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/mpl/int.hpp>
62 #include <boost/type_traits/is_integral.hpp>
63 #include <boost/math/tools/precision.hpp> // boost::math::tools::max_value().
64 #include <boost/math/tools/big_constant.hpp>
65 #include <boost/math/tools/cxx03_warn.hpp>
66
67 #include <limits>
68 #include <cmath>
69 #include <limits>
70 #include <exception>
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 typedef double lookup_t; // 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 <class 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 <class 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 <class T>
125 inline
lambert_w_halley_iterate(T w_est,const T z)126 T lambert_w_halley_iterate(T w_est, const T z)
127 {
128 BOOST_MATH_STD_USING
129 static const T max_diff = boost::math::tools::root_epsilon<T>() * fabs(w_est);
130
131 T w_new = lambert_w_halley_step(w_est, z);
132 T diff = fabs(w_est - w_new);
133 while (diff > max_diff)
134 {
135 w_est = w_new;
136 w_new = lambert_w_halley_step(w_est, z);
137 diff = fabs(w_est - w_new);
138 }
139 return w_new;
140 } // template <class T> lambert_w_halley_iterate(T w_est, T z)
141
142 // Two Halley function versions that either
143 // single step (if boost::false_type) or iterate (if boost::true_type).
144 // Selected at compile-time using parameter 3.
145 template <class T>
146 inline
lambert_w_maybe_halley_iterate(T z,T w,boost::false_type const &)147 T lambert_w_maybe_halley_iterate(T z, T w, boost::false_type const&)
148 {
149 return lambert_w_halley_step(z, w); // Single step.
150 }
151
152 template <class T>
153 inline
lambert_w_maybe_halley_iterate(T z,T w,boost::true_type const &)154 T lambert_w_maybe_halley_iterate(T z, T w, boost::true_type const&)
155 {
156 return lambert_w_halley_iterate(z, w); // Iterate steps.
157 }
158
159 //! maybe_reduce_to_double function,
160 //! Two versions that have a compile-time option to
161 //! reduce argument z to double precision (if true_type).
162 //! Version is selected at compile-time using parameter 2.
163
164 template <class T>
165 inline
maybe_reduce_to_double(const T & z,const boost::true_type &)166 double maybe_reduce_to_double(const T& z, const boost::true_type&)
167 {
168 return static_cast<double>(z); // Reduce to double precision.
169 }
170
171 template <class T>
172 inline
maybe_reduce_to_double(const T & z,const boost::false_type &)173 T maybe_reduce_to_double(const T& z, const boost::false_type&)
174 { // Don't reduce to double.
175 return z;
176 }
177
178 template <class T>
179 inline
must_reduce_to_double(const T & z,const boost::true_type &)180 double must_reduce_to_double(const T& z, const boost::true_type&)
181 {
182 return static_cast<double>(z); // Reduce to double precision.
183 }
184
185 template <class T>
186 inline
must_reduce_to_double(const T & z,const boost::false_type &)187 double must_reduce_to_double(const T& z, const boost::false_type&)
188 { // try a lexical_cast and hope for the best:
189 return boost::lexical_cast<double>(z);
190 }
191
192 //! \brief Schroeder method, fifth-order update formula,
193 //! \details See T. Fukushima page 80-81, and
194 //! A. Householder, The Numerical Treatment of a Single Nonlinear Equation,
195 //! McGraw-Hill, New York, 1970, section 4.4.
196 //! Fukushima algorithm switches to @c schroeder_update after pre-computed bisections,
197 //! chosen to ensure that the result will be achieve the +/- 10 epsilon target.
198 //! \param w Lambert w estimate from bisection or series.
199 //! \param y bracketing value from bisection.
200 //! \returns Refined estimate of Lambert w.
201
202 // Schroeder refinement, called unless NOT required by precision policy.
203 template<typename T>
204 inline
schroeder_update(const T w,const T y)205 T schroeder_update(const T w, const T y)
206 {
207 // Compute derivatives using 5th order Schroeder refinement.
208 // Since this is the final step, it will always use the highest precision type T.
209 // Example of Call:
210 // result = schroeder_update(w, y);
211 //where
212 // w is estimate of Lambert W (from bisection or series).
213 // y is z * e^-w.
214
215 BOOST_MATH_STD_USING // Aid argument dependent lookup of abs.
216 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
217 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
218 using boost::math::float_distance;
219 T fd = float_distance<T>(w, y);
220 std::cout << "Schroder ";
221 if (abs(fd) < 214748000.)
222 {
223 std::cout << " Distance = "<< static_cast<int>(fd);
224 }
225 else
226 {
227 std::cout << "Difference w - y = " << (w - y) << ".";
228 }
229 std::cout << std::endl;
230 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
231 // Fukushima equation 18, page 6.
232 const T f0 = w - y; // f0 = w - y.
233 const T f1 = 1 + y; // f1 = df/dW
234 const T f00 = f0 * f0;
235 const T f11 = f1 * f1;
236 const T f0y = f0 * y;
237 const T result =
238 w - 4 * f0 * (6 * f1 * (f11 + f0y) + f00 * y) /
239 (f11 * (24 * f11 + 36 * f0y) +
240 f00 * (6 * y * y + 8 * f1 * y + f0y)); // Fukushima Page 81, equation 21 from equation 20.
241
242 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
243 std::cout << "Schroeder refined " << w << " " << y << ", difference " << w-y << ", change " << w - result << ", to result " << result << std::endl;
244 std::cout.precision(saved_precision); // Restore.
245 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
246
247 return result;
248 } // template<typename T = double> T schroeder_update(const T w, const T y)
249
250 //! \brief Series expansion used near the singularity/branch point z = -exp(-1) = -3.6787944.
251 //! Wolfram InverseSeries[Series[sqrt[2(p Exp[1 + p] + 1)], {p,-1, 20}]]
252 //! Wolfram command used to obtain 40 series terms at 50 decimal digit precision was
253 //! N[InverseSeries[Series[Sqrt[2(p Exp[1 + p] + 1)], { p,-1,40 }]], 50]
254 //! -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 ...
255 //! Decimal values of specifications for built-in floating-point types below
256 //! are at least 21 digits precision == max_digits10 for long double.
257 //! Longer decimal digits strings are rationals evaluated using Wolfram.
258
259 template<typename T>
lambert_w_singularity_series(const T p)260 T lambert_w_singularity_series(const T p)
261 {
262 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES
263 std::size_t saved_precision = std::cout.precision(3);
264 std::cout << "Singularity_series Lambert_w p argument = " << p << std::endl;
265 std::cout
266 //<< "Argument Type = " << typeid(T).name()
267 //<< ", max_digits10 = " << std::numeric_limits<T>::max_digits10
268 //<< ", epsilon = " << std::numeric_limits<T>::epsilon()
269 << std::endl;
270 std::cout.precision(saved_precision);
271 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES
272
273 static const T q[] =
274 {
275 -static_cast<T>(1), // j0
276 +T(1), // j1
277 -T(1) / 3, // 1/3 j2
278 +T(11) / 72, // 0.152777777777777778, // 11/72 j3
279 -T(43) / 540, // 0.0796296296296296296, // 43/540 j4
280 +T(769) / 17280, // 0.0445023148148148148, j5
281 -T(221) / 8505, // 0.0259847148736037625, j6
282 //+T(0.0156356325323339212L), // j7
283 //+T(0.015635632532333921222810111699000587889476778365667L), // j7 from Wolfram N[680863/43545600, 50]
284 +T(680863uLL) / 43545600uLL, // +0.0156356325323339212, j7
285 //-T(0.00961689202429943171L), // j8
286 -T(1963uLL) / 204120uLL, // 0.00961689202429943171, j8
287 //-T(0.0096168920242994317068391142465216539290613364687439L), // j8 from Wolfram N[1963/204120, 50]
288 +T(226287557uLL) / 37623398400uLL, // 0.00601454325295611786, j9
289 -T(5776369uLL) / 1515591000uLL, // 0.00381129803489199923, j10
290 //+T(0.00244087799114398267L), j11 0.0024408779911439826658968585286437530215699919795550
291 +T(169709463197uLL) / 69528040243200uLL, // j11
292 // -T(0.00157693034468678425L), // j12 -0.0015769303446867842539234095399314115973161850314723
293 -T(1118511313uLL) / 709296588000uLL, // j12
294 +T(667874164916771uLL) / 650782456676352000uLL, // j13
295 //+T(0.00102626332050760715L), // j13 0.0010262633205076071544375481533906861056468041465973
296 -T(500525573uLL) / 744761417400uLL, // j14
297 // -T(0.000672061631156136204L), j14
298 //+T(1003663334225097487uLL) / 234281684403486720000uLL, // j15 0.00044247306181462090993020760858473726479232802068800 error C2177: constant too big
299 //+T(0.000442473061814620910L, // j15
300 BOOST_MATH_BIG_CONSTANT(T, 64, +0.000442473061814620910), // j15
301 // -T(0.000292677224729627445L), // j16
302 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000292677224729627445), // j16
303 //+T(0.000194387276054539318L), // j17
304 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000194387276054539318), // j17
305 //-T(0.000129574266852748819L), // j18
306 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000129574266852748819), // j18
307 //+T(0.0000866503580520812717L), // j19 N[+1150497127780071399782389/13277465363600276402995200000, 50] 0.000086650358052081271660451590462390293190597827783288
308 BOOST_MATH_BIG_CONSTANT(T, 64, +0.0000866503580520812717), // j19
309 //-T(0.0000581136075044138168L) // j20 N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913
310 // -T(2853534237182741069uLL) / 49102686267859224000000uLL // j20 // error C2177: constant too big,
311 // so must use BOOST_MATH_BIG_CONSTANT(T, ) format in hope of using suffix Q for quad or decimal digits string for others.
312 //-T(0.000058113607504413816772205464778828177256611844221913L), // j20 N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913
313 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000058113607504413816772205464778828177256611844221913) // j20 - last used by Fukushima
314 // More terms don't seem to give any improvement (worse in fact) and are not use for many z values.
315 //BOOST_MATH_BIG_CONSTANT(T, +0.000039076684867439051635395583044527492132109160553593), // j21
316 //BOOST_MATH_BIG_CONSTANT(T, -0.000026338064747231098738584082718649443078703982217219), // j22
317 //BOOST_MATH_BIG_CONSTANT(T, +0.000017790345805079585400736282075184540383274460464169), // j23
318 //BOOST_MATH_BIG_CONSTANT(T, -0.000012040352739559976942274116578992585158113153190354), // j24
319 //BOOST_MATH_BIG_CONSTANT(T, +8.1635319824966121713827512573558687050675701559448E-6), // j25
320 //BOOST_MATH_BIG_CONSTANT(T, -5.5442032085673591366657251660804575198155559225316E-6) // j26
321 // -T(5.5442032085673591366657251660804575198155559225316E-6L) // j26
322 // 21 to 26 Added for long double.
323 }; // static const T q[]
324
325 /*
326 // Temporary copy of original double values for comparison; these are reproduced well.
327 static const T q[] =
328 {
329 -1L, // j0
330 +1L, // j1
331 -0.333333333333333333L, // 1/3 j2
332 +0.152777777777777778L, // 11/72 j3
333 -0.0796296296296296296L, // 43/540
334 +0.0445023148148148148L,
335 -0.0259847148736037625L,
336 +0.0156356325323339212L,
337 -0.00961689202429943171L,
338 +0.00601454325295611786L,
339 -0.00381129803489199923L,
340 +0.00244087799114398267L,
341 -0.00157693034468678425L,
342 +0.00102626332050760715L,
343 -0.000672061631156136204L,
344 +0.000442473061814620910L,
345 -0.000292677224729627445L,
346 +0.000194387276054539318L,
347 -0.000129574266852748819L,
348 +0.0000866503580520812717L,
349 -0.0000581136075044138168L // j20
350 };
351 */
352
353 // Decide how many series terms to use, increasing as z approaches the singularity,
354 // balancing run-time versus computational noise from round-off.
355 // In practice, we truncate the series expansion at a certain order.
356 // If the order is too large, not only does the amount of computation increase,
357 // but also the round-off errors accumulate.
358 // See Fukushima equation 35, page 85 for logic of choice of number of series terms.
359
360 BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs.
361
362 const T absp = abs(p);
363
364 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS
365 {
366 int terms = 20; // Default to using all terms.
367 if (absp < 0.01159)
368 { // Very near singularity.
369 terms = 6;
370 }
371 else if (absp < 0.0766)
372 { // Near singularity.
373 terms = 10;
374 }
375 std::streamsize saved_precision = std::cout.precision(3);
376 std::cout << "abs(p) = " << absp << ", terms = " << terms << std::endl;
377 std::cout.precision(saved_precision);
378 }
379 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS
380
381 if (absp < 0.01159)
382 { // Only 6 near-singularity series terms are useful.
383 return
384 -1 +
385 p * (1 +
386 p * (q[2] +
387 p * (q[3] +
388 p * (q[4] +
389 p * (q[5] +
390 p * q[6]
391 )))));
392 }
393 else if (absp < 0.0766) // Use 10 near-singularity series terms.
394 { // Use 10 near-singularity series terms.
395 return
396 -1 +
397 p * (1 +
398 p * (q[2] +
399 p * (q[3] +
400 p * (q[4] +
401 p * (q[5] +
402 p * (q[6] +
403 p * (q[7] +
404 p * (q[8] +
405 p * (q[9] +
406 p * q[10]
407 )))))))));
408 }
409 else
410 { // Use all 20 near-singularity series terms.
411 return
412 -1 +
413 p * (1 +
414 p * (q[2] +
415 p * (q[3] +
416 p * (q[4] +
417 p * (q[5] +
418 p * (q[6] +
419 p * (q[7] +
420 p * (q[8] +
421 p * (q[9] +
422 p * (q[10] +
423 p * (q[11] +
424 p * (q[12] +
425 p * (q[13] +
426 p * (q[14] +
427 p * (q[15] +
428 p * (q[16] +
429 p * (q[17] +
430 p * (q[18] +
431 p * (q[19] +
432 p * q[20] // Last Fukushima term.
433 )))))))))))))))))));
434 // + // more terms for more precise T: long double ...
435 //// but makes almost no difference, so don't use more terms?
436 // p*q[21] +
437 // p*q[22] +
438 // p*q[23] +
439 // p*q[24] +
440 // p*q[25]
441 // )))))))))))))))))));
442 }
443 } // template<typename T = double> T lambert_w_singularity_series(const T p)
444
445
446 /////////////////////////////////////////////////////////////////////////////////////////////
447
448 //! \brief Series expansion used near zero (abs(z) < 0.05).
449 //! \details
450 //! Coefficients of the inverted series expansion of the Lambert W function around z = 0.
451 //! Tosio Fukushima always uses all 17 terms of a Taylor series computed using Wolfram with
452 //! InverseSeries[Series[z Exp[z],{z,0,17}]]
453 //! Tosio Fukushima / Journal of Computational and Applied Mathematics 244 (2013) page 86.
454
455 //! Decimal values of specifications for built-in floating-point types below
456 //! are 21 digits precision == max_digits10 for long double.
457 //! Care! Some coefficients might overflow some fixed_point types.
458
459 //! This version is intended to allow use by user-defined types
460 //! like Boost.Multiprecision quad and cpp_dec_float types.
461 //! The three specializations below for built-in float, double
462 //! (and perhaps long double) will be chosen in preference for these types.
463
464 //! This version uses rationals computed by Wolfram as far as possible,
465 //! limited by maximum size of uLL integers.
466 //! For higher term, uses decimal digit strings computed by Wolfram up to the maximum possible using uLL rationals,
467 //! and then higher coefficients are computed as necessary using function lambert_w0_small_z_series_term
468 //! until the precision required by the policy is achieved.
469 //! InverseSeries[Series[z Exp[z],{z,0,34}]] also computed.
470
471 // Series evaluation for LambertW(z) as z -> 0.
472 // See http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/
473 // http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/MainEq1.L.gif
474
475 //! \brief lambert_w0_small_z uses a tag_type to select a variant depending on the size of the type.
476 //! The Lambert W is computed by lambert_w0_small_z for small z.
477 //! The cutoff for z smallness determined by Tosio Fukushima by trial and error is (abs(z) < 0.05),
478 //! but the optimum might be a function of the size of the type of z.
479
480 //! \details
481 //! The tag_type selection is based on the value @c std::numeric_limits<T>::max_digits10.
482 //! This allows distinguishing between long double types that commonly vary between 64 and 80-bits,
483 //! and also compilers that have a float type using 64 bits and/or long double using 128-bits.
484 //! It assumes that max_digits10 is defined correctly or this might fail to make the correct selection.
485 //! causing very small differences in computing lambert_w that would be very difficult to detect and diagnose.
486 //! Cannot switch on @c std::numeric_limits<>::max() because comparison values may overflow the compiler limit.
487 //! Cannot switch on @c std::numeric_limits<long double>::max_exponent10()
488 //! because both 80 and 128 bit floating-point implementations use 11 bits for the exponent.
489 //! So must rely on @c std::numeric_limits<long double>::max_digits10.
490
491 //! Specialization of float zero series expansion used for small z (abs(z) < 0.05).
492 //! Specializations of lambert_w0_small_z for built-in types.
493 //! These specializations should be chosen in preference to T version.
494 //! For example: lambert_w0_small_z(0.001F) should use the float version.
495 //! (Parameter Policy is not used by built-in types when all terms are used during an inline computation,
496 //! but for the tag_type selection to work, they all must include Policy in their signature.
497
498 // Forward declaration of variants of lambert_w0_small_z.
499 template <class T, class Policy>
500 T lambert_w0_small_z(T x, const Policy&, boost::integral_constant<int, 0> const&); // for float (32-bit) type.
501
502 template <class T, class Policy>
503 T lambert_w0_small_z(T x, const Policy&, boost::integral_constant<int, 1> const&); // for double (64-bit) type.
504
505 template <class T, class Policy>
506 T lambert_w0_small_z(T x, const Policy&, boost::integral_constant<int, 2> const&); // for long double (double extended 80-bit) type.
507
508 template <class T, class Policy>
509 T lambert_w0_small_z(T x, const Policy&, boost::integral_constant<int, 3> const&); // for long double (128-bit) type.
510
511 template <class T, class Policy>
512 T lambert_w0_small_z(T x, const Policy&, boost::integral_constant<int, 4> const&); // for float128 quadmath Q type.
513
514 template <class T, class Policy>
515 T lambert_w0_small_z(T x, const Policy&, boost::integral_constant<int, 5> const&); // Generic multiprecision T.
516 // Set tag_type depending on max_digits10.
517 template <class T, class Policy>
518 T lambert_w0_small_z(T x, const Policy& pol)
519 { //std::numeric_limits<T>::max_digits10 == 36 ? 3 : // 128-bit long double.
520 typedef boost::integral_constant<int,
521 std::numeric_limits<T>::is_specialized == 0 ? 5 :
522 #ifndef BOOST_NO_CXX11_NUMERIC_LIMITS
523 std::numeric_limits<T>::max_digits10 <= 9 ? 0 : // for float 32-bit.
524 std::numeric_limits<T>::max_digits10 <= 17 ? 1 : // for double 64-bit.
525 std::numeric_limits<T>::max_digits10 <= 22 ? 2 : // for 80-bit double extended.
526 std::numeric_limits<T>::max_digits10 < 37 ? 4 // for both 128-bit long double (3) and 128-bit quad suffix Q type (4).
527 #else
528 std::numeric_limits<T>::radix != 2 ? 5 :
529 std::numeric_limits<T>::digits <= 24 ? 0 : // for float 32-bit.
530 std::numeric_limits<T>::digits <= 53 ? 1 : // for double 64-bit.
531 std::numeric_limits<T>::digits <= 64 ? 2 : // for 80-bit double extended.
532 std::numeric_limits<T>::digits <= 113 ? 4 // for both 128-bit long double (3) and 128-bit quad suffix Q type (4).
533 #endif
534 : 5 // All Generic multiprecision types.
535 > tag_type;
536 // std::cout << "\ntag type = " << tag_type << std::endl; // error C2275: 'tag_type': illegal use of this type as an expression.
537 return lambert_w0_small_z(x, pol, tag_type());
538 } // template <class T> T lambert_w0_small_z(T x)
539
540 //! Specialization of float (32-bit) series expansion used for small z (abs(z) < 0.05).
541 // Only 9 Coefficients are computed to 21 decimal digits precision, ample for 32-bit float used by most platforms.
542 // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction
543 // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],
544 // as proposed by Tosio Fukushima and implemented by Darko Veberic.
545
546 template <class T, class Policy>
547 T lambert_w0_small_z(T z, const Policy&, boost::integral_constant<int, 0> const&)
548 {
549 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
550 std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
551 std::cout << "\ntag_type 0 float lambert_w0_small_z called with z = " << z << " using " << 9 << " terms of precision "
552 << std::numeric_limits<float>::max_digits10 << " decimal digits. " << std::endl;
553 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
554 T result =
555 z * (1 - // j1 z^1 term = 1
556 z * (1 - // j2 z^2 term = -1
557 z * (static_cast<float>(3uLL) / 2uLL - // 3/2 // j3 z^3 term = 1.5.
558 z * (2.6666666666666666667F - // 8/3 // j4
559 z * (5.2083333333333333333F - // -125/24 // j5
560 z * (10.8F - // j6
561 z * (23.343055555555555556F - // j7
562 z * (52.012698412698412698F - // j8
563 z * 118.62522321428571429F)))))))); // j9
564
565 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
566 std::cout << "return w = " << result << std::endl;
567 std::cout.precision(prec); // Restore.
568 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
569
570 return result;
571 } // template <class T> T lambert_w0_small_z(T x, boost::integral_constant<int, 0> const&)
572
573 //! Specialization of double (64-bit double) series expansion used for small z (abs(z) < 0.05).
574 // 17 Coefficients are computed to 21 decimal digits precision suitable for 64-bit double used by most platforms.
575 // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction
576 // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], as proposed by Tosio Fukushima and implemented by Veberic.
577
578 template <class T, class Policy>
579 T lambert_w0_small_z(const T z, const Policy&, boost::integral_constant<int, 1> const&)
580 {
581 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
582 std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
583 std::cout << "\ntag_type 1 double lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision, "
584 << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl;
585 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
586 T result =
587 z * (1. - // j1 z^1
588 z * (1. - // j2 z^2
589 z * (1.5 - // 3/2 // j3 z^3
590 z * (2.6666666666666666667 - // 8/3 // j4
591 z * (5.2083333333333333333 - // -125/24 // j5
592 z * (10.8 - // j6
593 z * (23.343055555555555556 - // j7
594 z * (52.012698412698412698 - // j8
595 z * (118.62522321428571429 - // j9
596 z * (275.57319223985890653 - // j10
597 z * (649.78717234347442681 - // j11
598 z * (1551.1605194805194805 - // j12
599 z * (3741.4497029592385495 - // j13
600 z * (9104.5002411580189358 - // j14
601 z * (22324.308512706601434 - // j15
602 z * (55103.621972903835338 - // j16
603 z * 136808.86090394293563)))))))))))))))); // j17 z^17
604
605 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
606 std::cout << "return w = " << result << std::endl;
607 std::cout.precision(prec); // Restore.
608 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
609
610 return result;
611 } // T lambert_w0_small_z(const T z, boost::integral_constant<int, 1> const&)
612
613 //! Specialization of long double (80-bit double extended) series expansion used for small z (abs(z) < 0.05).
614 // 21 Coefficients are computed to 21 decimal digits precision suitable for 80-bit long double used by some
615 // platforms including GCC and Clang when generating for Intel X86 floating-point processors with 80-bit operations enabled (the default).
616 // (This is NOT used by Microsoft Visual Studio where double and long always both use only 64-bit type.
617 // Nor used for 128-bit float128.)
618 template <class T, class Policy>
619 T lambert_w0_small_z(const T z, const Policy&, boost::integral_constant<int, 2> const&)
620 {
621 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
622 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
623 std::cout << "\ntag_type 2 long double (80-bit double extended) lambert_w0_small_z called with z = " << z << " using " << 21 << " terms of precision, "
624 << std::numeric_limits<long double>::max_digits10 << " decimal digits. " << std::endl;
625 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
626 // T result =
627 // z * (1.L - // j1 z^1
628 // z * (1.L - // j2 z^2
629 // z * (1.5L - // 3/2 // j3
630 // z * (2.6666666666666666667L - // 8/3 // j4
631 // z * (5.2083333333333333333L - // -125/24 // j5
632 // z * (10.800000000000000000L - // j6
633 // z * (23.343055555555555556L - // j7
634 // z * (52.012698412698412698L - // j8
635 // z * (118.62522321428571429L - // j9
636 // z * (275.57319223985890653L - // j10
637 // z * (649.78717234347442681L - // j11
638 // z * (1551.1605194805194805L - // j12
639 // z * (3741.4497029592385495L - // j13
640 // z * (9104.5002411580189358L - // j14
641 // z * (22324.308512706601434L - // j15
642 // z * (55103.621972903835338L - // j16
643 // z * (136808.86090394293563L - // j17 z^17 last term used by Fukushima double.
644 // z * (341422.050665838363317L - // z^18
645 // z * (855992.9659966075514633L - // z^19
646 // z * (2.154990206091088289321e6L - // z^20
647 // z * 5.4455529223144624316423e6L // z^21
648 // ))))))))))))))))))));
649 //
650
651 T result =
652 z * (1.L - // z j1
653 z * (1.L - // z^2
654 z * (1.500000000000000000000000000000000L - // z^3
655 z * (2.666666666666666666666666666666666L - // z ^ 4
656 z * (5.208333333333333333333333333333333L - // z ^ 5
657 z * (10.80000000000000000000000000000000L - // z ^ 6
658 z * (23.34305555555555555555555555555555L - // z ^ 7
659 z * (52.01269841269841269841269841269841L - // z ^ 8
660 z * (118.6252232142857142857142857142857L - // z ^ 9
661 z * (275.5731922398589065255731922398589L - // z ^ 10
662 z * (649.7871723434744268077601410934744L - // z ^ 11
663 z * (1551.160519480519480519480519480519L - // z ^ 12
664 z * (3741.449702959238549516327294105071L - //z ^ 13
665 z * (9104.500241158018935796713574491352L - // z ^ 14
666 z * (22324.308512706601434280005708577137L - // z ^ 15
667 z * (55103.621972903835337697771560205422L - // z ^ 16
668 z * (136808.86090394293563342215789305736L - // z ^ 17
669 z * (341422.05066583836331735491399356945L - // z^18
670 z * (855992.9659966075514633630250633224L - // z^19
671 z * (2.154990206091088289321708745358647e6L // z^20 distance -5 without term 20
672 ))))))))))))))))))));
673
674 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
675 std::cout << "return w = " << result << std::endl;
676 std::cout.precision(precision); // Restore.
677 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
678 return result;
679 } // long double lambert_w0_small_z(const T z, boost::integral_constant<int, 1> const&)
680
681 //! Specialization of 128-bit long double series expansion used for small z (abs(z) < 0.05).
682 // 34 Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction
683 // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],
684 // and are suffixed by L as they are assumed of type long double.
685 // (This is NOT used for 128-bit quad boost::multiprecision::float128 type which required a suffix Q
686 // nor multiprecision type cpp_bin_float_quad that can only be initialised at full precision of the type
687 // constructed with a decimal digit string like "2.6666666666666666666666666666666666666666666666667".)
688
689 template <class T, class Policy>
690 T lambert_w0_small_z(const T z, const Policy&, boost::integral_constant<int, 3> const&)
691 {
692 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
693 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
694 std::cout << "\ntag_type 3 long double (128-bit) lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision, "
695 << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl;
696 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
697 T result =
698 z * (1.L - // j1
699 z * (1.L - // j2
700 z * (1.5L - // 3/2 // j3
701 z * (2.6666666666666666666666666666666666L - // 8/3 // j4
702 z * (5.2052083333333333333333333333333333L - // -125/24 // j5
703 z * (10.800000000000000000000000000000000L - // j6
704 z * (23.343055555555555555555555555555555L - // j7
705 z * (52.0126984126984126984126984126984126L - // j8
706 z * (118.625223214285714285714285714285714L - // j9
707 z * (275.57319223985890652557319223985890L - // * z ^ 10 - // j10
708 z * (649.78717234347442680776014109347442680776014109347L - // j11
709 z * (1551.1605194805194805194805194805194805194805194805L - // j12
710 z * (3741.4497029592385495163272941050718828496606274384L - // j13
711 z * (9104.5002411580189357967135744913522691300469078247L - // j14
712 z * (22324.308512706601434280005708577137148565719994291L - // j15
713 z * (55103.621972903835337697771560205422639285073147507L - // j16
714 z * 136808.86090394293563342215789305736395683485630576L // j17
715 ))))))))))))))));
716
717 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
718 std::cout << "return w = " << result << std::endl;
719 std::cout.precision(precision); // Restore.
720 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
721 return result;
722 } // T lambert_w0_small_z(const T z, boost::integral_constant<int, 3> const&)
723
724 //! Specialization of 128-bit quad series expansion used for small z (abs(z) < 0.05).
725 // 34 Taylor series coefficients used were computed by Wolfram to 50 decimal digits using instruction
726 // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],
727 // and are suffixed by Q as they are assumed of type quad.
728 // This could be used for 128-bit quad (which requires a suffix Q for full precision).
729 // But experiments with GCC 7.2.0 show that while this gives full 128-bit precision
730 // when the -f-ext-numeric-literals option is in force and the libquadmath library available,
731 // over the range -0.049 to +0.049,
732 // it is slightly slower than getting a double approximation followed by a single Halley step.
733
734 #ifdef BOOST_HAS_FLOAT128
735 template <class T, class Policy>
736 T lambert_w0_small_z(const T z, const Policy&, boost::integral_constant<int, 4> const&)
737 {
738 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
739 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
740 std::cout << "\ntag_type 4 128-bit quad float128 lambert_w0_small_z called with z = " << z << " using " << 34 << " terms of precision, "
741 << std::numeric_limits<float128>::max_digits10 << " max decimal digits." << std::endl;
742 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
743 T result =
744 z * (1.Q - // z j1
745 z * (1.Q - // z^2
746 z * (1.500000000000000000000000000000000Q - // z^3
747 z * (2.666666666666666666666666666666666Q - // z ^ 4
748 z * (5.208333333333333333333333333333333Q - // z ^ 5
749 z * (10.80000000000000000000000000000000Q - // z ^ 6
750 z * (23.34305555555555555555555555555555Q - // z ^ 7
751 z * (52.01269841269841269841269841269841Q - // z ^ 8
752 z * (118.6252232142857142857142857142857Q - // z ^ 9
753 z * (275.5731922398589065255731922398589Q - // z ^ 10
754 z * (649.7871723434744268077601410934744Q - // z ^ 11
755 z * (1551.160519480519480519480519480519Q - // z ^ 12
756 z * (3741.449702959238549516327294105071Q - //z ^ 13
757 z * (9104.500241158018935796713574491352Q - // z ^ 14
758 z * (22324.308512706601434280005708577137Q - // z ^ 15
759 z * (55103.621972903835337697771560205422Q - // z ^ 16
760 z * (136808.86090394293563342215789305736Q - // z ^ 17
761 z * (341422.05066583836331735491399356945Q - // z^18
762 z * (855992.9659966075514633630250633224Q - // z^19
763 z * (2.154990206091088289321708745358647e6Q - // 20
764 z * (5.445552922314462431642316420035073e6Q - // 21
765 z * (1.380733000216662949061923813184508e7Q - // 22
766 z * (3.511704498513923292853869855945334e7Q - // 23
767 z * (8.956800256102797693072819557780090e7Q - // 24
768 z * (2.290416846187949813964782641734774e8Q - // 25
769 z * (5.871035041171798492020292225245235e8Q - // 26
770 z * (1.508256053857792919641317138812957e9Q - // 27
771 z * (3.882630161293188940385873468413841e9Q - // 28
772 z * (1.001394313665482968013913601565723e10Q - // 29
773 z * (2.587356736265760638992878359024929e10Q - // 30
774 z * (6.696209709358073856946120522333454e10Q - // 31
775 z * (1.735711659599198077777078238043644e11Q - // 32
776 z * (4.505680465642353886756098108484670e11Q - // 33
777 z * (1.171223178256487391904047636564823e12Q //z^34
778 ))))))))))))))))))))))))))))))))));
779
780
781 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
782 std::cout << "return w = " << result << std::endl;
783 std::cout.precision(precision); // Restore.
784 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
785
786 return result;
787 } // T lambert_w0_small_z(const T z, boost::integral_constant<int, 4> const&) float128
788
789 #else
790
791 template <class T, class Policy>
lambert_w0_small_z(const T z,const Policy & pol,boost::integral_constant<int,4> const &)792 inline T lambert_w0_small_z(const T z, const Policy& pol, boost::integral_constant<int, 4> const&)
793 {
794 return lambert_w0_small_z(z, pol, boost::integral_constant<int, 5>());
795 }
796
797 #endif // BOOST_HAS_FLOAT128
798
799 //! Series functor to compute series term using pow and factorial.
800 //! \details Functor is called after evaluating polynomial with the coefficients as rationals below.
801 template <class T>
802 struct lambert_w0_small_z_series_term
803 {
804 typedef T result_type;
805 //! \param _z Lambert W argument z.
806 //! \param -term -pow<18>(z) / 6402373705728000uLL
807 //! \param _k number of terms == initially 18
808
809 // Note *after* evaluating N terms, its internal state has k = N and term = (-1)^N z^N.
810
lambert_w0_small_z_series_termboost::math::lambert_w_detail::lambert_w0_small_z_series_term811 lambert_w0_small_z_series_term(T _z, T _term, int _k)
812 : k(_k), z(_z), term(_term) { }
813
operator ()boost::math::lambert_w_detail::lambert_w0_small_z_series_term814 T operator()()
815 { // Called by sum_series until needs precision set by factor (policy::get_epsilon).
816 using std::pow;
817 ++k;
818 term *= -z / k;
819 //T t = pow(z, k) * pow(T(k), -1 + k) / factorial<T>(k); // (z^k * k(k-1)^k) / k!
820 T result = term * pow(T(k), -1 + k); // term * k^(k-1)
821 // std::cout << " k = " << k << ", term = " << term << ", result = " << result << std::endl;
822 return result; //
823 }
824 private:
825 int k;
826 T z;
827 T term;
828 }; // template <class T> struct lambert_w0_small_z_series_term
829
830 //! Generic variant for T a User-defined types like Boost.Multiprecision.
831 template <class T, class Policy>
lambert_w0_small_z(T z,const Policy & pol,boost::integral_constant<int,5> const &)832 inline T lambert_w0_small_z(T z, const Policy& pol, boost::integral_constant<int, 5> const&)
833 {
834 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
835 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
836 std::cout << "Generic lambert_w0_small_z called with z = " << z << " using as many terms needed for precision." << std::endl;
837 std::cout << "Argument z is of type " << typeid(T).name() << std::endl;
838 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
839
840 // First several terms of the series are tabulated and evaluated as a polynomial:
841 // this will save us a bunch of expensive calls to pow.
842 // Then our series functor is initialized "as if" it had already reached term 18,
843 // enough evaluation of built-in 64-bit double and float (and 80-bit long double?) types.
844
845 // Coefficients should be stored such that the coefficients for the x^i terms are in poly[i].
846 static const T coeff[] =
847 {
848 0, // z^0 Care: zeroth term needed by tools::evaluate_polynomial, but not in the Wolfram equation, so indexes are one different!
849 1, // z^1 term.
850 -1, // z^2 term
851 static_cast<T>(3uLL) / 2uLL, // z^3 term.
852 -static_cast<T>(8uLL) / 3uLL, // z^4
853 static_cast<T>(125uLL) / 24uLL, // z^5
854 -static_cast<T>(54uLL) / 5uLL, // z^6
855 static_cast<T>(16807uLL) / 720uLL, // z^7
856 -static_cast<T>(16384uLL) / 315uLL, // z^8
857 static_cast<T>(531441uLL) / 4480uLL, // z^9
858 -static_cast<T>(156250uLL) / 567uLL, // z^10
859 static_cast<T>(2357947691uLL) / 3628800uLL, // z^11
860 -static_cast<T>(2985984uLL) / 1925uLL, // z^12
861 static_cast<T>(1792160394037uLL) / 479001600uLL, // z^13
862 -static_cast<T>(7909306972uLL) / 868725uLL, // z^14
863 static_cast<T>(320361328125uLL) / 14350336uLL, // z^15
864 -static_cast<T>(35184372088832uLL) / 638512875uLL, // z^16
865 static_cast<T>(2862423051509815793uLL) / 20922789888000uLL, // z^17 term
866 -static_cast<T>(5083731656658uLL) / 14889875uLL,
867 // z^18 term. = 136808.86090394293563342215789305735851647769682393
868
869 // z^18 is biggest that can be computed as rational using the largest possible uLL integers,
870 // so higher terms cannot be potentially compiler-computed as uLL rationals.
871 // Wolfram (5083731656658 z ^ 18) / 14889875 or
872 // -341422.05066583836331735491399356945575432970390954 z^18
873
874 // See note below calling the functor to compute another term,
875 // sufficient for 80-bit long double precision.
876 // Wolfram -341422.05066583836331735491399356945575432970390954 z^19 term.
877 // (5480386857784802185939 z^19)/6402373705728000
878 // But now this variant is not used to compute long double
879 // as specializations are provided above.
880 }; // static const T coeff[]
881
882 /*
883 Table of 19 computed coefficients:
884
885 #0 0
886 #1 1
887 #2 -1
888 #3 1.5
889 #4 -2.6666666666666666666666666666666665382713370408509
890 #5 5.2083333333333333333333333333333330765426740817019
891 #6 -10.800000000000000000000000000000000616297582203915
892 #7 23.343055555555555555555555555555555076212991619177
893 #8 -52.012698412698412698412698412698412659282693193402
894 #9 118.62522321428571428571428571428571146835390992496
895 #10 -275.57319223985890652557319223985891400375196748314
896 #11 649.7871723434744268077601410934743969785223845882
897 #12 -1551.1605194805194805194805194805194947599566007429
898 #13 3741.4497029592385495163272941050719510009019331763
899 #14 -9104.5002411580189357967135744913524243896052869184
900 #15 22324.308512706601434280005708577137322392070452582
901 #16 -55103.621972903835337697771560205423203318720697224
902 #17 136808.86090394293563342215789305735851647769682393
903 136808.86090394293563342215789305735851647769682393 == Exactly same as Wolfram computed value.
904 #18 -341422.05066583836331735491399356947486381600607416
905 341422.05066583836331735491399356945575432970390954 z^19 Wolfram value differs at 36 decimal digit, as expected.
906 */
907
908 using boost::math::policies::get_epsilon; // for type T.
909 using boost::math::tools::sum_series;
910 using boost::math::tools::evaluate_polynomial;
911 // http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/roots/rational.html
912
913 // std::streamsize prec = std::cout.precision(std::numeric_limits <T>::max_digits10);
914
915 T result = evaluate_polynomial(coeff, z);
916 // template <std::size_t N, class T, class V>
917 // V evaluate_polynomial(const T(&poly)[N], const V& val);
918 // Size of coeff found from N
919 //std::cout << "evaluate_polynomial(coeff, z); == " << result << std::endl;
920 //std::cout << "result = " << result << std::endl;
921 // It's an artefact of the way I wrote the functor: *after* evaluating N
922 // terms, its internal state has k = N and term = (-1)^N z^N. So after
923 // evaluating 18 terms, we initialize the functor to the term we've just
924 // evaluated, and then when it's called, it increments itself to the next term.
925 // So 18!is 6402373705728000, which is where that comes from.
926
927 // The 19th coefficient of the polynomial is actually, 19 ^ 18 / 19!=
928 // 104127350297911241532841 / 121645100408832000 which after removing GCDs
929 // reduces down to Wolfram rational 5480386857784802185939 / 6402373705728000.
930 // Wolfram z^19 term +(5480386857784802185939 z^19) /6402373705728000
931 // +855992.96599660755146336302506332246623424823099755 z^19
932
933 //! Evaluate Functor.
934 lambert_w0_small_z_series_term<T> s(z, -pow<18>(z) / 6402373705728000uLL, 18);
935
936 // Temporary to list the coefficients.
937 //std::cout << " Table of coefficients" << std::endl;
938 //std::streamsize saved_precision = std::cout.precision(50);
939 //for (size_t i = 0; i != 19; i++)
940 //{
941 // std::cout << "#" << i << " " << coeff[i] << std::endl;
942 //}
943 //std::cout.precision(saved_precision);
944
945 boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); // Max iterations from policy.
946 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
947 std::cout << "max iter from policy = " << max_iter << std::endl;
948 // // max iter from policy = 1000000 is default.
949 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
950
951 result = sum_series(s, get_epsilon<T, Policy>(), max_iter, result);
952 // result == evaluate_polynomial.
953 //sum_series(Functor& func, int bits, boost::uintmax_t& max_terms, const U& init_value)
954 // std::cout << "sum_series(s, get_epsilon<T, Policy>(), max_iter, result); = " << result << std::endl;
955
956 //T epsilon = get_epsilon<T, Policy>();
957 //std::cout << "epsilon from policy = " << epsilon << std::endl;
958 // epsilon from policy = 1.93e-34 for T == quad
959 // 5.35e-51 for t = cpp_bin_float_50
960
961 // std::cout << " get eps = " << get_epsilon<T, Policy>() << std::endl; // quad eps = 1.93e-34, bin_float_50 eps = 5.35e-51
962 policies::check_series_iterations<T>("boost::math::lambert_w0_small_z<%1%>(%1%)", max_iter, pol);
963 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS
964 std::cout << "z = " << z << " needed " << max_iter << " iterations." << std::endl;
965 std::cout.precision(prec); // Restore.
966 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS
967 return result;
968 } // template <class T, class Policy> inline T lambert_w0_small_z_series(T z, const Policy& pol)
969
970 // Approximate lambert_w0 (used for z values that are outside range of lookup table or rational functions)
971 // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.
972 template <typename T>
973 inline
lambert_w0_approx(T z)974 T lambert_w0_approx(T z)
975 {
976 BOOST_MATH_STD_USING
977 T lz = log(z);
978 T llz = log(lz);
979 T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.
980 return w;
981 // std::cout << "w max " << max_w << std::endl; // double 703.227
982 }
983
984 //////////////////////////////////////////////////////////////////////////////////////////
985
986 //! \brief Lambert_w0 implementations for float, double and higher precisions.
987 //! 3rd parameter used to select which version is used.
988
989 //! /details Rational polynomials are provided for several range of argument z.
990 //! For very small values of z, and for z very near the branch singularity at -e^-1 (~= -0.367879),
991 //! two other series functions are used.
992
993 //! float precision polynomials are used for 32-bit (usually float) precision (for speed)
994 //! double precision polynomials are used for 64-bit (usually double) precision.
995 //! For higher precisions, a 64-bit double approximation is computed first,
996 //! and then refined using Halley iterations.
997
998 template <class T>
do_get_near_singularity_param(T z)999 inline T do_get_near_singularity_param(T z)
1000 {
1001 BOOST_MATH_STD_USING
1002 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
1003 const T p = sqrt(p2);
1004 return p;
1005 }
1006 template <class T, class Policy>
get_near_singularity_param(T z,const Policy)1007 inline T get_near_singularity_param(T z, const Policy)
1008 {
1009 typedef typename policies::evaluation<T, Policy>::type value_type;
1010 return static_cast<T>(do_get_near_singularity_param(static_cast<value_type>(z)));
1011 }
1012
1013 // Forward declarations:
1014
1015 //template <class T, class Policy> T lambert_w0_small_z(T z, const Policy& pol);
1016 //template <class T, class Policy>
1017 //T lambert_w0_imp(T w, const Policy& pol, const boost::integral_constant<int, 0>&); // 32 bit usually float.
1018 //template <class T, class Policy>
1019 //T lambert_w0_imp(T w, const Policy& pol, const boost::integral_constant<int, 1>&); // 64 bit usually double.
1020 //template <class T, class Policy>
1021 //T lambert_w0_imp(T w, const Policy& pol, const boost::integral_constant<int, 2>&); // 80-bit long double.
1022
1023 template <class T>
lambert_w_positive_rational_float(T z)1024 T lambert_w_positive_rational_float(T z)
1025 {
1026 BOOST_MATH_STD_USING
1027 if (z < 2)
1028 {
1029 if (z < 0.5)
1030 { // 0.05 < z < 0.5
1031 // Maximum Deviation Found: 2.993e-08
1032 // Expected Error Term : 2.993e-08
1033 // Maximum Relative Change in Control Points : 7.555e-04 Y offset : -8.196592331e-01
1034 static const T Y = 8.196592331e-01f;
1035 static const T P[] = {
1036 1.803388345e-01f,
1037 -4.820256838e-01f,
1038 -1.068349741e+00f,
1039 -3.506624319e-02f,
1040 };
1041 static const T Q[] = {
1042 1.000000000e+00f,
1043 2.871703469e+00f,
1044 1.690949264e+00f,
1045 };
1046 return z * (Y + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z));
1047 }
1048 else
1049 { // 0.5 < z < 2
1050 // Max error in interpolated form: 1.018e-08
1051 static const T Y = 5.503368378e-01f;
1052 static const T P[] = {
1053 4.493332766e-01f,
1054 2.543432707e-01f,
1055 -4.808788799e-01f,
1056 -1.244425316e-01f,
1057 };
1058 static const T Q[] = {
1059 1.000000000e+00f,
1060 2.780661241e+00f,
1061 1.830840318e+00f,
1062 2.407221031e-01f,
1063 };
1064 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
1065 }
1066 }
1067 else if (z < 6)
1068 {
1069 // 2 < z < 6
1070 // Max error in interpolated form: 2.944e-08
1071 static const T Y = 1.162393570e+00f;
1072 static const T P[] = {
1073 -1.144183394e+00f,
1074 -4.712732855e-01f,
1075 1.563162512e-01f,
1076 1.434010911e-02f,
1077 };
1078 static const T Q[] = {
1079 1.000000000e+00f,
1080 1.192626340e+00f,
1081 2.295580708e-01f,
1082 5.477869455e-03f,
1083 };
1084 return Y + boost::math::tools::evaluate_rational(P, Q, z);
1085 }
1086 else if (z < 18)
1087 {
1088 // 6 < z < 18
1089 // Max error in interpolated form: 5.893e-08
1090 static const T Y = 1.809371948e+00f;
1091 static const T P[] = {
1092 -1.689291769e+00f,
1093 -3.337812742e-01f,
1094 3.151434873e-02f,
1095 1.134178734e-03f,
1096 };
1097 static const T Q[] = {
1098 1.000000000e+00f,
1099 5.716915685e-01f,
1100 4.489521292e-02f,
1101 4.076716763e-04f,
1102 };
1103 return Y + boost::math::tools::evaluate_rational(P, Q, z);
1104 }
1105 else if (z < 9897.12905874) // 2.8 < log(z) < 9.2
1106 {
1107 // Max error in interpolated form: 1.771e-08
1108 static const T Y = -1.402973175e+00f;
1109 static const T P[] = {
1110 1.966174312e+00f,
1111 2.350864728e-01f,
1112 -5.098074353e-02f,
1113 -1.054818339e-02f,
1114 };
1115 static const T Q[] = {
1116 1.000000000e+00f,
1117 4.388208264e-01f,
1118 8.316639634e-02f,
1119 3.397187918e-03f,
1120 -1.321489743e-05f,
1121 };
1122 T log_w = log(z);
1123 return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);
1124 }
1125 else if (z < 7.896296e+13) // 9.2 < log(z) <= 32
1126 {
1127 // Max error in interpolated form: 5.821e-08
1128 static const T Y = -2.735729218e+00f;
1129 static const T P[] = {
1130 3.424903470e+00f,
1131 7.525631787e-02f,
1132 -1.427309584e-02f,
1133 -1.435974178e-05f,
1134 };
1135 static const T Q[] = {
1136 1.000000000e+00f,
1137 2.514005579e-01f,
1138 6.118994652e-03f,
1139 -1.357889535e-05f,
1140 7.312865624e-08f,
1141 };
1142 T log_w = log(z);
1143 return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);
1144 }
1145 else // 32 < log(z) < 100
1146 {
1147 // Max error in interpolated form: 1.491e-08
1148 static const T Y = -4.012863159e+00f;
1149 static const T P[] = {
1150 4.431629226e+00f,
1151 2.756690487e-01f,
1152 -2.992956930e-03f,
1153 -4.912259384e-05f,
1154 };
1155 static const T Q[] = {
1156 1.000000000e+00f,
1157 2.015434591e-01f,
1158 4.949426142e-03f,
1159 1.609659944e-05f,
1160 -5.111523436e-09f,
1161 };
1162 T log_w = log(z);
1163 return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);
1164 }
1165 }
1166
1167 template <class T, class Policy>
1168 T lambert_w_negative_rational_float(T z, const Policy& pol)
1169 {
1170 BOOST_MATH_STD_USING
1171 if (z > -0.27)
1172 {
1173 if (z < -0.051)
1174 {
1175 // -0.27 < z < -0.051
1176 // Max error in interpolated form: 5.080e-08
1177 static const T Y = 1.255809784e+00f;
1178 static const T P[] = {
1179 -2.558083412e-01f,
1180 -2.306524098e+00f,
1181 -5.630887033e+00f,
1182 -3.803974556e+00f,
1183 };
1184 static const T Q[] = {
1185 1.000000000e+00f,
1186 5.107680783e+00f,
1187 7.914062868e+00f,
1188 3.501498501e+00f,
1189 };
1190 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
1191 }
1192 else
1193 {
1194 // Very small z so use a series function.
1195 return lambert_w0_small_z(z, pol);
1196 }
1197 }
1198 else if (z > -0.3578794411714423215955237701)
1199 { // Very close to branch singularity.
1200 // Max error in interpolated form: 5.269e-08
1201 static const T Y = 1.220928431e-01f;
1202 static const T P[] = {
1203 -1.221787446e-01f,
1204 -6.816155875e+00f,
1205 7.144582035e+01f,
1206 1.128444390e+03f,
1207 };
1208 static const T Q[] = {
1209 1.000000000e+00f,
1210 6.480326790e+01f,
1211 1.869145243e+02f,
1212 -1.361804274e+03f,
1213 1.117826726e+03f,
1214 };
1215 T d = z + 0.367879441171442321595523770161460867445811f;
1216 return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));
1217 }
1218 else
1219 {
1220 // z is very close (within 0.01) of the singularity at e^-1.
1221 return lambert_w_singularity_series(get_near_singularity_param(z, pol));
1222 }
1223 }
1224
1225 //! Lambert_w0 @b 'float' implementation, selected when T is 32-bit precision.
1226 template <class T, class Policy>
lambert_w0_imp(T z,const Policy & pol,const boost::integral_constant<int,1> &)1227 inline T lambert_w0_imp(T z, const Policy& pol, const boost::integral_constant<int, 1>&)
1228 {
1229 static const char* function = "boost::math::lambert_w0<%1%>"; // For error messages.
1230 BOOST_MATH_STD_USING // Aid ADL of std functions.
1231
1232 if ((boost::math::isnan)(z))
1233 {
1234 return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol);
1235 }
1236 if ((boost::math::isinf)(z))
1237 {
1238 return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol);
1239 }
1240
1241 if (z >= 0.05) // Fukushima switch point.
1242 // if (z >= 0.045) // 34 terms makes 128-bit 'exact' below 0.045.
1243 { // Normal ranges using several rational polynomials.
1244 return lambert_w_positive_rational_float(z);
1245 }
1246 else if (z <= -0.3678794411714423215955237701614608674458111310f)
1247 {
1248 if (z < -0.3678794411714423215955237701614608674458111310f)
1249 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
1250 return -1;
1251 }
1252 else // z < 0.05
1253 {
1254 return lambert_w_negative_rational_float(z, pol);
1255 }
1256 } // T lambert_w0_imp(T z, const Policy& pol, const boost::integral_constant<int, 1>&) for 32-bit usually float.
1257
1258 template <class T>
lambert_w_positive_rational_double(T z)1259 T lambert_w_positive_rational_double(T z)
1260 {
1261 BOOST_MATH_STD_USING
1262 if (z < 2)
1263 {
1264 if (z < 0.5)
1265 {
1266 // Max error in interpolated form: 2.255e-17
1267 static const T offset = 8.19659233093261719e-01;
1268 static const T P[] = {
1269 1.80340766906685177e-01,
1270 3.28178241493119307e-01,
1271 -2.19153620687139706e+00,
1272 -7.24750929074563990e+00,
1273 -7.28395876262524204e+00,
1274 -2.57417169492512916e+00,
1275 -2.31606948888704503e-01
1276 };
1277 static const T Q[] = {
1278 1.00000000000000000e+00,
1279 7.36482529307436604e+00,
1280 2.03686007856430677e+01,
1281 2.62864592096657307e+01,
1282 1.59742041380858333e+01,
1283 4.03760534788374589e+00,
1284 2.91327346750475362e-01
1285 };
1286 return z * (offset + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z));
1287 }
1288 else
1289 {
1290 // Max error in interpolated form: 3.806e-18
1291 static const T offset = 5.50335884094238281e-01;
1292 static const T P[] = {
1293 4.49664083944098322e-01,
1294 1.90417666196776909e+00,
1295 1.99951368798255994e+00,
1296 -6.91217310299270265e-01,
1297 -1.88533935998617058e+00,
1298 -7.96743968047750836e-01,
1299 -1.02891726031055254e-01,
1300 -3.09156013592636568e-03
1301 };
1302 static const T Q[] = {
1303 1.00000000000000000e+00,
1304 6.45854489419584014e+00,
1305 1.54739232422116048e+01,
1306 1.72606164253337843e+01,
1307 9.29427055609544096e+00,
1308 2.29040824649748117e+00,
1309 2.21610620995418981e-01,
1310 5.70597669908194213e-03
1311 };
1312 return z * (offset + boost::math::tools::evaluate_rational(P, Q, z));
1313 }
1314 }
1315 else if (z < 6)
1316 {
1317 // 2 < z < 6
1318 // Max error in interpolated form: 1.216e-17
1319 static const T Y = 1.16239356994628906e+00;
1320 static const T P[] = {
1321 -1.16230494982099475e+00,
1322 -3.38528144432561136e+00,
1323 -2.55653717293161565e+00,
1324 -3.06755172989214189e-01,
1325 1.73149743765268289e-01,
1326 3.76906042860014206e-02,
1327 1.84552217624706666e-03,
1328 1.69434126904822116e-05,
1329 };
1330 static const T Q[] = {
1331 1.00000000000000000e+00,
1332 3.77187616711220819e+00,
1333 4.58799960260143701e+00,
1334 2.24101228462292447e+00,
1335 4.54794195426212385e-01,
1336 3.60761772095963982e-02,
1337 9.25176499518388571e-04,
1338 4.43611344705509378e-06,
1339 };
1340 return Y + boost::math::tools::evaluate_rational(P, Q, z);
1341 }
1342 else if (z < 18)
1343 {
1344 // 6 < z < 18
1345 // Max error in interpolated form: 1.985e-19
1346 static const T offset = 1.80937194824218750e+00;
1347 static const T P[] =
1348 {
1349 -1.80690935424793635e+00,
1350 -3.66995929380314602e+00,
1351 -1.93842957940149781e+00,
1352 -2.94269984375794040e-01,
1353 1.81224710627677778e-03,
1354 2.48166798603547447e-03,
1355 1.15806592415397245e-04,
1356 1.43105573216815533e-06,
1357 3.47281483428369604e-09
1358 };
1359 static const T Q[] = {
1360 1.00000000000000000e+00,
1361 2.57319080723908597e+00,
1362 1.96724528442680658e+00,
1363 5.84501352882650722e-01,
1364 7.37152837939206240e-02,
1365 3.97368430940416778e-03,
1366 8.54941838187085088e-05,
1367 6.05713225608426678e-07,
1368 8.17517283816615732e-10
1369 };
1370 return offset + boost::math::tools::evaluate_rational(P, Q, z);
1371 }
1372 else if (z < 9897.12905874) // 2.8 < log(z) < 9.2
1373 {
1374 // Max error in interpolated form: 1.195e-18
1375 static const T Y = -1.40297317504882812e+00;
1376 static const T P[] = {
1377 1.97011826279311924e+00,
1378 1.05639945701546704e+00,
1379 3.33434529073196304e-01,
1380 3.34619153200386816e-02,
1381 -5.36238353781326675e-03,
1382 -2.43901294871308604e-03,
1383 -2.13762095619085404e-04,
1384 -4.85531936495542274e-06,
1385 -2.02473518491905386e-08,
1386 };
1387 static const T Q[] = {
1388 1.00000000000000000e+00,
1389 8.60107275833921618e-01,
1390 4.10420467985504373e-01,
1391 1.18444884081994841e-01,
1392 2.16966505556021046e-02,
1393 2.24529766630769097e-03,
1394 9.82045090226437614e-05,
1395 1.36363515125489502e-06,
1396 3.44200749053237945e-09,
1397 };
1398 T log_w = log(z);
1399 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
1400 }
1401 else if (z < 7.896296e+13) // 9.2 < log(z) <= 32
1402 {
1403 // Max error in interpolated form: 6.529e-18
1404 static const T Y = -2.73572921752929688e+00;
1405 static const T P[] = {
1406 3.30547638424076217e+00,
1407 1.64050071277550167e+00,
1408 4.57149576470736039e-01,
1409 4.03821227745424840e-02,
1410 -4.99664976882514362e-04,
1411 -1.28527893803052956e-04,
1412 -2.95470325373338738e-06,
1413 -1.76662025550202762e-08,
1414 -1.98721972463709290e-11,
1415 };
1416 static const T Q[] = {
1417 1.00000000000000000e+00,
1418 6.91472559412458759e-01,
1419 2.48154578891676774e-01,
1420 4.60893578284335263e-02,
1421 3.60207838982301946e-03,
1422 1.13001153242430471e-04,
1423 1.33690948263488455e-06,
1424 4.97253225968548872e-09,
1425 3.39460723731970550e-12,
1426 };
1427 T log_w = log(z);
1428 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
1429 }
1430 else if (z < 2.6881171e+43) // 32 < log(z) < 100
1431 {
1432 // Max error in interpolated form: 2.015e-18
1433 static const T Y = -4.01286315917968750e+00;
1434 static const T P[] = {
1435 5.07714858354309672e+00,
1436 -3.32994414518701458e+00,
1437 -8.61170416909864451e-01,
1438 -4.01139705309486142e-02,
1439 -1.85374201771834585e-04,
1440 1.08824145844270666e-05,
1441 1.17216905810452396e-07,
1442 2.97998248101385990e-10,
1443 1.42294856434176682e-13,
1444 };
1445 static const T Q[] = {
1446 1.00000000000000000e+00,
1447 -4.85840770639861485e-01,
1448 -3.18714850604827580e-01,
1449 -3.20966129264610534e-02,
1450 -1.06276178044267895e-03,
1451 -1.33597828642644955e-05,
1452 -6.27900905346219472e-08,
1453 -9.35271498075378319e-11,
1454 -2.60648331090076845e-14,
1455 };
1456 T log_w = log(z);
1457 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
1458 }
1459 else // 100 < log(z) < 710
1460 {
1461 // Max error in interpolated form: 5.277e-18
1462 static const T Y = -5.70115661621093750e+00;
1463 static const T P[] = {
1464 6.42275660145116698e+00,
1465 1.33047964073367945e+00,
1466 6.72008923401652816e-02,
1467 1.16444069958125895e-03,
1468 7.06966760237470501e-06,
1469 5.48974896149039165e-09,
1470 -7.00379652018853621e-11,
1471 -1.89247635913659556e-13,
1472 -1.55898770790170598e-16,
1473 -4.06109208815303157e-20,
1474 -2.21552699006496737e-24,
1475 };
1476 static const T Q[] = {
1477 1.00000000000000000e+00,
1478 3.34498588416632854e-01,
1479 2.51519862456384983e-02,
1480 6.81223810622416254e-04,
1481 7.94450897106903537e-06,
1482 4.30675039872881342e-08,
1483 1.10667669458467617e-10,
1484 1.31012240694192289e-13,
1485 6.53282047177727125e-17,
1486 1.11775518708172009e-20,
1487 3.78250395617836059e-25,
1488 };
1489 T log_w = log(z);
1490 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
1491 }
1492 }
1493
1494 template <class T, class Policy>
1495 T lambert_w_negative_rational_double(T z, const Policy& pol)
1496 {
1497 BOOST_MATH_STD_USING
1498 if (z > -0.1)
1499 {
1500 if (z < -0.051)
1501 {
1502 // -0.1 < z < -0.051
1503 // Maximum Deviation Found: 4.402e-22
1504 // Expected Error Term : 4.240e-22
1505 // Maximum Relative Change in Control Points : 4.115e-03
1506 static const T Y = 1.08633995056152344e+00;
1507 static const T P[] = {
1508 -8.63399505615014331e-02,
1509 -1.64303871814816464e+00,
1510 -7.71247913918273738e+00,
1511 -1.41014495545382454e+01,
1512 -1.02269079949257616e+01,
1513 -2.17236002836306691e+00,
1514 };
1515 static const T Q[] = {
1516 1.00000000000000000e+00,
1517 7.44775406945739243e+00,
1518 2.04392643087266541e+01,
1519 2.51001961077774193e+01,
1520 1.31256080849023319e+01,
1521 2.11640324843601588e+00,
1522 };
1523 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
1524 }
1525 else
1526 {
1527 // Very small z > 0.051:
1528 return lambert_w0_small_z(z, pol);
1529 }
1530 }
1531 else if (z > -0.2)
1532 {
1533 // -0.2 < z < -0.1
1534 // Maximum Deviation Found: 2.898e-20
1535 // Expected Error Term : 2.873e-20
1536 // Maximum Relative Change in Control Points : 3.779e-04
1537 static const T Y = 1.20359611511230469e+00;
1538 static const T P[] = {
1539 -2.03596115108465635e-01,
1540 -2.95029082937201859e+00,
1541 -1.54287922188671648e+01,
1542 -3.81185809571116965e+01,
1543 -4.66384358235575985e+01,
1544 -2.59282069989642468e+01,
1545 -4.70140451266553279e+00,
1546 };
1547 static const T Q[] = {
1548 1.00000000000000000e+00,
1549 9.57921436074599929e+00,
1550 3.60988119290234377e+01,
1551 6.73977699505546007e+01,
1552 6.41104992068148823e+01,
1553 2.82060127225153607e+01,
1554 4.10677610657724330e+00,
1555 };
1556 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
1557 }
1558 else if (z > -0.3178794411714423215955237)
1559 {
1560 // Max error in interpolated form: 6.996e-18
1561 static const T Y = 3.49680423736572266e-01;
1562 static const T P[] = {
1563 -3.49729841718749014e-01,
1564 -6.28207407760709028e+01,
1565 -2.57226178029669171e+03,
1566 -2.50271008623093747e+04,
1567 1.11949239154711388e+05,
1568 1.85684566607844318e+06,
1569 4.80802490427638643e+06,
1570 2.76624752134636406e+06,
1571 };
1572 static const T Q[] = {
1573 1.00000000000000000e+00,
1574 1.82717661215113000e+02,
1575 8.00121119810280100e+03,
1576 1.06073266717010129e+05,
1577 3.22848993926057721e+05,
1578 -8.05684814514171256e+05,
1579 -2.59223192927265737e+06,
1580 -5.61719645211570871e+05,
1581 6.27765369292636844e+04,
1582 };
1583 T d = z + 0.367879441171442321595523770161460867445811;
1584 return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));
1585 }
1586 else if (z > -0.3578794411714423215955237701)
1587 {
1588 // Max error in interpolated form: 1.404e-17
1589 static const T Y = 5.00126481056213379e-02;
1590 static const T P[] = {
1591 -5.00173570682372162e-02,
1592 -4.44242461870072044e+01,
1593 -9.51185533619946042e+03,
1594 -5.88605699015429386e+05,
1595 -1.90760843597427751e+06,
1596 5.79797663818311404e+08,
1597 1.11383352508459134e+10,
1598 5.67791253678716467e+10,
1599 6.32694500716584572e+10,
1600 };
1601 static const T Q[] = {
1602 1.00000000000000000e+00,
1603 9.08910517489981551e+02,
1604 2.10170163753340133e+05,
1605 1.67858612416470327e+07,
1606 4.90435561733227953e+08,
1607 4.54978142622939917e+09,
1608 2.87716585708739168e+09,
1609 -4.59414247951143131e+10,
1610 -1.72845216404874299e+10,
1611 };
1612 T d = z + 0.36787944117144232159552377016146086744581113103176804;
1613 return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));
1614 }
1615 else
1616 { // z is very close (within 0.01) of the singularity at -e^-1,
1617 // so use a series expansion from R. M. Corless et al.
1618 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
1619 const T p = sqrt(p2);
1620 return lambert_w_detail::lambert_w_singularity_series(p);
1621 }
1622 }
1623
1624 //! Lambert_w0 @b 'double' implementation, selected when T is 64-bit precision.
1625 template <class T, class Policy>
lambert_w0_imp(T z,const Policy & pol,const boost::integral_constant<int,2> &)1626 inline T lambert_w0_imp(T z, const Policy& pol, const boost::integral_constant<int, 2>&)
1627 {
1628 static const char* function = "boost::math::lambert_w0<%1%>";
1629 BOOST_MATH_STD_USING // Aid ADL of std functions.
1630
1631 // Detect unusual case of 32-bit double with a wider/64-bit long double
1632 BOOST_STATIC_ASSERT_MSG(std::numeric_limits<double>::digits >= 53,
1633 "Our double precision coefficients will be truncated, "
1634 "please file a bug report with details of your platform's floating point types "
1635 "- or possibly edit the coefficients to have "
1636 "an appropriate size-suffix for 64-bit floats on your platform - L?");
1637
1638 if ((boost::math::isnan)(z))
1639 {
1640 return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol);
1641 }
1642 if ((boost::math::isinf)(z))
1643 {
1644 return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol);
1645 }
1646
1647 if (z >= 0.05)
1648 {
1649 return lambert_w_positive_rational_double(z);
1650 }
1651 else if (z <= -0.36787944117144232159552377016146086744581113103176804) // Precision is max_digits10(cpp_bin_float_50).
1652 {
1653 if (z < -0.36787944117144232159552377016146086744581113103176804)
1654 {
1655 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
1656 }
1657 return -1;
1658 }
1659 else
1660 {
1661 return lambert_w_negative_rational_double(z, pol);
1662 }
1663 } // T lambert_w0_imp(T z, const Policy& pol, const boost::integral_constant<int, 2>&) 64-bit precision, usually double.
1664
1665 //! lambert_W0 implementation for extended precision types including
1666 //! long double (80-bit and 128-bit), ???
1667 //! quad float128, Boost.Multiprecision types like cpp_bin_float_quad, cpp_bin_float_50...
1668
1669 template <class T, class Policy>
lambert_w0_imp(T z,const Policy & pol,const boost::integral_constant<int,0> &)1670 inline T lambert_w0_imp(T z, const Policy& pol, const boost::integral_constant<int, 0>&)
1671 {
1672 static const char* function = "boost::math::lambert_w0<%1%>";
1673 BOOST_MATH_STD_USING // Aid ADL of std functions.
1674
1675 // Filter out special cases first:
1676 if ((boost::math::isnan)(z))
1677 {
1678 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
1679 }
1680 if (fabs(z) <= 0.05f)
1681 {
1682 // Very small z:
1683 return lambert_w0_small_z(z, pol);
1684 }
1685 if (z > (std::numeric_limits<double>::max)())
1686 {
1687 if ((boost::math::isinf)(z))
1688 {
1689 return policies::raise_overflow_error<T>(function, 0, pol);
1690 // Or might return infinity if available else max_value,
1691 // but other Boost.Math special functions raise overflow.
1692 }
1693 // z is larger than the largest double, so cannot use the polynomial to get an approximation,
1694 // so use the asymptotic approximation and Halley iterate:
1695
1696 T w = lambert_w0_approx(z); // Make an inline function as also used elsewhere.
1697 //T lz = log(z);
1698 //T llz = log(lz);
1699 //T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.
1700 return lambert_w_halley_iterate(w, z);
1701 }
1702 if (z < -0.3578794411714423215955237701)
1703 { // Very close to branch point so rational polynomials are not usable.
1704 if (z <= -boost::math::constants::exp_minus_one<T>())
1705 {
1706 if (z == -boost::math::constants::exp_minus_one<T>())
1707 { // Exactly at the branch point singularity.
1708 return -1;
1709 }
1710 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
1711 }
1712 // z is very close (within 0.01) of the branch singularity at -e^-1
1713 // so use a series approximation proposed by Corless et al.
1714 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
1715 const T p = sqrt(p2);
1716 T w = lambert_w_detail::lambert_w_singularity_series(p);
1717 return lambert_w_halley_iterate(w, z);
1718 }
1719
1720 // Phew! If we get here we are in the normal range of the function,
1721 // so get a double precision approximation first, then iterate to full precision of T.
1722 // We define a tag_type that is:
1723 // true_type if there are so many digits precision wanted that iteration is necessary.
1724 // false_type if a single Halley step is sufficient.
1725
1726 typedef typename policies::precision<T, Policy>::type precision_type;
1727 typedef boost::integral_constant<bool,
1728 (precision_type::value == 0) || (precision_type::value > 113) ?
1729 true // Unknown at compile-time, variable/arbitrary, or more than float128 or cpp_bin_quad 128-bit precision.
1730 : false // float, double, float128, cpp_bin_quad 128-bit, so single Halley step.
1731 > tag_type;
1732
1733 // For speed, we also cast z to type double when that is possible
1734 // if (boost::is_constructible<double, T>() == true).
1735 T w = lambert_w0_imp(maybe_reduce_to_double(z, boost::is_constructible<double, T>()), pol, boost::integral_constant<int, 2>());
1736
1737 return lambert_w_maybe_halley_iterate(w, z, tag_type());
1738
1739 } // T lambert_w0_imp(T z, const Policy& pol, const boost::integral_constant<int, 0>&) all extended precision types.
1740
1741 // Lambert w-1 implementation
1742 // ==============================================================================================
1743
1744 //! Lambert W for W-1 branch, -max(z) < z <= -1/e.
1745 // TODO is -max(z) allowed?
1746 template<typename T, class Policy>
1747 T lambert_wm1_imp(const T z, const Policy& pol)
1748 {
1749 // Catch providing an integer value as parameter x to lambert_w, for example, lambert_w(1).
1750 // Need to ensure it is a floating-point type (of the desired type, float 1.F, double 1., or long double 1.L),
1751 // or static_casted integer, for example: static_cast<float>(1) or static_cast<cpp_dec_float_50>(1).
1752 // Want to allow fixed_point types too, so do not just test for floating-point.
1753 // Integral types should be promoted to double by user Lambert w functions.
1754 // If integral type provided to user function lambert_w0 or lambert_wm1,
1755 // then should already have been promoted to double.
1756 BOOST_STATIC_ASSERT_MSG(!boost::is_integral<T>::value,
1757 "Must be floating-point or fixed type (not integer type), for example: lambert_wm1(1.), not lambert_wm1(1)!");
1758
1759 BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs.
1760
1761 const char* function = "boost::math::lambert_wm1<RealType>(<RealType>)"; // Used for error messages.
1762
1763 // Check for edge and corner cases first:
1764 if ((boost::math::isnan)(z))
1765 {
1766 return policies::raise_domain_error(function,
1767 "Argument z is NaN!",
1768 z, pol);
1769 } // isnan
1770
1771 if ((boost::math::isinf)(z))
1772 {
1773 return policies::raise_domain_error(function,
1774 "Argument z is infinite!",
1775 z, pol);
1776 } // isinf
1777
1778 if (z == static_cast<T>(0))
1779 { // z is exactly zero so return -std::numeric_limits<T>::infinity();
1780 if (std::numeric_limits<T>::has_infinity)
1781 {
1782 return -std::numeric_limits<T>::infinity();
1783 }
1784 else
1785 {
1786 return -tools::max_value<T>();
1787 }
1788 }
1789 if (std::numeric_limits<T>::has_denorm)
1790 { // All real types except arbitrary precision.
1791 if (!(boost::math::isnormal)(z))
1792 { // Almost zero - might also just return infinity like z == 0 or max_value?
1793 return policies::raise_overflow_error(function,
1794 "Argument z = %1% is denormalized! (must be z > (std::numeric_limits<RealType>::min)() or z == 0)",
1795 z, pol);
1796 }
1797 }
1798
1799 if (z > static_cast<T>(0))
1800 { //
1801 return policies::raise_domain_error(function,
1802 "Argument z = %1% is out of range (z <= 0) for Lambert W-1 branch! (Try Lambert W0 branch?)",
1803 z, pol);
1804 }
1805 if (z > -boost::math::tools::min_value<T>())
1806 { // z is denormalized, so cannot be computed.
1807 // -std::numeric_limits<T>::min() is smallest for type T,
1808 // for example, for double: lambert_wm1(-2.2250738585072014e-308) = -714.96865723796634
1809 return policies::raise_overflow_error(function,
1810 "Argument z = %1% is too small (z < -std::numeric_limits<T>::min so denormalized) for Lambert W-1 branch!",
1811 z, pol);
1812 }
1813 if (z == -boost::math::constants::exp_minus_one<T>()) // == singularity/branch point z = -exp(-1) = -3.6787944.
1814 { // At singularity, so return exactly -1.
1815 return -static_cast<T>(1);
1816 }
1817 // z is too negative for the W-1 (or W0) branch.
1818 if (z < -boost::math::constants::exp_minus_one<T>()) // > singularity/branch point z = -exp(-1) = -3.6787944.
1819 {
1820 return policies::raise_domain_error(function,
1821 "Argument z = %1% is out of range (z < -exp(-1) = -3.6787944... <= 0) for Lambert W-1 (or W0) branch!",
1822 z, pol);
1823 }
1824 if (z < static_cast<T>(-0.35))
1825 { // Close to singularity/branch point z = -0.3678794411714423215955237701614608727 but on W-1 branch.
1826 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
1827 if (p2 == 0)
1828 { // At the singularity at branch point.
1829 return -1;
1830 }
1831 if (p2 > 0)
1832 {
1833 T w_series = lambert_w_singularity_series(T(-sqrt(p2)));
1834 if (boost::math::tools::digits<T>() > 53)
1835 { // Multiprecision, so try a Halley refinement.
1836 w_series = lambert_w_detail::lambert_w_halley_iterate(w_series, z);
1837 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN
1838 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1839 std::cout << "Lambert W-1 Halley updated to " << w_series << std::endl;
1840 std::cout.precision(saved_precision);
1841 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN
1842 }
1843 return w_series;
1844 }
1845 // Should not get here.
1846 return policies::raise_domain_error(function,
1847 "Argument z = %1% is out of range for Lambert W-1 branch. (Should not get here - please report!)",
1848 z, pol);
1849 } // if (z < -0.35)
1850
1851 using lambert_w_lookup::wm1es;
1852 using lambert_w_lookup::wm1zs;
1853 using lambert_w_lookup::noof_wm1zs; // size == 64
1854
1855 // std::cout <<" Wm1zs[63] (== G[64]) = " << " " << wm1zs[63] << std::endl; // Wm1zs[63] (== G[64]) = -1.0264389699511283e-26
1856 // Check that z argument value is not smaller than lookup_table G[64]
1857 // std::cout << "(z > wm1zs[63]) = " << std::boolalpha << (z > wm1zs[63]) << std::endl;
1858
1859 if (z >= wm1zs[63]) // wm1zs[63] = -1.0264389699511282259046957018510946438e-26L W = 64.00000000000000000
1860 { // z >= -1.0264389699511303e-26 (but z != 0 and z >= std::numeric_limits<T>::min() and so NOT denormalized).
1861
1862 // Some info on Lambert W-1 values for extreme values of z.
1863 // std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1864 // std::cout << "-std::numeric_limits<float>::min() = " << -(std::numeric_limits<float>::min)() << std::endl;
1865 // std::cout << "-std::numeric_limits<double>::min() = " << -(std::numeric_limits<double>::min)() << std::endl;
1866 // -std::numeric_limits<float>::min() = -1.1754943508222875e-38
1867 // -std::numeric_limits<double>::min() = -2.2250738585072014e-308
1868 // N[productlog(-1, -1.1754943508222875 * 10^-38 ), 50] = -91.856775324595479509567756730093823993834155027858
1869 // N[productlog(-1, -2.2250738585072014e-308 * 10^-308 ), 50] = -1424.8544521230553853558132180518404363617968042942
1870 // N[productlog(-1, -1.4325445274604020119111357113179868158* 10^-27), 37] = -65.99999999999999999999999999999999955
1871
1872 // R.M.Corless, G.H.Gonnet, D.E.G.Hare, D.J.Jeffrey, and D.E.Knuth,
1873 // On the Lambert W function, Adv.Comput.Math., vol. 5, pp. 329, 1996.
1874 // Francois Chapeau-Blondeau and Abdelilah Monir
1875 // Numerical Evaluation of the Lambert W Function
1876 // IEEE Transactions On Signal Processing, VOL. 50, NO. 9, Sep 2002
1877 // https://pdfs.semanticscholar.org/7a5a/76a9369586dd0dd34dda156d8f2779d1fd59.pdf
1878 // Estimate Lambert W using ln(-z) ...
1879 // This is roughly the power of ten * ln(10) ~= 2.3. n ~= 10^n
1880 // and improve by adding a second term -ln(ln(-z))
1881 T guess; // bisect lowest possible Gk[=64] (for lookup_t type)
1882 T lz = log(-z);
1883 T llz = log(-lz);
1884 guess = lz - llz + (llz / lz); // Chapeau-Blondeau equation 20, page 2162.
1885 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY
1886 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1887 std::cout << "z = " << z << ", guess = " << guess << ", ln(-z) = " << lz << ", ln(-ln(-z) = " << llz << ", llz/lz = " << (llz / lz) << std::endl;
1888 // z = -1.0000000000000001e-30, guess = -73.312782616731482, ln(-z) = -69.077552789821368, ln(-ln(-z) = 4.2352298269101114, llz/lz = -0.061311231447304194
1889 // z = -9.9999999999999999e-91, guess = -212.56650048504233, ln(-z) = -207.23265836946410, ln(-ln(-z) = 5.3338421155782205, llz/lz = -0.025738424423764311
1890 // >z = -2.2250738585072014e-308, guess = -714.95942238244606, ln(-z) = -708.39641853226408, ln(-ln(-z) = 6.5630038501819854, llz/lz = -0.0092645920821846622
1891 int d10 = policies::digits_base10<T, Policy>(); // policy template parameter digits10
1892 int d2 = policies::digits<T, Policy>(); // digits base 2 from policy.
1893 std::cout << "digits10 = " << d10 << ", digits2 = " << d2 // For example: digits10 = 1, digits2 = 5
1894 << std::endl;
1895 std::cout.precision(saved_precision);
1896 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY
1897 if (policies::digits<T, Policy>() < 12)
1898 { // 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.
1899 return guess;
1900 }
1901 T result = lambert_w_detail::lambert_w_halley_iterate(guess, z);
1902 return result;
1903
1904 // Was Fukushima
1905 // G[k=64] == g[63] == -1.02643897e-26
1906 //return policies::raise_domain_error(function,
1907 // "Argument z = %1% is too small (< -1.02643897e-26) ! (Should not occur, please report.",
1908 // z, pol);
1909 } // Z too small so use approximation and Halley.
1910 // Else Use a lookup table to find the nearest integer part of Lambert W-1 as starting point for Bisection.
1911
1912 if (boost::math::tools::digits<T>() > 53)
1913 { // T is more precise than 64-bit double (or long double, or ?),
1914 // so compute an approximate value using only one Schroeder refinement,
1915 // (avoiding any double-precision Halley refinement from policy double_digits2<50> 53 - 3 = 50
1916 // because are next going to use Halley refinement at full/high precision using this as an approximation).
1917 using boost::math::policies::precision;
1918 using boost::math::policies::digits10;
1919 using boost::math::policies::digits2;
1920 using boost::math::policies::policy;
1921 // Compute a 50-bit precision approximate W0 in a double (no Halley refinement).
1922 T double_approx(static_cast<T>(lambert_wm1_imp(must_reduce_to_double(z, boost::is_constructible<double, T>()), policy<digits2<50> >())));
1923 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN
1924 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1925 std::cout << "Lambert_wm1 Argument Type " << typeid(T).name() << " approximation double = " << double_approx << std::endl;
1926 std::cout.precision(saved_precision);
1927 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1
1928 // Perform additional Halley refinement(s) to ensure that
1929 // get a near as possible to correct result (usually +/- one epsilon).
1930 T result = lambert_w_halley_iterate(double_approx, z);
1931 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1
1932 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1933 std::cout << "Result " << typeid(T).name() << " precision Halley refinement = " << result << std::endl;
1934 std::cout.precision(saved_precision);
1935 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1
1936 return result;
1937 } // digits > 53 - higher precision than double.
1938 else // T is double or less precision.
1939 { // Use a lookup table to find the nearest integer part of Lambert W as starting point for Bisection.
1940 using namespace boost::math::lambert_w_detail::lambert_w_lookup;
1941 // Bracketing sequence n = (2, 4, 8, 16, 32, 64) for W-1 branch. (0 is -infinity)
1942 // Since z is probably quite small, start with lowest n (=2).
1943 int n = 2;
1944 if (wm1zs[n - 1] > z)
1945 {
1946 goto bisect;
1947 }
1948 for (int j = 1; j <= 5; ++j)
1949 {
1950 n *= 2;
1951 if (wm1zs[n - 1] > z)
1952 {
1953 goto overshot;
1954 }
1955 }
1956 // else z < g[63] == -1.0264389699511303e-26, so Lambert W-1 integer part > 64.
1957 // This should not now occur (should be caught by test and code above) so should be a logic_error?
1958 return policies::raise_domain_error(function,
1959 "Argument z = %1% is too small (< -1.026439e-26) (logic error - please report!)",
1960 z, pol);
1961 overshot:
1962 {
1963 int nh = n / 2;
1964 for (int j = 1; j <= 5; ++j)
1965 {
1966 nh /= 2; // halve step size.
1967 if (nh <= 0)
1968 {
1969 break; // goto bisect;
1970 }
1971 if (wm1zs[n - nh - 1] > z)
1972 {
1973 n -= nh;
1974 }
1975 }
1976 }
1977 bisect:
1978 --n;
1979 // g[n] now holds lambert W of floor integer n and g[n+1] the ceil part;
1980 // these are used as initial values for bisection.
1981 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP
1982 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
1983 std::cout << "Result lookup W-1(" << z << ") bisection between wm1zs[" << n - 1 << "] = " << wm1zs[n - 1] << " and wm1zs[" << n << "] = " << wm1zs[n]
1984 << ", bisect mean = " << (wm1zs[n - 1] + wm1zs[n]) / 2 << std::endl;
1985 std::cout.precision(saved_precision);
1986 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP
1987
1988 // Compute bisections is the number of bisections computed from n,
1989 // such that a single application of the fifth-order Schroeder update formula
1990 // after the bisections is enough to evaluate Lambert W-1 with (near?) 53-bit accuracy.
1991 // Fukushima established these by trial and error?
1992 int bisections = 11; // Assume maximum number of bisections will be needed (most common case).
1993 if (n >= 8)
1994 {
1995 bisections = 8;
1996 }
1997 else if (n >= 3)
1998 {
1999 bisections = 9;
2000 }
2001 else if (n >= 2)
2002 {
2003 bisections = 10;
2004 }
2005 // Bracketing, Fukushima section 2.3, page 82:
2006 // (Avoiding using exponential function for speed).
2007 // Only use @c lookup_t precision, default double, for bisection (again for speed),
2008 // and use later Halley refinement for higher precisions.
2009 using lambert_w_lookup::halves;
2010 using lambert_w_lookup::sqrtwm1s;
2011
2012 typedef typename mpl::if_c<boost::is_constructible<lookup_t, T>::value, lookup_t, T>::type calc_type;
2013
2014 calc_type w = -static_cast<calc_type>(n); // Equation 25,
2015 calc_type y = static_cast<calc_type>(z * wm1es[n - 1]); // Equation 26,
2016 // Perform the bisections fractional bisections for necessary precision.
2017 for (int j = 0; j < bisections; ++j)
2018 { // Equation 27.
2019 calc_type wj = w - halves[j]; // Subtract 1/2, 1/4, 1/8 ...
2020 calc_type yj = y * sqrtwm1s[j]; // Multiply by sqrt(1/e), ...
2021 if (wj < yj)
2022 {
2023 w = wj;
2024 y = yj;
2025 }
2026 } // for j
2027 return static_cast<T>(schroeder_update(w, y)); // Schroeder 5th order method refinement.
2028
2029 // else // Perform additional Halley refinement(s) to ensure that
2030 // // get a near as possible to correct result (usually +/- epsilon).
2031 // {
2032 // // result = lambert_w_halley_iterate(result, z);
2033 // result = lambert_w_halley_step(result, z); // Just one Halley step should be enough.
2034 //#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_HALLEY
2035 // std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
2036 // std::cout << "Halley refinement estimate = " << result << std::endl;
2037 // std::cout.precision(saved_precision);
2038 //#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W1_HALLEY
2039 // return result; // Halley
2040 // } // Schroeder or Schroeder and Halley.
2041 }
2042 } // template<typename T = double> T lambert_wm1_imp(const T z)
2043 } // namespace lambert_w_detail
2044
2045 ///////////////////////////// User Lambert w functions. //////////////////////////////
2046
2047 //! Lambert W0 using User-defined policy.
2048 template <class T, class Policy>
2049 inline
2050 typename boost::math::tools::promote_args<T>::type
lambert_w0(T z,const Policy & pol)2051 lambert_w0(T z, const Policy& pol)
2052 {
2053 // Promote integer or expression template arguments to double,
2054 // without doing any other internal promotion like float to double.
2055 typedef typename tools::promote_args<T>::type result_type;
2056
2057 // Work out what precision has been selected,
2058 // based on the Policy and the number type.
2059 typedef typename policies::precision<result_type, Policy>::type precision_type;
2060 // and then select the correct implementation based on that precision (not the type T):
2061 typedef boost::integral_constant<int,
2062 (precision_type::value == 0) || (precision_type::value > 53) ?
2063 0 // either variable precision (0), or greater than 64-bit precision.
2064 : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision.
2065 : 2 // 64-bit (probably double) precision.
2066 > tag_type;
2067
2068 return lambert_w_detail::lambert_w0_imp(result_type(z), pol, tag_type()); //
2069 } // lambert_w0(T z, const Policy& pol)
2070
2071 //! Lambert W0 using default policy.
2072 template <class T>
2073 inline
2074 typename tools::promote_args<T>::type
lambert_w0(T z)2075 lambert_w0(T z)
2076 {
2077 // Promote integer or expression template arguments to double,
2078 // without doing any other internal promotion like float to double.
2079 typedef typename tools::promote_args<T>::type result_type;
2080
2081 // Work out what precision has been selected, based on the Policy and the number type.
2082 // For the default policy version, we want the *default policy* precision for T.
2083 typedef typename policies::precision<result_type, policies::policy<> >::type precision_type;
2084 // and then select the correct implementation based on that (not the type T):
2085 typedef boost::integral_constant<int,
2086 (precision_type::value == 0) || (precision_type::value > 53) ?
2087 0 // either variable precision (0), or greater than 64-bit precision.
2088 : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision.
2089 : 2 // 64-bit (probably double) precision.
2090 > tag_type;
2091 return lambert_w_detail::lambert_w0_imp(result_type(z), policies::policy<>(), tag_type());
2092 } // lambert_w0(T z) using default policy.
2093
2094 //! W-1 branch (-max(z) < z <= -1/e).
2095
2096 //! Lambert W-1 using User-defined policy.
2097 template <class T, class Policy>
2098 inline
2099 typename tools::promote_args<T>::type
lambert_wm1(T z,const Policy & pol)2100 lambert_wm1(T z, const Policy& pol)
2101 {
2102 // Promote integer or expression template arguments to double,
2103 // without doing any other internal promotion like float to double.
2104 typedef typename tools::promote_args<T>::type result_type;
2105 return lambert_w_detail::lambert_wm1_imp(result_type(z), pol); //
2106 }
2107
2108 //! Lambert W-1 using default policy.
2109 template <class T>
2110 inline
2111 typename tools::promote_args<T>::type
lambert_wm1(T z)2112 lambert_wm1(T z)
2113 {
2114 typedef typename tools::promote_args<T>::type result_type;
2115 return lambert_w_detail::lambert_wm1_imp(result_type(z), policies::policy<>());
2116 } // lambert_wm1(T z)
2117
2118 // First derivative of Lambert W0 and W-1.
2119 template <class T, class Policy>
2120 inline typename tools::promote_args<T>::type
lambert_w0_prime(T z,const Policy & pol)2121 lambert_w0_prime(T z, const Policy& pol)
2122 {
2123 typedef typename tools::promote_args<T>::type result_type;
2124 using std::numeric_limits;
2125 if (z == 0)
2126 {
2127 return static_cast<result_type>(1);
2128 }
2129 // This is the sensible choice if we regard the Lambert-W function as complex analytic.
2130 // Of course on the real line, it's just undefined.
2131 if (z == - boost::math::constants::exp_minus_one<result_type>())
2132 {
2133 return numeric_limits<result_type>::has_infinity ? numeric_limits<result_type>::infinity() : boost::math::tools::max_value<result_type>();
2134 }
2135 // if z < -1/e, we'll let lambert_w0 do the error handling:
2136 result_type w = lambert_w0(result_type(z), pol);
2137 // If w ~ -1, then presumably this can get inaccurate.
2138 // Is there an accurate way to evaluate 1 + W(-1/e + eps)?
2139 // Yes: This is discussed in the Princeton Companion to Applied Mathematics,
2140 // 'The Lambert-W function', Section 1.3: Series and Generating Functions.
2141 // 1 + W(-1/e + x) ~ sqrt(2ex).
2142 // Nick is not convinced this formula is more accurate than the naive one.
2143 // However, for z != -1/e, we never get rounded to w = -1 in any precision I've tested (up to cpp_bin_float_100).
2144 return w / (z * (1 + w));
2145 } // lambert_w0_prime(T z)
2146
2147 template <class T>
2148 inline typename tools::promote_args<T>::type
lambert_w0_prime(T z)2149 lambert_w0_prime(T z)
2150 {
2151 return lambert_w0_prime(z, policies::policy<>());
2152 }
2153
2154 template <class T, class Policy>
2155 inline typename tools::promote_args<T>::type
lambert_wm1_prime(T z,const Policy & pol)2156 lambert_wm1_prime(T z, const Policy& pol)
2157 {
2158 using std::numeric_limits;
2159 typedef typename tools::promote_args<T>::type result_type;
2160 //if (z == 0)
2161 //{
2162 // return static_cast<result_type>(1);
2163 //}
2164 //if (z == - boost::math::constants::exp_minus_one<result_type>())
2165 if (z == 0 || z == - boost::math::constants::exp_minus_one<result_type>())
2166 {
2167 return numeric_limits<result_type>::has_infinity ? -numeric_limits<result_type>::infinity() : -boost::math::tools::max_value<result_type>();
2168 }
2169
2170 result_type w = lambert_wm1(z, pol);
2171 return w/(z*(1+w));
2172 } // lambert_wm1_prime(T z)
2173
2174 template <class T>
2175 inline typename tools::promote_args<T>::type
lambert_wm1_prime(T z)2176 lambert_wm1_prime(T z)
2177 {
2178 return lambert_wm1_prime(z, policies::policy<>());
2179 }
2180
2181 }} //boost::math namespaces
2182
2183 #endif // #ifdef BOOST_MATH_SF_LAMBERT_W_HPP
2184
2185