465973
288 -T(500525573uLL) / 744761417400uLL, // j14
289 // -T(0.000672061631156136204L), j14
290 //+T(1003663334225097487uLL) / 234281684403486720000uLL, // j15 0.00044247306181462090993020760858473726479232802068800 error C2177: constant too big
291 //+T(0.000442473061814620910L, // j15
292 BOOST_MATH_BIG_CONSTANT(T, 64, +0.000442473061814620910), // j15
293 // -T(0.000292677224729627445L), // j16
294 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000292677224729627445), // j16
295 //+T(0.000194387276054539318L), // j17
296 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000194387276054539318), // j17
297 //-T(0.000129574266852748819L), // j18
298 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000129574266852748819), // j18
299 //+T(0.0000866503580520812717L), // j19 N[+1150497127780071399782389/13277465363600276402995200000, 50] 0.000086650358052081271660451590462390293190597827783288
300 BOOST_MATH_BIG_CONSTANT(T, 64, +0.0000866503580520812717), // j19
301 //-T(0.0000581136075044138168L) // j20 N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913
302 // -T(2853534237182741069uLL) / 49102686267859224000000uLL // j20 // error C2177: constant too big,
303 // so must use BOOST_MATH_BIG_CONSTANT(T, ) format in hope of using suffix Q for quad or decimal digits string for others.
304 //-T(0.000058113607504413816772205464778828177256611844221913L), // j20 N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913
305 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000058113607504413816772205464778828177256611844221913) // j20 - last used by Fukushima
306 // More terms don't seem to give any improvement (worse in fact) and are not use for many z values.
307 //BOOST_MATH_BIG_CONSTANT(T, +0.000039076684867439051635395583044527492132109160553593), // j21
308 //BOOST_MATH_BIG_CONSTANT(T, -0.000026338064747231098738584082718649443078703982217219), // j22
309 //BOOST_MATH_BIG_CONSTANT(T, +0.000017790345805079585400736282075184540383274460464169), // j23
310 //BOOST_MATH_BIG_CONSTANT(T, -0.000012040352739559976942274116578992585158113153190354), // j24
311 //BOOST_MATH_BIG_CONSTANT(T, +8.1635319824966121713827512573558687050675701559448E-6), // j25
312 //BOOST_MATH_BIG_CONSTANT(T, -5.5442032085673591366657251660804575198155559225316E-6) // j26
313 // -T(5.5442032085673591366657251660804575198155559225316E-6L) // j26
314 // 21 to 26 Added for long double.
315 }; // static const T q[]
316
317 /*
318 // Temporary copy of original double values for comparison; these are reproduced well.
319 static const T q[] =
320 {
321 -1L, // j0
322 +1L, // j1
323 -0.333333333333333333L, // 1/3 j2
324 +0.152777777777777778L, // 11/72 j3
325 -0.0796296296296296296L, // 43/540
326 +0.0445023148148148148L,
327 -0.0259847148736037625L,
328 +0.0156356325323339212L,
329 -0.00961689202429943171L,
330 +0.00601454325295611786L,
331 -0.00381129803489199923L,
332 +0.00244087799114398267L,
333 -0.00157693034468678425L,
334 +0.00102626332050760715L,
335 -0.000672061631156136204L,
336 +0.000442473061814620910L,
337 -0.000292677224729627445L,
338 +0.000194387276054539318L,
339 -0.000129574266852748819L,
340 +0.0000866503580520812717L,
341 -0.0000581136075044138168L // j20
342 };
343 */
344
345 // Decide how many series terms to use, increasing as z approaches the singularity,
346 // balancing run-time versus computational noise from round-off.
347 // In practice, we truncate the series expansion at a certain order.
348 // If the order is too large, not only does the amount of computation increase,
349 // but also the round-off errors accumulate.
350 // See Fukushima equation 35, page 85 for logic of choice of number of series terms.
351
352 BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs.
353
354 const T absp = abs(p);
355
356 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS
357 {
358 int terms = 20; // Default to using all terms.
359 if (absp < 0.01159)
360 { // Very near singularity.
361 terms = 6;
362 }
363 else if (absp < 0.0766)
364 { // Near singularity.
365 terms = 10;
366 }
367 std::streamsize saved_precision = std::cout.precision(3);
368 std::cout << "abs(p) = " << absp << ", terms = " << terms << std::endl;
369 std::cout.precision(saved_precision);
370 }
371 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS
372
373 if (absp < 0.01159)
374 { // Only 6 near-singularity series terms are useful.
375 return
376 -1 +
377 p * (1 +
378 p * (q[2] +
379 p * (q[3] +
380 p * (q[4] +
381 p * (q[5] +
382 p * q[6]
383 )))));
384 }
385 else if (absp < 0.0766) // Use 10 near-singularity series terms.
386 { // Use 10 near-singularity series terms.
387 return
388 -1 +
389 p * (1 +
390 p * (q[2] +
391 p * (q[3] +
392 p * (q[4] +
393 p * (q[5] +
394 p * (q[6] +
395 p * (q[7] +
396 p * (q[8] +
397 p * (q[9] +
398 p * q[10]
399 )))))))));
400 }
401 else
402 { // Use all 20 near-singularity series terms.
403 return
404 -1 +
405 p * (1 +
406 p * (q[2] +
407 p * (q[3] +
408 p * (q[4] +
409 p * (q[5] +
410 p * (q[6] +
411 p * (q[7] +
412 p * (q[8] +
413 p * (q[9] +
414 p * (q[10] +
415 p * (q[11] +
416 p * (q[12] +
417 p * (q[13] +
418 p * (q[14] +
419 p * (q[15] +
420 p * (q[16] +
421 p * (q[17] +
422 p * (q[18] +
423 p * (q[19] +
424 p * q[20] // Last Fukushima term.
425 )))))))))))))))))));
426 // + // more terms for more precise T: long double ...
427 //// but makes almost no difference, so don't use more terms?
428 // p*q[21] +
429 // p*q[22] +
430 // p*q[23] +
431 // p*q[24] +
432 // p*q[25]
433 // )))))))))))))))))));
434 }
435 } // template<typename T = double> T lambert_w_singularity_series(const T p)
436
437
438 /////////////////////////////////////////////////////////////////////////////////////////////
439
440 //! \brief Series expansion used near zero (abs(z) < 0.05).
441 //! \details
442 //! Coefficients of the inverted series expansion of the Lambert W function around z = 0.
443 //! Tosio Fukushima always uses all 17 terms of a Taylor series computed using Wolfram with
444 //! InverseSeries[Series[z Exp[z],{z,0,17}]]
445 //! Tosio Fukushima / Journal of Computational and Applied Mathematics 244 (2013) page 86.
446
447 //! Decimal values of specifications for built-in floating-point types below
448 //! are 21 digits precision == max_digits10 for long double.
449 //! Care! Some coefficients might overflow some fixed_point types.
450
451 //! This version is intended to allow use by user-defined types
452 //! like Boost.Multiprecision quad and cpp_dec_float types.
453 //! The three specializations below for built-in float, double
454 //! (and perhaps long double) will be chosen in preference for these types.
455
456 //! This version uses rationals computed by Wolfram as far as possible,
457 //! limited by maximum size of uLL integers.
458 //! For higher term, uses decimal digit strings computed by Wolfram up to the maximum possible using uLL rationals,
459 //! and then higher coefficients are computed as necessary using function lambert_w0_small_z_series_term
460 //! until the precision required by the policy is achieved.
461 //! InverseSeries[Series[z Exp[z],{z,0,34}]] also computed.
462
463 // Series evaluation for LambertW(z) as z -> 0.
464 // See http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/
465 // http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/MainEq1.L.gif
466
467 //! \brief lambert_w0_small_z uses a tag_type to select a variant depending on the size of the type.
468 //! The Lambert W is computed by lambert_w0_small_z for small z.
469 //! The cutoff for z smallness determined by Tosio Fukushima by trial and error is (abs(z) < 0.05),
470 //! but the optimum might be a function of the size of the type of z.
471
472 //! \details
473 //! The tag_type selection is based on the value @c std::numeric_limits<T>::max_digits10.
474 //! This allows distinguishing between long double types that commonly vary between 64 and 80-bits,
475 //! and also compilers that have a float type using 64 bits and/or long double using 128-bits.
476 //! It assumes that max_digits10 is defined correctly or this might fail to make the correct selection.
477 //! causing very small differences in computing lambert_w that would be very difficult to detect and diagnose.
478 //! Cannot switch on @c std::numeric_limits<>::max() because comparison values may overflow the compiler limit.
479 //! Cannot switch on @c std::numeric_limits<long double>::max_exponent10()
480 //! because both 80 and 128 bit floating-point implementations use 11 bits for the exponent.
481 //! So must rely on @c std::numeric_limits<long double>::max_digits10.
482
483 //! Specialization of float zero series expansion used for small z (abs(z) < 0.05).
484 //! Specializations of lambert_w0_small_z for built-in types.
485 //! These specializations should be chosen in preference to T version.
486 //! For example: lambert_w0_small_z(0.001F) should use the float version.
487 //! (Parameter Policy is not used by built-in types when all terms are used during an inline computation,
488 //! but for the tag_type selection to work, they all must include Policy in their signature.
489
490 // Forward declaration of variants of lambert_w0_small_z.
491 template <typename T, typename Policy>
492 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 0> const&); // for float (32-bit) type.
493
494 template <typename T, typename Policy>
495 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 1> const&); // for double (64-bit) type.
496
497 template <typename T, typename Policy>
498 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 2> const&); // for long double (double extended 80-bit) type.
499
500 template <typename T, typename Policy>
501 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 3> const&); // for long double (128-bit) type.
502
503 template <typename T, typename Policy>
504 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 4> const&); // for float128 quadmath Q type.
505
506 template <typename T, typename Policy>
507 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 5> const&); // Generic multiprecision T.
508 // Set tag_type depending on max_digits10.
509 template <typename T, typename Policy>
510 T lambert_w0_small_z(T x, const Policy& pol)
511 { //std::numeric_limits<T>::max_digits10 == 36 ? 3 : // 128-bit long double.
512 using tag_type = std::integral_constant<int,
513 std::numeric_limits<T>::is_specialized == 0 ? 5 :
514 #ifndef BOOST_NO_CXX11_NUMERIC_LIMITS
515 std::numeric_limits<T>::max_digits10 <= 9 ? 0 : // for float 32-bit.
516 std::numeric_limits<T>::max_digits10 <= 17 ? 1 : // for double 64-bit.
517 std::numeric_limits<T>::max_digits10 <= 22 ? 2 : // for 80-bit double extended.
518 std::numeric_limits<T>::max_digits10 < 37 ? 4 // for both 128-bit long double (3) and 128-bit quad suffix Q type (4).
519 #else
520 std::numeric_limits<T>::radix != 2 ? 5 :
521 std::numeric_limits<T>::digits <= 24 ? 0 : // for float 32-bit.
522 std::numeric_limits<T>::digits <= 53 ? 1 : // for double 64-bit.
523 std::numeric_limits<T>::digits <= 64 ? 2 : // for 80-bit double extended.
524 std::numeric_limits<T>::digits <= 113 ? 4 // for both 128-bit long double (3) and 128-bit quad suffix Q type (4).
525 #endif
526 : 5>; // All Generic multiprecision types.
527 // std::cout << "\ntag type = " << tag_type << std::endl; // error C2275: 'tag_type': illegal use of this type as an expression.
528 return lambert_w0_small_z(x, pol, tag_type());
529 } // template <typename T> T lambert_w0_small_z(T x)
530
531 //! Specialization of float (32-bit) series expansion used for small z (abs(z) < 0.05).
532 // Only 9 Coefficients are computed to 21 decimal digits precision, ample for 32-bit float used by most platforms.
533 // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction
534 // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],
535 // as proposed by Tosio Fukushima and implemented by Darko Veberic.
536
537 template <typename T, typename Policy>
538 T lambert_w0_small_z(T z, const Policy&, std::integral_constant<int, 0> const&)
539 {
540 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
541 std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
542 std::cout << "\ntag_type 0 float lambert_w0_small_z called with z = " << z << " using " << 9 << " terms of precision "
543 << std::numeric_limits<float>::max_digits10 << " decimal digits. " << std::endl;
544 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
545 T result =
546 z * (1 - // j1 z^1 term = 1
547 z * (1 - // j2 z^2 term = -1
548 z * (static_cast<float>(3uLL) / 2uLL - // 3/2 // j3 z^3 term = 1.5.
549 z * (2.6666666666666666667F - // 8/3 // j4
550 z * (5.2083333333333333333F - // -125/24 // j5
551 z * (10.8F - // j6
552 z * (23.343055555555555556F - // j7
553 z * (52.012698412698412698F - // j8
554 z * 118.62522321428571429F)))))))); // j9
555
556 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
557 std::cout << "return w = " << result << std::endl;
558 std::cout.precision(prec); // Restore.
559 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
560
561 return result;
562 } // template <typename T> T lambert_w0_small_z(T x, std::integral_constant<int, 0> const&)
563
564 //! Specialization of double (64-bit double) series expansion used for small z (abs(z) < 0.05).
565 // 17 Coefficients are computed to 21 decimal digits precision suitable for 64-bit double used by most platforms.
566 // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction
567 // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], as proposed by Tosio Fukushima and implemented by Veberic.
568
569 template <typename T, typename Policy>
570 T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 1> const&)
571 {
572 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
573 std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
574 std::cout << "\ntag_type 1 double lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision, "
575 << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl;
576 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
577 T result =
578 z * (1. - // j1 z^1
579 z * (1. - // j2 z^2
580 z * (1.5 - // 3/2 // j3 z^3
581 z * (2.6666666666666666667 - // 8/3 // j4
582 z * (5.2083333333333333333 - // -125/24 // j5
583 z * (10.8 - // j6
584 z * (23.343055555555555556 - // j7
585 z * (52.012698412698412698 - // j8
586 z * (118.62522321428571429 - // j9
587 z * (275.57319223985890653 - // j10
588 z * (649.78717234347442681 - // j11
589 z * (1551.1605194805194805 - // j12
590 z * (3741.4497029592385495 - // j13
591 z * (9104.5002411580189358 - // j14
592 z * (22324.308512706601434 - // j15
593 z * (55103.621972903835338 - // j16
594 z * 136808.86090394293563)))))))))))))))); // j17 z^17
595
596 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
597 std::cout << "return w = " << result << std::endl;
598 std::cout.precision(prec); // Restore.
599 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
600
601 return result;
602 } // T lambert_w0_small_z(const T z, std::integral_constant<int, 1> const&)
603
604 //! Specialization of long double (80-bit double extended) series expansion used for small z (abs(z) < 0.05).
605 // 21 Coefficients are computed to 21 decimal digits precision suitable for 80-bit long double used by some
606 // platforms including GCC and Clang when generating for Intel X86 floating-point processors with 80-bit operations enabled (the default).
607 // (This is NOT used by Microsoft Visual Studio where double and long always both use only 64-bit type.
608 // Nor used for 128-bit float128.)
609 template <typename T, typename Policy>
610 T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 2> const&)
611 {
612 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
613 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
614 std::cout << "\ntag_type 2 long double (80-bit double extended) lambert_w0_small_z called with z = " << z << " using " << 21 << " terms of precision, "
615 << std::numeric_limits<long double>::max_digits10 << " decimal digits. " << std::endl;
616 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
617 // T result =
618 // z * (1.L - // j1 z^1
619 // z * (1.L - // j2 z^2
620 // z * (1.5L - // 3/2 // j3
621 // z * (2.6666666666666666667L - // 8/3 // j4
622 // z * (5.2083333333333333333L - // -125/24 // j5
623 // z * (10.800000000000000000L - // j6
624 // z * (23.343055555555555556L - // j7
625 // z * (52.012698412698412698L - // j8
626 // z * (118.62522321428571429L - // j9
627 // z * (275.57319223985890653L - // j10
628 // z * (649.78717234347442681L - // j11
629 // z * (1551.1605194805194805L - // j12
630 // z * (3741.4497029592385495L - // j13
631 // z * (9104.5002411580189358L - // j14
632 // z * (22324.308512706601434L - // j15
633 // z * (55103.621972903835338L - // j16
634 // z * (136808.86090394293563L - // j17 z^17 last term used by Fukushima double.
635 // z * (341422.050665838363317L - // z^18
636 // z * (855992.9659966075514633L - // z^19
637 // z * (2.154990206091088289321e6L - // z^20
638 // z * 5.4455529223144624316423e6L // z^21
639 // ))))))))))))))))))));
640 //
641
642 T result =
643 z * (1.L - // z j1
644 z * (1.L - // z^2
645 z * (1.500000000000000000000000000000000L - // z^3
646 z * (2.666666666666666666666666666666666L - // z ^ 4
647 z * (5.208333333333333333333333333333333L - // z ^ 5
648 z * (10.80000000000000000000000000000000L - // z ^ 6
649 z * (23.34305555555555555555555555555555L - // z ^ 7
650 z * (52.01269841269841269841269841269841L - // z ^ 8
651 z * (118.6252232142857142857142857142857L - // z ^ 9
652 z * (275.5731922398589065255731922398589L - // z ^ 10
653 z * (649.7871723434744268077601410934744L - // z ^ 11
654 z * (1551.160519480519480519480519480519L - // z ^ 12
655 z * (3741.449702959238549516327294105071L - //z ^ 13
656 z * (9104.500241158018935796713574491352L - // z ^ 14
657 z * (22324.308512706601434280005708577137L - // z ^ 15
658 z * (55103.621972903835337697771560205422L - // z ^ 16
659 z * (136808.86090394293563342215789305736L - // z ^ 17
660 z * (341422.05066583836331735491399356945L - // z^18
661 z * (855992.9659966075514633630250633224L - // z^19
662 z * (2.154990206091088289321708745358647e6L // z^20 distance -5 without term 20
663 ))))))))))))))))))));
664
665 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
666 std::cout << "return w = " << result << std::endl;
667 std::cout.precision(precision); // Restore.
668 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
669 return result;
670 } // long double lambert_w0_small_z(const T z, std::integral_constant<int, 1> const&)
671
672 //! Specialization of 128-bit long double series expansion used for small z (abs(z) < 0.05).
673 // 34 Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction
674 // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],
675 // and are suffixed by L as they are assumed of type long double.
676 // (This is NOT used for 128-bit quad boost::multiprecision::float128 type which required a suffix Q
677 // nor multiprecision type cpp_bin_float_quad that can only be initialised at full precision of the type
678 // constructed with a decimal digit string like "2.6666666666666666666666666666666666666666666666667".)
679
680 template <typename T, typename Policy>
681 T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 3> const&)
682 {
683 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
684 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
685 std::cout << "\ntag_type 3 long double (128-bit) lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision, "
686 << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl;
687 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
688 T result =
689 z * (1.L - // j1
690 z * (1.L - // j2
691 z * (1.5L - // 3/2 // j3
692 z * (2.6666666666666666666666666666666666L - // 8/3 // j4
693 z * (5.2052083333333333333333333333333333L - // -125/24 // j5
694 z * (10.800000000000000000000000000000000L - // j6
695 z * (23.343055555555555555555555555555555L - // j7
696 z * (52.0126984126984126984126984126984126L - // j8
697 z * (118.625223214285714285714285714285714L - // j9
698 z * (275.57319223985890652557319223985890L - // * z ^ 10 - // j10
699 z * (649.78717234347442680776014109347442680776014109347L - // j11
700 z * (1551.1605194805194805194805194805194805194805194805L - // j12
701 z * (3741.4497029592385495163272941050718828496606274384L - // j13
702 z * (9104.5002411580189357967135744913522691300469078247L - // j14
703 z * (22324.308512706601434280005708577137148565719994291L - // j15
704 z * (55103.621972903835337697771560205422639285073147507L - // j16
705 z * 136808.86090394293563342215789305736395683485630576L // j17
706 ))))))))))))))));
707
708 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
709 std::cout << "return w = " << result << std::endl;
710 std::cout.precision(precision); // Restore.
711 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
712 return result;
713 } // T lambert_w0_small_z(const T z, std::integral_constant<int, 3> const&)
714
715 //! Specialization of 128-bit quad series expansion used for small z (abs(z) < 0.05).
716 // 34 Taylor series coefficients used were computed by Wolfram to 50 decimal digits using instruction
717 // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],
718 // and are suffixed by Q as they are assumed of type quad.
719 // This could be used for 128-bit quad (which requires a suffix Q for full precision).
720 // But experiments with GCC 7.2.0 show that while this gives full 128-bit precision
721 // when the -f-ext-numeric-literals option is in force and the libquadmath library available,
722 // over the range -0.049 to +0.049,
723 // it is slightly slower than getting a double approximation followed by a single Halley step.
724
725 #ifdef BOOST_HAS_FLOAT128
726 template <typename T, typename Policy>
727 T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 4> const&)
728 {
729 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
730 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
731 std::cout << "\ntag_type 4 128-bit quad float128 lambert_w0_small_z called with z = " << z << " using " << 34 << " terms of precision, "
732 << std::numeric_limits<float128>::max_digits10 << " max decimal digits." << std::endl;
733 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
734 T result =
735 z * (1.Q - // z j1
736 z * (1.Q - // z^2
737 z * (1.500000000000000000000000000000000Q - // z^3
738 z * (2.666666666666666666666666666666666Q - // z ^ 4
739 z * (5.208333333333333333333333333333333Q - // z ^ 5
740 z * (10.80000000000000000000000000000000Q - // z ^ 6
741 z * (23.34305555555555555555555555555555Q - // z ^ 7
742 z * (52.01269841269841269841269841269841Q - // z ^ 8
743 z * (118.6252232142857142857142857142857Q - // z ^ 9
744 z * (275.5731922398589065255731922398589Q - // z ^ 10
745 z * (649.7871723434744268077601410934744Q - // z ^ 11
746 z * (1551.160519480519480519480519480519Q - // z ^ 12
747 z * (3741.449702959238549516327294105071Q - //z ^ 13
748 z * (9104.500241158018935796713574491352Q - // z ^ 14
749 z * (22324.308512706601434280005708577137Q - // z ^ 15
750 z * (55103.621972903835337697771560205422Q - // z ^ 16
751 z * (136808.86090394293563342215789305736Q - // z ^ 17
752 z * (341422.05066583836331735491399356945Q - // z^18
753 z * (855992.9659966075514633630250633224Q - // z^19
754 z * (2.154990206091088289321708745358647e6Q - // 20
755 z * (5.445552922314462431642316420035073e6Q - // 21
756 z * (1.380733000216662949061923813184508e7Q - // 22
757 z * (3.511704498513923292853869855945334e7Q - // 23
758 z * (8.956800256102797693072819557780090e7Q - // 24
759 z * (2.290416846187949813964782641734774e8Q - // 25
760 z * (5.871035041171798492020292225245235e8Q - // 26
761 z * (1.508256053857792919641317138812957e9Q - // 27
762 z * (3.882630161293188940385873468413841e9Q - // 28
763 z * (1.001394313665482968013913601565723e10Q - // 29
764 z * (2.587356736265760638992878359024929e10Q - // 30
765 z * (6.696209709358073856946120522333454e10Q - // 31
766 z * (1.735711659599198077777078238043644e11Q - // 32
767 z * (4.505680465642353886756098108484670e11Q - // 33
768 z * (1.171223178256487391904047636564823e12Q //z^34
769 ))))))))))))))))))))))))))))))))));
770
771
772 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
773 std::cout << "return w = " << result << std::endl;
774 std::cout.precision(precision); // Restore.
775 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
776
777 return result;
778 } // T lambert_w0_small_z(const T z, std::integral_constant<int, 4> const&) float128
779
780 #else
781
782 template <typename T, typename Policy>
lambert_w0_small_z(const T z,const Policy & pol,std::integral_constant<int,4> const &)783 inline T lambert_w0_small_z(const T z, const Policy& pol, std::integral_constant<int, 4> const&)
784 {
785 return lambert_w0_small_z(z, pol, std::integral_constant<int, 5>());
786 }
787
788 #endif // BOOST_HAS_FLOAT128
789
790 //! Series functor to compute series term using pow and factorial.
791 //! \details Functor is called after evaluating polynomial with the coefficients as rationals below.
792 template <typename T>
793 struct lambert_w0_small_z_series_term
794 {
795 using result_type = T;
796 //! \param _z Lambert W argument z.
797 //! \param -term -pow<18>(z) / 6402373705728000uLL
798 //! \param _k number of terms == initially 18
799
800 // Note *after* evaluating N terms, its internal state has k = N and term = (-1)^N z^N.
801
lambert_w0_small_z_series_termboost::math::lambert_w_detail::lambert_w0_small_z_series_term802 lambert_w0_small_z_series_term(T _z, T _term, int _k)
803 : k(_k), z(_z), term(_term) { }
804
operator ()boost::math::lambert_w_detail::lambert_w0_small_z_series_term805 T operator()()
806 { // Called by sum_series until needs precision set by factor (policy::get_epsilon).
807 using std::pow;
808 ++k;
809 term *= -z / k;
810 //T t = pow(z, k) * pow(T(k), -1 + k) / factorial<T>(k); // (z^k * k(k-1)^k) / k!
811 T result = term * pow(T(k), -1 + k); // term * k^(k-1)
812 // std::cout << " k = " << k << ", term = " << term << ", result = " << result << std::endl;
813 return result; //
814 }
815 private:
816 int k;
817 T z;
818 T term;
819 }; // template <typename T> struct lambert_w0_small_z_series_term
820
821 //! Generic variant for T a User-defined types like Boost.Multiprecision.
822 template <typename T, typename Policy>
lambert_w0_small_z(T z,const Policy & pol,std::integral_constant<int,5> const &)823 inline T lambert_w0_small_z(T z, const Policy& pol, std::integral_constant<int, 5> const&)
824 {
825 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
826 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
827 std::cout << "Generic lambert_w0_small_z called with z = " << z << " using as many terms needed for precision." << std::endl;
828 std::cout << "Argument z is of type " << typeid(T).name() << std::endl;
829 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
830
831 // First several terms of the series are tabulated and evaluated as a polynomial:
832 // this will save us a bunch of expensive calls to pow.
833 // Then our series functor is initialized "as if" it had already reached term 18,
834 // enough evaluation of built-in 64-bit double and float (and 80-bit long double?) types.
835
836 // Coefficients should be stored such that the coefficients for the x^i terms are in poly[i].
837 static const T coeff[] =
838 {
839 0, // z^0 Care: zeroth term needed by tools::evaluate_polynomial, but not in the Wolfram equation, so indexes are one different!
840 1, // z^1 term.
841 -1, // z^2 term
842 static_cast<T>(3uLL) / 2uLL, // z^3 term.
843 -static_cast<T>(8uLL) / 3uLL, // z^4
844 static_cast<T>(125uLL) / 24uLL, // z^5
845 -static_cast<T>(54uLL) / 5uLL, // z^6
846 static_cast<T>(16807uLL) / 720uLL, // z^7
847 -static_cast<T>(16384uLL) / 315uLL, // z^8
848 static_cast<T>(531441uLL) / 4480uLL, // z^9
849 -static_cast<T>(156250uLL) / 567uLL, // z^10
850 static_cast<T>(2357947691uLL) / 3628800uLL, // z^11
851 -static_cast<T>(2985984uLL) / 1925uLL, // z^12
852 static_cast<T>(1792160394037uLL) / 479001600uLL, // z^13
853 -static_cast<T>(7909306972uLL) / 868725uLL, // z^14
854 static_cast<T>(320361328125uLL) / 14350336uLL, // z^15
855 -static_cast<T>(35184372088832uLL) / 638512875uLL, // z^16
856 static_cast<T>(2862423051509815793uLL) / 20922789888000uLL, // z^17 term
857 -static_cast<T>(5083731656658uLL) / 14889875uLL,
858 // z^18 term. = 136808.86090394293563342215789305735851647769682393
859
860 // z^18 is biggest that can be computed as rational using the largest possible uLL integers,
861 // so higher terms cannot be potentially compiler-computed as uLL rationals.
862 // Wolfram (5083731656658 z ^ 18) / 14889875 or
863 // -341422.05066583836331735491399356945575432970390954 z^18
864
865 // See note below calling the functor to compute another term,
866 // sufficient for 80-bit long double precision.
867 // Wolfram -341422.05066583836331735491399356945575432970390954 z^19 term.
868 // (5480386857784802185939 z^19)/6402373705728000
869 // But now this variant is not used to compute long double
870 // as specializations are provided above.
871 }; // static const T coeff[]
872
873 /*
874 Table of 19 computed coefficients:
875
876 #0 0
877 #1 1
878 #2 -1
879 #3 1.5
880 #4 -2.6666666666666666666666666666666665382713370408509
881 #5 5.2083333333333333333333333333333330765426740817019
882 #6 -10.800000000000000000000000000000000616297582203915
883 #7 23.343055555555555555555555555555555076212991619177
884 #8 -52.012698412698412698412698412698412659282693193402
885 #9 118.62522321428571428571428571428571146835390992496
886 #10 -275.57319223985890652557319223985891400375196748314
887 #11 649.7871723434744268077601410934743969785223845882
888 #12 -1551.1605194805194805194805194805194947599566007429
889 #13 3741.4497029592385495163272941050719510009019331763
890 #14 -9104.5002411580189357967135744913524243896052869184
891 #15 22324.308512706601434280005708577137322392070452582
892 #16 -55103.621972903835337697771560205423203318720697224
893 #17 136808.86090394293563342215789305735851647769682393
894 136808.86090394293563342215789305735851647769682393 == Exactly same as Wolfram computed value.
895 #18 -341422.05066583836331735491399356947486381600607416
896 341422.05066583836331735491399356945575432970390954 z^19 Wolfram value differs at 36 decimal digit, as expected.
897 */
898
899 using boost::math::policies::get_epsilon; // for type T.
900 using boost::math::tools::sum_series;
901 using boost::math::tools::evaluate_polynomial;
902 // http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/roots/rational.html
903
904 // std::streamsize prec = std::cout.precision(std::numeric_limits <T>::max_digits10);
905
906 T result = evaluate_polynomial(coeff, z);
907 // template <std::size_t N, typename T, typename V>
908 // V evaluate_polynomial(const T(&poly)[N], const V& val);
909 // Size of coeff found from N
910 //std::cout << "evaluate_polynomial(coeff, z); == " << result << std::endl;
911 //std::cout << "result = " << result << std::endl;
912 // It's an artefact of the way I wrote the functor: *after* evaluating N
913 // terms, its internal state has k = N and term = (-1)^N z^N. So after
914 // evaluating 18 terms, we initialize the functor to the term we've just
915 // evaluated, and then when it's called, it increments itself to the next term.
916 // So 18!is 6402373705728000, which is where that comes from.
917
918 // The 19th coefficient of the polynomial is actually, 19 ^ 18 / 19!=
919 // 104127350297911241532841 / 121645100408832000 which after removing GCDs
920 // reduces down to Wolfram rational 5480386857784802185939 / 6402373705728000.
921 // Wolfram z^19 term +(5480386857784802185939 z^19) /6402373705728000
922 // +855992.96599660755146336302506332246623424823099755 z^19
923
924 //! Evaluate Functor.
925 lambert_w0_small_z_series_term<T> s(z, -pow<18>(z) / 6402373705728000uLL, 18);
926
927 // Temporary to list the coefficients.
928 //std::cout << " Table of coefficients" << std::endl;
929 //std::streamsize saved_precision = std::cout.precision(50);
930 //for (size_t i = 0; i != 19; i++)
931 //{
932 // std::cout << "#" << i << " " << coeff[i] << std::endl;
933 //}
934 //std::cout.precision(saved_precision);
935
936 std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); // Max iterations from policy.
937 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
938 std::cout << "max iter from policy = " << max_iter << std::endl;
939 // // max iter from policy = 1000000 is default.
940 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
941
942 result = sum_series(s, get_epsilon<T, Policy>(), max_iter, result);
943 // result == evaluate_polynomial.
944 //sum_series(Functor& func, int bits, std::uintmax_t& max_terms, const U& init_value)
945 // std::cout << "sum_series(s, get_epsilon<T, Policy>(), max_iter, result); = " << result << std::endl;
946
947 //T epsilon = get_epsilon<T, Policy>();
948 //std::cout << "epsilon from policy = " << epsilon << std::endl;
949 // epsilon from policy = 1.93e-34 for T == quad
950 // 5.35e-51 for t = cpp_bin_float_50
951
952 // std::cout << " get eps = " << get_epsilon<T, Policy>() << std::endl; // quad eps = 1.93e-34, bin_float_50 eps = 5.35e-51
953 policies::check_series_iterations<T>("boost::math::lambert_w0_small_z<%1%>(%1%)", max_iter, pol);
954 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS
955 std::cout << "z = " << z << " needed " << max_iter << " iterations." << std::endl;
956 std::cout.precision(prec); // Restore.
957 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS
958 return result;
959 } // template <typename T, typename Policy> inline T lambert_w0_small_z_series(T z, const Policy& pol)
960
961 // Approximate lambert_w0 (used for z values that are outside range of lookup table or rational functions)
962 // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.
963 template <typename T>
lambert_w0_approx(T z)964 inline T lambert_w0_approx(T z)
965 {
966 BOOST_MATH_STD_USING
967 T lz = log(z);
968 T llz = log(lz);
969 T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.
970 return w;
971 // std::cout << "w max " << max_w << std::endl; // double 703.227
972 }
973
974 //////////////////////////////////////////////////////////////////////////////////////////
975
976 //! \brief Lambert_w0 implementations for float, double and higher precisions.
977 //! 3rd parameter used to select which version is used.
978
979 //! /details Rational polynomials are provided for several range of argument z.
980 //! For very small values of z, and for z very near the branch singularity at -e^-1 (~= -0.367879),
981 //! two other series functions are used.
982
983 //! float precision polynomials are used for 32-bit (usually float) precision (for speed)
984 //! double precision polynomials are used for 64-bit (usually double) precision.
985 //! For higher precisions, a 64-bit double approximation is computed first,
986 //! and then refined using Halley iterations.
987
988 template <typename T>
do_get_near_singularity_param(T z)989 inline T do_get_near_singularity_param(T z)
990 {
991 BOOST_MATH_STD_USING
992 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
993 const T p = sqrt(p2);
994 return p;
995 }
996 template <typename T, typename Policy>
get_near_singularity_param(T z,const Policy)997 inline T get_near_singularity_param(T z, const Policy)
998 {
999 using value_type = typename policies::evaluation<T, Policy>::type;
1000 return static_cast<T>(do_get_near_singularity_param(static_cast<value_type>(z)));
1001 }
1002
1003 // Forward declarations:
1004
1005 //template <typename T, typename Policy> T lambert_w0_small_z(T z, const Policy& pol);
1006 //template <typename T, typename Policy>
1007 //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 0>&); // 32 bit usually float.
1008 //template <typename T, typename Policy>
1009 //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 1>&); // 64 bit usually double.
1010 //template <typename T, typename Policy>
1011 //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 2>&); // 80-bit long double.
1012
1013 template <typename T>
lambert_w_positive_rational_float(T z)1014 T lambert_w_positive_rational_float(T z)
1015 {
1016 BOOST_MATH_STD_USING
1017 if (z < 2)
1018 {
1019 if (z < 0.5)
1020 { // 0.05 < z < 0.5
1021 // Maximum Deviation Found: 2.993e-08
1022 // Expected Error Term : 2.993e-08
1023 // Maximum Relative Change in Control Points : 7.555e-04 Y offset : -8.196592331e-01
1024 static const T Y = 8.196592331e-01f;
1025 static const T P[] = {
1026 1.803388345e-01f,
1027 -4.820256838e-01f,
1028 -1.068349741e+00f,
1029 -3.506624319e-02f,
1030 };
1031 static const T Q[] = {
1032 1.000000000e+00f,
1033 2.871703469e+00f,
1034 1.690949264e+00f,
1035 };
1036 return z * (Y + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z));
1037 }
1038 else
1039 { // 0.5 < z < 2
1040 // Max error in interpolated form: 1.018e-08
1041 static const T Y = 5.503368378e-01f;
1042 static const T P[] = {
1043 4.493332766e-01f,
1044 2.543432707e-01f,
1045 -4.808788799e-01f,
1046 -1.244425316e-01f,
1047 };
1048 static const T Q[] = {
1049 1.000000000e+00f,
1050 2.780661241e+00f,
1051 1.830840318e+00f,
1052 2.407221031e-01f,
1053 };
1054 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
1055 }
1056 }
1057 else if (z < 6)
1058 {
1059 // 2 < z < 6
1060 // Max error in interpolated form: 2.944e-08
1061 static const T Y = 1.162393570e+00f;
1062 static const T P[] = {
1063 -1.144183394e+00f,
1064 -4.712732855e-01f,
1065 1.563162512e-01f,
1066 1.434010911e-02f,
1067 };
1068 static const T Q[] = {
1069 1.000000000e+00f,
1070 1.192626340e+00f,
1071 2.295580708e-01f,
1072 5.477869455e-03f,
1073 };
1074 return Y + boost::math::tools::evaluate_rational(P, Q, z);
1075 }
1076 else if (z < 18)
1077 {
1078 // 6 < z < 18
1079 // Max error in interpolated form: 5.893e-08
1080 static const T Y = 1.809371948e+00f;
1081 static const T P[] = {
1082 -1.689291769e+00f,
1083 -3.337812742e-01f,
1084 3.151434873e-02f,
1085 1.134178734e-03f,
1086 };
1087 static const T Q[] = {
1088 1.000000000e+00f,
1089 5.716915685e-01f,
1090 4.489521292e-02f,
1091 4.076716763e-04f,
1092 };
1093 return Y + boost::math::tools::evaluate_rational(P, Q, z);
1094 }
1095 else if (z < 9897.12905874) // 2.8 < log(z) < 9.2
1096 {
1097 // Max error in interpolated form: 1.771e-08
1098 static const T Y = -1.402973175e+00f;
1099 static const T P[] = {
1100 1.966174312e+00f,
1101 2.350864728e-01f,
1102 -5.098074353e-02f,
1103 -1.054818339e-02f,
1104 };
1105 static const T Q[] = {
1106 1.000000000e+00f,
1107 4.388208264e-01f,
1108 8.316639634e-02f,
1109 3.397187918e-03f,
1110 -1.321489743e-05f,
1111 };
1112 T log_w = log(z);
1113 return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);
1114 }
1115 else if (z < 7.896296e+13) // 9.2 < log(z) <= 32
1116 {
1117 // Max error in interpolated form: 5.821e-08
1118 static const T Y = -2.735729218e+00f;
1119 static const T P[] = {
1120 3.424903470e+00f,
1121 7.525631787e-02f,
1122 -1.427309584e-02f,
1123 -1.435974178e-05f,
1124 };
1125 static const T Q[] = {
1126 1.000000000e+00f,
1127 2.514005579e-01f,
1128 6.118994652e-03f,
1129 -1.357889535e-05f,
1130 7.312865624e-08f,
1131 };
1132 T log_w = log(z);
1133 return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);
1134 }
1135 else // 32 < log(z) < 100
1136 {
1137 // Max error in interpolated form: 1.491e-08
1138 static const T Y = -4.012863159e+00f;
1139 static const T P[] = {
1140 4.431629226e+00f,
1141 2.756690487e-01f,
1142 -2.992956930e-03f,
1143 -4.912259384e-05f,
1144 };
1145 static const T Q[] = {
1146 1.000000000e+00f,
1147 2.015434591e-01f,
1148 4.949426142e-03f,
1149 1.609659944e-05f,
1150 -5.111523436e-09f,
1151 };
1152 T log_w = log(z);
1153 return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);
1154 }
1155 }
1156
1157 template <typename T, typename Policy>
1158 T lambert_w_negative_rational_float(T z, const Policy& pol)
1159 {
1160 BOOST_MATH_STD_USING
1161 if (z > -0.27)
1162 {
1163 if (z < -0.051)
1164 {
1165 // -0.27 < z < -0.051
1166 // Max error in interpolated form: 5.080e-08
1167 static const T Y = 1.255809784e+00f;
1168 static const T P[] = {
1169 -2.558083412e-01f,
1170 -2.306524098e+00f,
1171 -5.630887033e+00f,
1172 -3.803974556e+00f,
1173 };
1174 static const T Q[] = {
1175 1.000000000e+00f,
1176 5.107680783e+00f,
1177 7.914062868e+00f,
1178 3.501498501e+00f,
1179 };
1180 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
1181 }
1182 else
1183 {
1184 // Very small z so use a series function.
1185 return lambert_w0_small_z(z, pol);
1186 }
1187 }
1188 else if (z > -0.3578794411714423215955237701)
1189 { // Very close to branch singularity.
1190 // Max error in interpolated form: 5.269e-08
1191 static const T Y = 1.220928431e-01f;
1192 static const T P[] = {
1193 -1.221787446e-01f,
1194 -6.816155875e+00f,
1195 7.144582035e+01f,
1196 1.128444390e+03f,
1197 };
1198 static const T Q[] = {
1199 1.000000000e+00f,
1200 6.480326790e+01f,
1201 1.869145243e+02f,
1202 -1.361804274e+03f,
1203 1.117826726e+03f,
1204 };
1205 T d = z + 0.367879441171442321595523770161460867445811f;
1206 return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));
1207 }
1208 else
1209 {
1210 // z is very close (within 0.01) of the singularity at e^-1.
1211 return lambert_w_singularity_series(get_near_singularity_param(z, pol));
1212 }
1213 }
1214
1215 //! Lambert_w0 @b 'float' implementation, selected when T is 32-bit precision.
1216 template <typename T, typename Policy>
lambert_w0_imp(T z,const Policy & pol,const std::integral_constant<int,1> &)1217 inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 1>&)
1218 {
1219 static const char* function = "boost::math::lambert_w0<%1%>"; // For error messages.
1220 BOOST_MATH_STD_USING // Aid ADL of std functions.
1221
1222 if ((boost::math::isnan)(z))
1223 {
1224 return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol);
1225 }
1226 if ((boost::math::isinf)(z))
1227 {
1228 return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol);
1229 }
1230
1231 if (z >= 0.05) // Fukushima switch point.
1232 // if (z >= 0.045) // 34 terms makes 128-bit 'exact' below 0.045.
1233 { // Normal ranges using several rational polynomials.
1234 return lambert_w_positive_rational_float(z);
1235 }
1236 else if (z <= -0.3678794411714423215955237701614608674458111310f)
1237 {
1238 if (z < -0.3678794411714423215955237701614608674458111310f)
1239 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
1240 return -1;
1241 }
1242 else // z < 0.05
1243 {
1244 return lambert_w_negative_rational_float(z, pol);
1245 }
1246 } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 1>&) for 32-bit usually float.
1247
1248 template <typename T>
lambert_w_positive_rational_double(T z)1249 T lambert_w_positive_rational_double(T z)
1250 {
1251 BOOST_MATH_STD_USING
1252 if (z < 2)
1253 {
1254 if (z < 0.5)
1255 {
1256 // Max error in interpolated form: 2.255e-17
1257 static const T offset = 8.19659233093261719e-01;
1258 static const T P[] = {
1259 1.80340766906685177e-01,
1260 3.28178241493119307e-01,
1261 -2.19153620687139706e+00,
1262 -7.24750929074563990e+00,
1263 -7.28395876262524204e+00,
1264 -2.57417169492512916e+00,
1265 -2.31606948888704503e-01
1266 };
1267 static const T Q[] = {
1268 1.00000000000000000e+00,
1269 7.36482529307436604e+00,
1270 2.03686007856430677e+01,
1271 2.62864592096657307e+01,
1272 1.59742041380858333e+01,
1273 4.03760534788374589e+00,
1274 2.91327346750475362e-01
1275 };
1276 return z * (offset + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z));
1277 }
1278 else
1279 {
1280 // Max error in interpolated form: 3.806e-18
1281 static const T offset = 5.50335884094238281e-01;
1282 static const T P[] = {
1283 4.49664083944098322e-01,
1284 1.90417666196776909e+00,
1285 1.99951368798255994e+00,
1286 -6.91217310299270265e-01,
1287 -1.88533935998617058e+00,
1288 -7.96743968047750836e-01,
1289 -1.02891726031055254e-01,
1290 -3.09156013592636568e-03
1291 };
1292 static const T Q[] = {
1293 1.00000000000000000e+00,
1294 6.45854489419584014e+00,
1295 1.54739232422116048e+01,
1296 1.72606164253337843e+01,
1297 9.29427055609544096e+00,
1298 2.29040824649748117e+00,
1299 2.21610620995418981e-01,
1300 5.70597669908194213e-03
1301 };
1302 return z * (offset + boost::math::tools::evaluate_rational(P, Q, z));
1303 }
1304 }
1305 else if (z < 6)
1306 {
1307 // 2 < z < 6
1308 // Max error in interpolated form: 1.216e-17
1309 static const T Y = 1.16239356994628906e+00;
1310 static const T P[] = {
1311 -1.16230494982099475e+00,
1312 -3.38528144432561136e+00,
1313 -2.55653717293161565e+00,
1314 -3.06755172989214189e-01,
1315 1.73149743765268289e-01,
1316 3.76906042860014206e-02,
1317 1.84552217624706666e-03,
1318 1.69434126904822116e-05,
1319 };
1320 static const T Q[] = {
1321 1.00000000000000000e+00,
1322 3.77187616711220819e+00,
1323 4.58799960260143701e+00,
1324 2.24101228462292447e+00,
1325 4.54794195426212385e-01,
1326 3.60761772095963982e-02,
1327 9.25176499518388571e-04,
1328 4.43611344705509378e-06,
1329 };
1330 return Y + boost::math::tools::evaluate_rational(P, Q, z);
1331 }
1332 else if (z < 18)
1333 {
1334 // 6 < z < 18
1335 // Max error in interpolated form: 1.985e-19
1336 static const T offset = 1.80937194824218750e+00;
1337 static const T P[] =
1338 {
1339 -1.80690935424793635e+00,
1340 -3.66995929380314602e+00,
1341 -1.93842957940149781e+00,
1342 -2.94269984375794040e-01,
1343 1.81224710627677778e-03,
1344 2.48166798603547447e-03,
1345 1.15806592415397245e-04,
1346 1.43105573216815533e-06,
1347 3.47281483428369604e-09
1348 };
1349 static const T Q[] = {
1350 1.00000000000000000e+00,
1351 2.57319080723908597e+00,
1352 1.96724528442680658e+00,
1353 5.84501352882650722e-01,
1354 7.37152837939206240e-02,
1355 3.97368430940416778e-03,
1356 8.54941838187085088e-05,
1357 6.05713225608426678e-07,
1358 8.17517283816615732e-10
1359 };
1360 return offset + boost::math::tools::evaluate_rational(P, Q, z);
1361 }
1362 else if (z < 9897.12905874) // 2.8 < log(z) < 9.2
1363 {
1364 // Max error in interpolated form: 1.195e-18
1365 static const T Y = -1.40297317504882812e+00;
1366 static const T P[] = {
1367 1.97011826279311924e+00,
1368 1.05639945701546704e+00,
1369 3.33434529073196304e-01,
1370 3.34619153200386816e-02,
1371 -5.36238353781326675e-03,
1372 -2.43901294871308604e-03,
1373 -2.13762095619085404e-04,
1374 -4.85531936495542274e-06,
1375 -2.02473518491905386e-08,
1376 };
1377 static const T Q[] = {
1378 1.00000000000000000e+00,
1379 8.60107275833921618e-01,
1380 4.10420467985504373e-01,
1381 1.18444884081994841e-01,
1382 2.16966505556021046e-02,
1383 2.24529766630769097e-03,
1384 9.82045090226437614e-05,
1385 1.36363515125489502e-06,
1386 3.44200749053237945e-09,
1387 };
1388 T log_w = log(z);
1389 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
1390 }
1391 else if (z < 7.896296e+13) // 9.2 < log(z) <= 32
1392 {
1393 // Max error in interpolated form: 6.529e-18
1394 static const T Y = -2.73572921752929688e+00;
1395 static const T P[] = {
1396 3.30547638424076217e+00,
1397 1.64050071277550167e+00,
1398 4.57149576470736039e-01,
1399 4.03821227745424840e-02,
1400 -4.99664976882514362e-04,
1401 -1.28527893803052956e-04,
1402 -2.95470325373338738e-06,
1403 -1.76662025550202762e-08,
1404 -1.98721972463709290e-11,
1405 };
1406 static const T Q[] = {
1407 1.00000000000000000e+00,
1408 6.91472559412458759e-01,
1409 2.48154578891676774e-01,
1410 4.60893578284335263e-02,
1411 3.60207838982301946e-03,
1412 1.13001153242430471e-04,
1413 1.33690948263488455e-06,
1414 4.97253225968548872e-09,
1415 3.39460723731970550e-12,
1416 };
1417 T log_w = log(z);
1418 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
1419 }
1420 else if (z < 2.6881171e+43) // 32 < log(z) < 100
1421 {
1422 // Max error in interpolated form: 2.015e-18
1423 static const T Y = -4.01286315917968750e+00;
1424 static const T P[] = {
1425 5.07714858354309672e+00,
1426 -3.32994414518701458e+00,
1427 -8.61170416909864451e-01,
1428 -4.01139705309486142e-02,
1429 -1.85374201771834585e-04,
1430 1.08824145844270666e-05,
1431 1.17216905810452396e-07,
1432 2.97998248101385990e-10,
1433 1.42294856434176682e-13,
1434 };
1435 static const T Q[] = {
1436 1.00000000000000000e+00,
1437 -4.85840770639861485e-01,
1438 -3.18714850604827580e-01,
1439 -3.20966129264610534e-02,
1440 -1.06276178044267895e-03,
1441 -1.33597828642644955e-05,
1442 -6.27900905346219472e-08,
1443 -9.35271498075378319e-11,
1444 -2.60648331090076845e-14,
1445 };
1446 T log_w = log(z);
1447 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
1448 }
1449 else // 100 < log(z) < 710
1450 {
1451 // Max error in interpolated form: 5.277e-18
1452 static const T Y = -5.70115661621093750e+00;
1453 static const T P[] = {
1454 6.42275660145116698e+00,
1455 1.33047964073367945e+00,
1456 6.72008923401652816e-02,
1457 1.16444069958125895e-03,
1458 7.06966760237470501e-06,
1459 5.48974896149039165e-09,
1460 -7.00379652018853621e-11,
1461 -1.89247635913659556e-13,
1462 -1.55898770790170598e-16,
1463 -4.06109208815303157e-20,
1464 -2.21552699006496737e-24,
1465 };
1466 static const T Q[] = {
1467 1.00000000000000000e+00,
1468 3.34498588416632854e-01,
1469 2.51519862456384983e-02,
1470 6.81223810622416254e-04,
1471 7.94450897106903537e-06,
1472 4.30675039872881342e-08,
1473 1.10667669458467617e-10,
1474 1.31012240694192289e-13,
1475 6.53282047177727125e-17,
1476 1.11775518708172009e-20,
1477 3.78250395617836059e-25,
1478 };
1479 T log_w = log(z);
1480 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
1481 }
1482 }
1483
1484 template <typename T, typename Policy>
1485 T lambert_w_negative_rational_double(T z, const Policy& pol)
1486 {
1487 BOOST_MATH_STD_USING
1488 if (z > -0.1)
1489 {
1490 if (z < -0.051)
1491 {
1492 // -0.1 < z < -0.051
1493 // Maximum Deviation Found: 4.402e-22
1494 // Expected Error Term : 4.240e-22
1495 // Maximum Relative Change in Control Points : 4.115e-03
1496 static const T Y = 1.08633995056152344e+00;
1497 static const T P[] = {
1498 -8.63399505615014331e-02,
1499 -1.64303871814816464e+00,
1500 -7.71247913918273738e+00,
1501 -1.41014495545382454e+01,
1502 -1.02269079949257616e+01,
1503 -2.17236002836306691e+00,
1504 };
1505 static const T Q[] = {
1506 1.00000000000000000e+00,
1507 7.44775406945739243e+00,
1508 2.04392643087266541e+01,
1509 2.51001961077774193e+01,
1510 1.31256080849023319e+01,
1511 2.11640324843601588e+00,
1512 };
1513 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
1514 }
1515 else
1516 {
1517 // Very small z > 0.051:
1518 return lambert_w0_small_z(z, pol);
1519 }
1520 }
1521 else if (z > -0.2)
1522 {
1523 // -0.2 < z < -0.1
1524 // Maximum Deviation Found: 2.898e-20
1525 // Expected Error Term : 2.873e-20
1526 // Maximum Relative Change in Control Points : 3.779e-04
1527 static const T Y = 1.20359611511230469e+00;
1528 static const T P[] = {
1529 -2.03596115108465635e-01,
1530 -2.95029082937201859e+00,
1531 -1.54287922188671648e+01,
1532 -3.81185809571116965e+01,
1533 -4.66384358235575985e+01,
1534 -2.59282069989642468e+01,
1535 -4.70140451266553279e+00,
1536 };
1537 static const T Q[] = {
1538 1.00000000000000000e+00,
1539 9.57921436074599929e+00,
1540 3.60988119290234377e+01,
1541 6.73977699505546007e+01,
1542 6.41104992068148823e+01,
1543 2.82060127225153607e+01,
1544 4.10677610657724330e+00,
1545 };
1546 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
1547 }
1548 else if (z > -0.3178794411714423215955237)
1549 {
1550 // Max error in interpolated form: 6.996e-18
1551 static const T Y = 3.49680423736572266e-01;
1552 static const T P[] = {
1553 -3.49729841718749014e-01,
1554 -6.28207407760709028e+01,
1555 -2.57226178029669171e+03,
1556 -2.50271008623093747e+04,
1557 1.11949239154711388e+05,
1558 1.85684566607844318e+06,
1559 4.80802490427638643e+06,
1560 2.76624752134636406e+06,
1561 };
1562 static const T Q[] = {
1563 1.00000000000000000e+00,
1564 1.82717661215113000e+02,
1565 8.00121119810280100e+03,
1566 1.06073266717010129e+05,
1567 3.22848993926057721e+05,
1568 -8.05684814514171256e+05,
1569 -2.59223192927265737e+06,
1570 -5.61719645211570871e+05,
1571 6.27765369292636844e+04,
1572 };
1573 T d = z + 0.367879441171442321595523770161460867445811;
1574 return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));
1575 }
1576 else if (z > -0.3578794411714423215955237701)
1577 {
1578 // Max error in interpolated form: 1.404e-17
1579 static const T Y = 5.00126481056213379e-02;
1580 static const T P[] = {
1581 -5.00173570682372162e-02,
1582 -4.44242461870072044e+01,
1583 -9.51185533619946042e+03,
1584 -5.88605699015429386e+05,
1585 -1.90760843597427751e+06,
1586 5.79797663818311404e+08,
1587 1.11383352508459134e+10,
1588 5.67791253678716467e+10,
1589 6.32694500716584572e+10,
1590 };
1591 static const T Q[] = {
1592 1.00000000000000000e+00,
1593 9.08910517489981551e+02,
1594 2.10170163753340133e+05,
1595 1.67858612416470327e+07,
1596 4.90435561733227953e+08,
1597 4.54978142622939917e+09,
1598 2.87716585708739168e+09,
1599 -4.59414247951143131e+10,
1600 -1.72845216404874299e+10,
1601 };
1602 T d = z + 0.36787944117144232159552377016146086744581113103176804;
1603 return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));
1604 }
1605 else
1606 { // z is very close (within 0.01) of the singularity at -e^-1,
1607 // so use a series expansion from R. M. Corless et al.
1608 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
1609 const T p = sqrt(p2);
1610 return lambert_w_detail::lambert_w_singularity_series(p);
1611 }
1612 }
1613
1614 //! Lambert_w0 @b 'double' implementation, selected when T is 64-bit precision.
1615 template <typename T, typename Policy>
lambert_w0_imp(T z,const Policy & pol,const std::integral_constant<int,2> &)1616 inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 2>&)
1617 {
1618 static const char* function = "boost::math::lambert_w0<%1%>";
1619 BOOST_MATH_STD_USING // Aid ADL of std functions.
1620
1621 // Detect unusual case of 32-bit double with a wider/64-bit long double
1622 static_assert(std::numeric_limits<double>::digits >= 53,
1623 "Our double precision coefficients will be truncated, "
1624 "please file a bug report with details of your platform's floating point types "
1625 "- or possibly edit the coefficients to have "
1626 "an appropriate size-suffix for 64-bit floats on your platform - L?");
1627
1628 if ((boost::math::isnan)(z))
1629 {
1630 return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol);
1631 }
1632 if ((boost::math::isinf)(z))
1633 {
1634 return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol);
1635 }
1636
1637 if (z >= 0.05)
1638 {
1639 return lambert_w_positive_rational_double(z);
1640 }
1641 else if (z <= -0.36787944117144232159552377016146086744581113103176804) // Precision is max_digits10(cpp_bin_float_50).
1642 {
1643 if (z < -0.36787944117144232159552377016146086744581113103176804)
1644 {
1645 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
1646 }
1647 return -1;
1648 }
1649 else
1650 {
1651 return lambert_w_negative_rational_double(z, pol);
1652 }
1653 } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 2>&) 64-bit precision, usually double.
1654
1655 //! lambert_W0 implementation for extended precision types including
1656 //! long double (80-bit and 128-bit), ???
1657 //! quad float128, Boost.Multiprecision types like cpp_bin_float_quad, cpp_bin_float_50...
1658
1659 template <typename T, typename Policy>
lambert_w0_imp(T z,const Policy & pol,const std::integral_constant<int,0> &)1660 inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 0>&)
1661 {
1662 static const char* function = "boost::math::lambert_w0<%1%>";
1663 BOOST_MATH_STD_USING // Aid ADL of std functions.
1664
1665 // Filter out special cases first:
1666 if ((boost::math::isnan)(z))
1667 {
1668 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
1669 }
1670 if (fabs(z) <= 0.05f)
1671 {
1672 // Very small z:
1673 return lambert_w0_small_z(z, pol);
1674 }
1675 if (z > (std::numeric_limits<double>::max)())
1676 {
1677 if ((boost::math::isinf)(z))
1678 {
1679 return policies::raise_overflow_error<T>(function, 0, pol);
1680 // Or might return infinity if available else max_value,
1681 // but other Boost.Math special functions raise overflow.
1682 }
1683 // z is larger than the largest double, so cannot use the polynomial to get an approximation,
1684 // so use the asymptotic approximation and Halley iterate:
1685
1686 T w = lambert_w0_approx(z); // Make an inline function as also used elsewhere.
1687 //T lz = log(z);
1688 //T llz = log(lz);
1689 //T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.
1690 return lambert_w_halley_iterate(w, z);
1691 }
1692 if (z < -0.3578794411714423215955237701)
1693 { // Very close to branch point so rational polynomials are not usable.
1694 if (z <= -boost::math::constants::exp_minus_one<T>())
1695 {
1696 if (z == -boost::math::constants::exp_minus_one<T>())
1697 { // Exactly at the branch point singularity.
1698 return -1;
1699 }
1700 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
1701 }
1702 // z is very close (within 0.01) of the branch singularity at -e^-1
1703 // so use a series approximation proposed by Corless et al.
1704 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
1705 const T p = sqrt(p2);
1706 T w = lambert_w_detail::lambert_w_singularity_series(p);
1707 return lambert_w_halley_iterate(w, z);
1708 }
1709
1710 // Phew! If we get here we are in the normal range of the function,
1711 // so get a double precision approximation first, then iterate to full precision of T.
1712 // We define a tag_type that is:
1713 // true_type if there are so many digits precision wanted that iteration is necessary.
1714 // false_type if a single Halley step is sufficient.
1715
1716 using precision_type = typename policies::precision<T, Policy>::type;
1717 using tag_type = std::integral_constant<bool,
1718 (precision_type::value == 0) || (precision_type::value > 113) ?
1719 true // Unknown at compile-time, variable/arbitrary, or more than float128 or cpp_bin_quad 128-bit precision.
1720 : false // float, double, float128, cpp_bin_quad 128-bit, so single Halley step.
1721 >;
1722
1723 // For speed, we also cast z to type double when that is possible
1724 // if (std::is_constructible<double, T>() == true).
1725 T w = lambert_w0_imp(maybe_reduce_to_double(z, std::is_constructible<double, T>()), pol, std::integral_constant<int, 2>());
1726
1727 return lambert_w_maybe_halley_iterate(w, z, tag_type());
1728
1729 } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 0>&) all extended precision types.
1730
1731 // Lambert w-1 implementation
1732 // ==============================================================================================
1733
1734 //! Lambert W for W-1 branch, -max(z) < z <= -1/e.
1735 // TODO is -max(z) allowed?
1736 template<typename T, typename Policy>
1737 T lambert_wm1_imp(const T z, const Policy& pol)
1738 {
1739 // Catch providing an integer value as parameter x to lambert_w, for example, lambert_w(1).
1740 // Need to ensure it is a floating-point type (of the desired type, float 1.F, double 1., or long double 1.L),
1741 // or static_casted integer, for example: static_cast<float>(1) or static_cast<cpp_dec_float_50>(1).
1742 // Want to allow fixed_point types too, so do not just test for floating-point.
1743 // Integral types should be promoted to double by user Lambert w functions.
1744 // If integral type provided to user function lambert_w0 or lambert_wm1,
1745 // then should already have been promoted to double.
1746 static_assert(!std::is_integral<T>::value,
1747 "Must be floating-point or fixed type (not integer type), for example: lambert_wm1(1.), not lambert_wm1(1)!");
1748
1749 BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs.
1750
1751 const char* function = "boost::math::lambert_wm1<RealType>(<RealType>)"; // Used for error messages.
1752
1753 // Check for edge and corner cases first:
1754 if ((boost::math::isnan)(z))
1755 {
1756 return policies::raise_domain_error(function,
1757 "Argument z is NaN!",
1758 z, pol);
1759 } // isnan
1760
1761 if ((boost::math::isinf)(z))
1762 {
1763 return policies::raise_domain_error(function,
1764 "Argument z is infinite!",
1765 z, pol);
1766 } // isinf
1767
1768 if (z == static_cast<T>(0))
1769 { // z is exactly zero so return -std::numeric_limits<T>::infinity();
1770 if (std::numeric_limits<T>::has_infinity)
1771 {
1772 return -std::numeric_limits<T>::infinity();
1773 }
1774 else
1775 {
1776 return -tools::max_value<T>();
1777 }
1778 }
1779 if (std::numeric_limits<T>::has_denorm)
1780 { // All real types except arbitrary precision.
1781 if (!(boost::math::isnormal)(z))
1782 { // Almost zero - might also just return infinity like z == 0 or max_value?
1783 return policies::raise_overflow_error(function,
1784 "Argument z = %1% is denormalized! (must be z > (std::numeric_limits<RealType>::min)() or z == 0)",
1785 z, pol);
1786 }
1787 }
1788
1789 if (z > static_cast<T>(0))
1790 { //
1791 return policies::raise_domain_error(function,
1792 "Argument z = %1% is out of range (z <= 0) for Lambert W-1 branch! (Try Lambert W0 branch?)",
1793 z, pol);
1794 }
1795 if (z > -boost::math::tools::min_value<T>())
1796 { // z is denormalized, so cannot be computed.
1797 // -std::numeric_limits<T>::min() is smallest for type T,
1798 // for example, for double: lambert_wm1(-2.2250738585072014e-308) = -714.96865723796634
1799 return policies::raise_overflow_error(function,
1800 "Argument z = %1% is too small (z < -std::numeric_limits<T>::min so denormalized) for Lambert W-1 branch!",
1801 z, pol);
1802 }
1803 if (z == -boost::math::constants::exp_minus_one<T>()) // == singularity/branch point z = -exp(-1) = -3.6787944.
1804 { // At singularity, so return exactly -1.
1805 return -static_cast<T>(1);
1806 }
1807 // z is too negative for the W-1 (or W0) branch.
1808 if (z < -boost::math::constants::exp_minus_one<T>()) // > singularity/branch point z = -exp(-1) = -3.6787944.
1809 {
1810 return policies::raise_domain_error(function,
1811 "Argument z = %1% is out of range (z < -exp(-1) = -3.6787944... <= 0) for Lambert W-1 (or W0) branch!",
1812 z, pol);
1813 }
1814 if (z < static_cast<T>(-0.35))
1815 { // Close to singularity/branch point z = -0.3678794411714423215955237701614608727 but on W-1 branch.
1816 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
1817 if (p2 == 0)
1818 { // At the singularity at branch point.
1819 return -1;
1820 }
1821 if (p2 > 0)
1822 {
1823 T w_series = lambert_w_singularity_series(T(-sqrt(p2)));
1824 if (boost::math::tools::digits<T>() > 53)
1825 { // Multiprecision, so try a Halley refinement.
1826 w_series = lambert_w_detail::lambert_w_halley_iterate(w_series, z);
1827 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN
1828 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1829 std::cout << "Lambert W-1 Halley updated to " << w_series << std::endl;
1830 std::cout.precision(saved_precision);
1831 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN
1832 }
1833 return w_series;
1834 }
1835 // Should not get here.
1836 return policies::raise_domain_error(function,
1837 "Argument z = %1% is out of range for Lambert W-1 branch. (Should not get here - please report!)",
1838 z, pol);
1839 } // if (z < -0.35)
1840
1841 using lambert_w_lookup::wm1es;
1842 using lambert_w_lookup::wm1zs;
1843 using lambert_w_lookup::noof_wm1zs; // size == 64
1844
1845 // std::cout <<" Wm1zs[63] (== G[64]) = " << " " << wm1zs[63] << std::endl; // Wm1zs[63] (== G[64]) = -1.0264389699511283e-26
1846 // Check that z argument value is not smaller than lookup_table G[64]
1847 // std::cout << "(z > wm1zs[63]) = " << std::boolalpha << (z > wm1zs[63]) << std::endl;
1848
1849 if (z >= wm1zs[63]) // wm1zs[63] = -1.0264389699511282259046957018510946438e-26L W = 64.00000000000000000
1850 { // z >= -1.0264389699511303e-26 (but z != 0 and z >= std::numeric_limits<T>::min() and so NOT denormalized).
1851
1852 // Some info on Lambert W-1 values for extreme values of z.
1853 // std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1854 // std::cout << "-std::numeric_limits<float>::min() = " << -(std::numeric_limits<float>::min)() << std::endl;
1855 // std::cout << "-std::numeric_limits<double>::min() = " << -(std::numeric_limits<double>::min)() << std::endl;
1856 // -std::numeric_limits<float>::min() = -1.1754943508222875e-38
1857 // -std::numeric_limits<double>::min() = -2.2250738585072014e-308
1858 // N[productlog(-1, -1.1754943508222875 * 10^-38 ), 50] = -91.856775324595479509567756730093823993834155027858
1859 // N[productlog(-1, -2.2250738585072014e-308 * 10^-308 ), 50] = -1424.8544521230553853558132180518404363617968042942
1860 // N[productlog(-1, -1.4325445274604020119111357113179868158* 10^-27), 37] = -65.99999999999999999999999999999999955
1861
1862 // R.M.Corless, G.H.Gonnet, D.E.G.Hare, D.J.Jeffrey, and D.E.Knuth,
1863 // On the Lambert W function, Adv.Comput.Math., vol. 5, pp. 329, 1996.
1864 // Francois Chapeau-Blondeau and Abdelilah Monir
1865 // Numerical Evaluation of the Lambert W Function
1866 // IEEE Transactions On Signal Processing, VOL. 50, NO. 9, Sep 2002
1867 // https://pdfs.semanticscholar.org/7a5a/76a9369586dd0dd34dda156d8f2779d1fd59.pdf
1868 // Estimate Lambert W using ln(-z) ...
1869 // This is roughly the power of ten * ln(10) ~= 2.3. n ~= 10^n
1870 // and improve by adding a second term -ln(ln(-z))
1871 T guess; // bisect lowest possible Gk[=64] (for lookup_t type)
1872 T lz = log(-z);
1873 T llz = log(-lz);
1874 guess = lz - llz + (llz / lz); // Chapeau-Blondeau equation 20, page 2162.
1875 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY
1876 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1877 std::cout << "z = " << z << ", guess = " << guess << ", ln(-z) = " << lz << ", ln(-ln(-z) = " << llz << ", llz/lz = " << (llz / lz) << std::endl;
1878 // z = -1.0000000000000001e-30, guess = -73.312782616731482, ln(-z) = -69.077552789821368, ln(-ln(-z) = 4.2352298269101114, llz/lz = -0.061311231447304194
1879 // z = -9.9999999999999999e-91, guess = -212.56650048504233, ln(-z) = -207.23265836946410, ln(-ln(-z) = 5.3338421155782205, llz/lz = -0.025738424423764311
1880 // >z = -2.2250738585072014e-308, guess = -714.95942238244606, ln(-z) = -708.39641853226408, ln(-ln(-z) = 6.5630038501819854, llz/lz = -0.0092645920821846622
1881 int d10 = policies::digits_base10<T, Policy>(); // policy template parameter digits10
1882 int d2 = policies::digits<T, Policy>(); // digits base 2 from policy.
1883 std::cout << "digits10 = " << d10 << ", digits2 = " << d2 // For example: digits10 = 1, digits2 = 5
1884 << std::endl;
1885 std::cout.precision(saved_precision);
1886 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY
1887 if (policies::digits<T, Policy>() < 12)
1888 { // 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.
1889 return guess;
1890 }
1891 T result = lambert_w_detail::lambert_w_halley_iterate(guess, z);
1892 return result;
1893
1894 // Was Fukushima
1895 // G[k=64] == g[63] == -1.02643897e-26
1896 //return policies::raise_domain_error(function,
1897 // "Argument z = %1% is too small (< -1.02643897e-26) ! (Should not occur, please report.",
1898 // z, pol);
1899 } // Z too small so use approximation and Halley.
1900 // Else Use a lookup table to find the nearest integer part of Lambert W-1 as starting point for Bisection.
1901
1902 if (boost::math::tools::digits<T>() > 53)
1903 { // T is more precise than 64-bit double (or long double, or ?),
1904 // so compute an approximate value using only one Schroeder refinement,
1905 // (avoiding any double-precision Halley refinement from policy double_digits2<50> 53 - 3 = 50
1906 // because are next going to use Halley refinement at full/high precision using this as an approximation).
1907 using boost::math::policies::precision;
1908 using boost::math::policies::digits10;
1909 using boost::math::policies::digits2;
1910 using boost::math::policies::policy;
1911 // Compute a 50-bit precision approximate W0 in a double (no Halley refinement).
1912 T double_approx(static_cast<T>(lambert_wm1_imp(must_reduce_to_double(z, std::is_constructible<double, T>()), policy<digits2<50>>())));
1913 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN
1914 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1915 std::cout << "Lambert_wm1 Argument Type " << typeid(T).name() << " approximation double = " << double_approx << std::endl;
1916 std::cout.precision(saved_precision);
1917 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1
1918 // Perform additional Halley refinement(s) to ensure that
1919 // get a near as possible to correct result (usually +/- one epsilon).
1920 T result = lambert_w_halley_iterate(double_approx, z);
1921 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1
1922 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1923 std::cout << "Result " << typeid(T).name() << " precision Halley refinement = " << result << std::endl;
1924 std::cout.precision(saved_precision);
1925 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1
1926 return result;
1927 } // digits > 53 - higher precision than double.
1928 else // T is double or less precision.
1929 { // Use a lookup table to find the nearest integer part of Lambert W as starting point for Bisection.
1930 using namespace boost::math::lambert_w_detail::lambert_w_lookup;
1931 // Bracketing sequence n = (2, 4, 8, 16, 32, 64) for W-1 branch. (0 is -infinity)
1932 // Since z is probably quite small, start with lowest n (=2).
1933 int n = 2;
1934 if (wm1zs[n - 1] > z)
1935 {
1936 goto bisect;
1937 }
1938 for (int j = 1; j <= 5; ++j)
1939 {
1940 n *= 2;
1941 if (wm1zs[n - 1] > z)
1942 {
1943 goto overshot;
1944 }
1945 }
1946 // else z < g[63] == -1.0264389699511303e-26, so Lambert W-1 integer part > 64.
1947 // This should not now occur (should be caught by test and code above) so should be a logic_error?
1948 return policies::raise_domain_error(function,
1949 "Argument z = %1% is too small (< -1.026439e-26) (logic error - please report!)",
1950 z, pol);
1951 overshot:
1952 {
1953 int nh = n / 2;
1954 for (int j = 1; j <= 5; ++j)
1955 {
1956 nh /= 2; // halve step size.
1957 if (nh <= 0)
1958 {
1959 break; // goto bisect;
1960 }
1961 if (wm1zs[n - nh - 1] > z)
1962 {
1963 n -= nh;
1964 }
1965 }
1966 }
1967 bisect:
1968 --n;
1969 // g[n] now holds lambert W of floor integer n and g[n+1] the ceil part;
1970 // these are used as initial values for bisection.
1971 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP
1972 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1973 std::cout << "Result lookup W-1(" << z << ") bisection between wm1zs[" << n - 1 << "] = " << wm1zs[n - 1] << " and wm1zs[" << n << "] = " << wm1zs[n]
1974 << ", bisect mean = " << (wm1zs[n - 1] + wm1zs[n]) / 2 << std::endl;
1975 std::cout.precision(saved_precision);
1976 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP
1977
1978 // Compute bisections is the number of bisections computed from n,
1979 // such that a single application of the fifth-order Schroeder update formula
1980 // after the bisections is enough to evaluate Lambert W-1 with (near?) 53-bit accuracy.
1981 // Fukushima established these by trial and error?
1982 int bisections = 11; // Assume maximum number of bisections will be needed (most common case).
1983 if (n >= 8)
1984 {
1985 bisections = 8;
1986 }
1987 else if (n >= 3)
1988 {
1989 bisections = 9;
1990 }
1991 else if (n >= 2)
1992 {
1993 bisections = 10;
1994 }
1995 // Bracketing, Fukushima section 2.3, page 82:
1996 // (Avoiding using exponential function for speed).
1997 // Only use @c lookup_t precision, default double, for bisection (again for speed),
1998 // and use later Halley refinement for higher precisions.
1999 using lambert_w_lookup::halves;
2000 using lambert_w_lookup::sqrtwm1s;
2001
2002 using calc_type = typename std::conditional<std::is_constructible<lookup_t, T>::value, lookup_t, T>::type;
2003
2004 calc_type w = -static_cast<calc_type>(n); // Equation 25,
2005 calc_type y = static_cast<calc_type>(z * wm1es[n - 1]); // Equation 26,
2006 // Perform the bisections fractional bisections for necessary precision.
2007 for (int j = 0; j < bisections; ++j)
2008 { // Equation 27.
2009 calc_type wj = w - halves[j]; // Subtract 1/2, 1/4, 1/8 ...
2010 calc_type yj = y * sqrtwm1s[j]; // Multiply by sqrt(1/e), ...
2011 if (wj < yj)
2012 {
2013 w = wj;
2014 y = yj;
2015 }
2016 } // for j
2017 return static_cast<T>(schroeder_update(w, y)); // Schroeder 5th order method refinement.
2018
2019 // else // Perform additional Halley refinement(s) to ensure that
2020 // // get a near as possible to correct result (usually +/- epsilon).
2021 // {
2022 // // result = lambert_w_halley_iterate(result, z);
2023 // result = lambert_w_halley_step(result, z); // Just one Halley step should be enough.
2024 //#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_HALLEY
2025 // std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
2026 // std::cout << "Halley refinement estimate = " << result << std::endl;
2027 // std::cout.precision(saved_precision);
2028 //#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W1_HALLEY
2029 // return result; // Halley
2030 // } // Schroeder or Schroeder and Halley.
2031 }
2032 } // template<typename T = double> T lambert_wm1_imp(const T z)
2033 } // namespace lambert_w_detail
2034
2035 ///////////////////////////// User Lambert w functions. //////////////////////////////
2036
2037 //! Lambert W0 using User-defined policy.
2038 template <typename T, typename Policy>
2039 inline
2040 typename boost::math::tools::promote_args<T>::type
lambert_w0(T z,const Policy & pol)2041 lambert_w0(T z, const Policy& pol)
2042 {
2043 // Promote integer or expression template arguments to double,
2044 // without doing any other internal promotion like float to double.
2045 using result_type = typename tools::promote_args<T>::type;
2046
2047 // Work out what precision has been selected,
2048 // based on the Policy and the number type.
2049 using precision_type = typename policies::precision<result_type, Policy>::type;
2050 // and then select the correct implementation based on that precision (not the type T):
2051 using tag_type = std::integral_constant<int,
2052 (precision_type::value == 0) || (precision_type::value > 53) ?
2053 0 // either variable precision (0), or greater than 64-bit precision.
2054 : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision.
2055 : 2 // 64-bit (probably double) precision.
2056 >;
2057
2058 return lambert_w_detail::lambert_w0_imp(result_type(z), pol, tag_type()); //
2059 } // lambert_w0(T z, const Policy& pol)
2060
2061 //! Lambert W0 using default policy.
2062 template <typename T>
2063 inline
2064 typename tools::promote_args<T>::type
lambert_w0(T z)2065 lambert_w0(T z)
2066 {
2067 // Promote integer or expression template arguments to double,
2068 // without doing any other internal promotion like float to double.
2069 using result_type = typename tools::promote_args<T>::type;
2070
2071 // Work out what precision has been selected, based on the Policy and the number type.
2072 // For the default policy version, we want the *default policy* precision for T.
2073 using precision_type = typename policies::precision<result_type, policies::policy<>>::type;
2074 // and then select the correct implementation based on that (not the type T):
2075 using tag_type = std::integral_constant<int,
2076 (precision_type::value == 0) || (precision_type::value > 53) ?
2077 0 // either variable precision (0), or greater than 64-bit precision.
2078 : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision.
2079 : 2 // 64-bit (probably double) precision.
2080 >;
2081 return lambert_w_detail::lambert_w0_imp(result_type(z), policies::policy<>(), tag_type());
2082 } // lambert_w0(T z) using default policy.
2083
2084 //! W-1 branch (-max(z) < z <= -1/e).
2085
2086 //! Lambert W-1 using User-defined policy.
2087 template <typename T, typename Policy>
2088 inline
2089 typename tools::promote_args<T>::type
lambert_wm1(T z,const Policy & pol)2090 lambert_wm1(T z, const Policy& pol)
2091 {
2092 // Promote integer or expression template arguments to double,
2093 // without doing any other internal promotion like float to double.
2094 using result_type = typename tools::promote_args<T>::type;
2095 return lambert_w_detail::lambert_wm1_imp(result_type(z), pol); //
2096 }
2097
2098 //! Lambert W-1 using default policy.
2099 template <typename T>
2100 inline
2101 typename tools::promote_args<T>::type
lambert_wm1(T z)2102 lambert_wm1(T z)
2103 {
2104 using result_type = typename tools::promote_args<T>::type;
2105 return lambert_w_detail::lambert_wm1_imp(result_type(z), policies::policy<>());
2106 } // lambert_wm1(T z)
2107
2108 // First derivative of Lambert W0 and W-1.
2109 template <typename T, typename Policy>
2110 inline typename tools::promote_args<T>::type
lambert_w0_prime(T z,const Policy & pol)2111 lambert_w0_prime(T z, const Policy& pol)
2112 {
2113 using result_type = typename tools::promote_args<T>::type;
2114 using std::numeric_limits;
2115 if (z == 0)
2116 {
2117 return static_cast<result_type>(1);
2118 }
2119 // This is the sensible choice if we regard the Lambert-W function as complex analytic.
2120 // Of course on the real line, it's just undefined.
2121 if (z == - boost::math::constants::exp_minus_one<result_type>())
2122 {
2123 return numeric_limits<result_type>::has_infinity ? numeric_limits<result_type>::infinity() : boost::math::tools::max_value<result_type>();
2124 }
2125 // if z < -1/e, we'll let lambert_w0 do the error handling:
2126 result_type w = lambert_w0(result_type(z), pol);
2127 // If w ~ -1, then presumably this can get inaccurate.
2128 // Is there an accurate way to evaluate 1 + W(-1/e + eps)?
2129 // Yes: This is discussed in the Princeton Companion to Applied Mathematics,
2130 // 'The Lambert-W function', Section 1.3: Series and Generating Functions.
2131 // 1 + W(-1/e + x) ~ sqrt(2ex).
2132 // Nick is not convinced this formula is more accurate than the naive one.
2133 // However, for z != -1/e, we never get rounded to w = -1 in any precision I've tested (up to cpp_bin_float_100).
2134 return w / (z * (1 + w));
2135 } // lambert_w0_prime(T z)
2136
2137 template <typename T>
2138 inline typename tools::promote_args<T>::type
lambert_w0_prime(T z)2139 lambert_w0_prime(T z)
2140 {
2141 return lambert_w0_prime(z, policies::policy<>());
2142 }
2143
2144 template <typename T, typename Policy>
2145 inline typename tools::promote_args<T>::type
lambert_wm1_prime(T z,const Policy & pol)2146 lambert_wm1_prime(T z, const Policy& pol)
2147 {
2148 using std::numeric_limits;
2149 using result_type = typename tools::promote_args<T>::type;
2150 //if (z == 0)
2151 //{
2152 // return static_cast<result_type>(1);
2153 //}
2154 //if (z == - boost::math::constants::exp_minus_one<result_type>())
2155 if (z == 0 || z == - boost::math::constants::exp_minus_one<result_type>())
2156 {
2157 return numeric_limits<result_type>::has_infinity ? -numeric_limits<result_type>::infinity() : -boost::math::tools::max_value<result_type>();
2158 }
2159
2160 result_type w = lambert_wm1(z, pol);
2161 return w/(z*(1+w));
2162 } // lambert_wm1_prime(T z)
2163
2164 template <typename T>
2165 inline typename tools::promote_args<T>::type
lambert_wm1_prime(T z)2166 lambert_wm1_prime(T z)
2167 {
2168 return lambert_wm1_prime(z, policies::policy<>());
2169 }
2170
2171 }} //boost::math namespaces
2172
2173 #endif // #ifdef BOOST_MATH_SF_LAMBERT_W_HPP
2174
2175