boost/math/special_functions/lambert_w.hpp
// Copyright John Maddock 2017. // Copyright Paul A. Bristow 2016, 2017, 2018. // Copyright Nicholas Thompson 2018 // Distributed under the Boost Software License, Version 1.0. // (See accompanying file LICENSE_1_0.txt or // copy at http ://www.boost.org/LICENSE_1_0.txt). #ifndef BOOST_MATH_SF_LAMBERT_W_HPP #define BOOST_MATH_SF_LAMBERT_W_HPP #ifdef _MSC_VER #pragma warning(disable : 4127) #endif /* Implementation of an algorithm for the Lambert W0 and W-1 real-only functions. This code is based in part on the algorithm by Toshio Fukushima, "Precise and fast computation of Lambert W-functions without transcendental function evaluations", J.Comp.Appl.Math. 244 (2013) 77-89, and on a C/C++ version by Darko Veberic, darko.veberic@ijs.si based on the Fukushima algorithm and Toshio Fukushima's FORTRAN version of his algorithm. First derivative of Lambert_w is derived from Princeton Companion to Applied Mathematics, 'The Lambert-W function', Section 1.3: Series and Generating Functions. */ /* TODO revise this list of macros. Some macros that will show some (or much) diagnostic values if #defined. //[boost_math_instrument_lambert_w_macros // #define-able macros BOOST_MATH_INSTRUMENT_LAMBERT_W_HALLEY // Halley refinement diagnostics. BOOST_MATH_INSTRUMENT_LAMBERT_W_PRECISION // Precision. BOOST_MATH_INSTRUMENT_LAMBERT_WM1 // W1 branch diagnostics. BOOST_MATH_INSTRUMENT_LAMBERT_WM1_HALLEY // Halley refinement diagnostics only for W-1 branch. BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY // K > 64, z > -1.0264389699511303e-26 BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP // Show results from W-1 lookup table. BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER // Schroeder refinement diagnostics. BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS // Number of terms used for near-singularity series. BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES // Show evaluation of series near branch singularity. BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS // Show evaluation of series for small z. //] [/boost_math_instrument_lambert_w_macros] */ #include <boost/math/policies/error_handling.hpp> #include <boost/math/policies/policy.hpp> #include <boost/math/tools/promotion.hpp> #include <boost/math/special_functions/fpclassify.hpp> #include <boost/math/special_functions/log1p.hpp> // for log (1 + x) #include <boost/math/constants/constants.hpp> // For exp_minus_one == 3.67879441171442321595523770161460867e-01. #include <boost/math/special_functions/pow.hpp> // powers with compile time exponent, used in arbitrary precision code. #include <boost/math/tools/series.hpp> // series functor. //#include <boost/math/tools/polynomial.hpp> // polynomial. #include <boost/math/tools/rational.hpp> // evaluate_polynomial. #include <boost/math/tools/precision.hpp> // boost::math::tools::max_value(). #include <boost/math/tools/big_constant.hpp> #include <boost/math/tools/cxx03_warn.hpp> #ifndef BOOST_MATH_STANDALONE #include <boost/lexical_cast.hpp> #endif #include <limits> #include <cmath> #include <limits> #include <exception> #include <type_traits> #include <cstdint> // Needed for testing and diagnostics only. #include <iostream> #include <typeinfo> #include <boost/math/special_functions/next.hpp> // For float_distance. using lookup_t = double; // Type for lookup table (double or float, or even long double?) //#include "J:\Cpp\Misc\lambert_w_lookup_table_generator\lambert_w_lookup_table.ipp" // #include "lambert_w_lookup_table.ipp" // Boost.Math version. #include <boost/math/special_functions/detail/lambert_w_lookup_table.ipp> #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128) // // This is the only way we can avoid // warning: non-standard suffix on floating constant [-Wpedantic] // when building with -Wall -pedantic. Neither __extension__ // nor #pragma diagnostic ignored work :( // #pragma GCC system_header #endif namespace boost { namespace math { namespace lambert_w_detail { //! \brief Applies a single Halley step to make a better estimate of Lambert W. //! \details Used the simplified formulae obtained from //! 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 //! [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)] //! \tparam T floating-point (or fixed-point) type. //! \param w_est Lambert W estimate. //! \param z Argument z for Lambert_w function. //! \returns New estimate of Lambert W, hopefully improved. //! template <typename T> inline T lambert_w_halley_step(T w_est, const T z) { BOOST_MATH_STD_USING T e = exp(w_est); w_est -= 2 * (w_est + 1) * (e * w_est - z) / (z * (w_est + 2) + e * (w_est * (w_est + 2) + 2)); return w_est; } // template <typename T> lambert_w_halley_step(T w_est, T z) //! \brief Halley iterate to refine Lambert_w estimate, //! taking at least one Halley_step. //! Repeat Halley steps until the *last step* had fewer than half the digits wrong, //! the step we've just taken should have been sufficient to have completed the iteration. //! \tparam T floating-point (or fixed-point) type. //! \param z Argument z for Lambert_w function. //! \param w_est Lambert w estimate. template <typename T> inline T lambert_w_halley_iterate(T w_est, const T z) { BOOST_MATH_STD_USING static const T max_diff = boost::math::tools::root_epsilon<T>() * fabs(w_est); T w_new = lambert_w_halley_step(w_est, z); T diff = fabs(w_est - w_new); while (diff > max_diff) { w_est = w_new; w_new = lambert_w_halley_step(w_est, z); diff = fabs(w_est - w_new); } return w_new; } // template <typename T> lambert_w_halley_iterate(T w_est, T z) // Two Halley function versions that either // single step (if std::false_type) or iterate (if std::true_type). // Selected at compile-time using parameter 3. template <typename T> inline T lambert_w_maybe_halley_iterate(T z, T w, std::false_type const&) { return lambert_w_halley_step(z, w); // Single step. } template <typename T> inline T lambert_w_maybe_halley_iterate(T z, T w, std::true_type const&) { return lambert_w_halley_iterate(z, w); // Iterate steps. } //! maybe_reduce_to_double function, //! Two versions that have a compile-time option to //! reduce argument z to double precision (if true_type). //! Version is selected at compile-time using parameter 2. template <typename T> inline double maybe_reduce_to_double(const T& z, const std::true_type&) { return static_cast<double>(z); // Reduce to double precision. } template <typename T> inline T maybe_reduce_to_double(const T& z, const std::false_type&) { // Don't reduce to double. return z; } template <typename T> inline double must_reduce_to_double(const T& z, const std::true_type&) { return static_cast<double>(z); // Reduce to double precision. } template <typename T> inline double must_reduce_to_double(const T& z, const std::false_type&) { // try a lexical_cast and hope for the best: #ifndef BOOST_MATH_STANDALONE return boost::lexical_cast<double>(z); #else static_assert(sizeof(T) == 0, "Unsupported in standalone mode: don't know how to cast your number type to a double."); return 0.0; #endif } //! \brief Schroeder method, fifth-order update formula, //! \details See T. Fukushima page 80-81, and //! A. Householder, The Numerical Treatment of a Single Nonlinear Equation, //! McGraw-Hill, New York, 1970, section 4.4. //! Fukushima algorithm switches to @c schroeder_update after pre-computed bisections, //! chosen to ensure that the result will be achieve the +/- 10 epsilon target. //! \param w Lambert w estimate from bisection or series. //! \param y bracketing value from bisection. //! \returns Refined estimate of Lambert w. // Schroeder refinement, called unless NOT required by precision policy. template<typename T> inline T schroeder_update(const T w, const T y) { // Compute derivatives using 5th order Schroeder refinement. // Since this is the final step, it will always use the highest precision type T. // Example of Call: // result = schroeder_update(w, y); //where // w is estimate of Lambert W (from bisection or series). // y is z * e^-w. BOOST_MATH_STD_USING // Aid argument dependent lookup of abs. #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); using boost::math::float_distance; T fd = float_distance<T>(w, y); std::cout << "Schroder "; if (abs(fd) < 214748000.) { std::cout << " Distance = "<< static_cast<int>(fd); } else { std::cout << "Difference w - y = " << (w - y) << "."; } std::cout << std::endl; #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER // Fukushima equation 18, page 6. const T f0 = w - y; // f0 = w - y. const T f1 = 1 + y; // f1 = df/dW const T f00 = f0 * f0; const T f11 = f1 * f1; const T f0y = f0 * y; const T result = w - 4 * f0 * (6 * f1 * (f11 + f0y) + f00 * y) / (f11 * (24 * f11 + 36 * f0y) + f00 * (6 * y * y + 8 * f1 * y + f0y)); // Fukushima Page 81, equation 21 from equation 20. #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER std::cout << "Schroeder refined " << w << " " << y << ", difference " << w-y << ", change " << w - result << ", to result " << result << std::endl; std::cout.precision(saved_precision); // Restore. #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER return result; } // template<typename T = double> T schroeder_update(const T w, const T y) //! \brief Series expansion used near the singularity/branch point z = -exp(-1) = -3.6787944. //! Wolfram InverseSeries[Series[sqrt[2(p Exp[1 + p] + 1)], {p,-1, 20}]] //! Wolfram command used to obtain 40 series terms at 50 decimal digit precision was //! N[InverseSeries[Series[Sqrt[2(p Exp[1 + p] + 1)], { p,-1,40 }]], 50] //! -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 ... //! Decimal values of specifications for built-in floating-point types below //! are at least 21 digits precision == max_digits10 for long double. //! Longer decimal digits strings are rationals evaluated using Wolfram. template<typename T> T lambert_w_singularity_series(const T p) { #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES std::size_t saved_precision = std::cout.precision(3); std::cout << "Singularity_series Lambert_w p argument = " << p << std::endl; std::cout //<< "Argument Type = " << typeid(T).name() //<< ", max_digits10 = " << std::numeric_limits<T>::max_digits10 //<< ", epsilon = " << std::numeric_limits<T>::epsilon() << std::endl; std::cout.precision(saved_precision); #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES static const T q[] = { -static_cast<T>(1), // j0 +T(1), // j1 -T(1) / 3, // 1/3 j2 +T(11) / 72, // 0.152777777777777778, // 11/72 j3 -T(43) / 540, // 0.0796296296296296296, // 43/540 j4 +T(769) / 17280, // 0.0445023148148148148, j5 -T(221) / 8505, // 0.0259847148736037625, j6 //+T(0.0156356325323339212L), // j7 //+T(0.015635632532333921222810111699000587889476778365667L), // j7 from Wolfram N[680863/43545600, 50] +T(680863uLL) / 43545600uLL, // +0.0156356325323339212, j7 //-T(0.00961689202429943171L), // j8 -T(1963uLL) / 204120uLL, // 0.00961689202429943171, j8 //-T(0.0096168920242994317068391142465216539290613364687439L), // j8 from Wolfram N[1963/204120, 50] +T(226287557uLL) / 37623398400uLL, // 0.00601454325295611786, j9 -T(5776369uLL) / 1515591000uLL, // 0.00381129803489199923, j10 //+T(0.00244087799114398267L), j11 0.0024408779911439826658968585286437530215699919795550 +T(169709463197uLL) / 69528040243200uLL, // j11 // -T(0.00157693034468678425L), // j12 -0.0015769303446867842539234095399314115973161850314723 -T(1118511313uLL) / 709296588000uLL, // j12 +T(667874164916771uLL) / 650782456676352000uLL, // j13 //+T(0.00102626332050760715L), // j13 0.0010262633205076071544375481533906861056468041465973 -T(500525573uLL) / 744761417400uLL, // j14 // -T(0.000672061631156136204L), j14 //+T(1003663334225097487uLL) / 234281684403486720000uLL, // j15 0.00044247306181462090993020760858473726479232802068800 error C2177: constant too big //+T(0.000442473061814620910L, // j15 BOOST_MATH_BIG_CONSTANT(T, 64, +0.000442473061814620910), // j15 // -T(0.000292677224729627445L), // j16 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000292677224729627445), // j16 //+T(0.000194387276054539318L), // j17 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000194387276054539318), // j17 //-T(0.000129574266852748819L), // j18 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000129574266852748819), // j18 //+T(0.0000866503580520812717L), // j19 N[+1150497127780071399782389/13277465363600276402995200000, 50] 0.000086650358052081271660451590462390293190597827783288 BOOST_MATH_BIG_CONSTANT(T, 64, +0.0000866503580520812717), // j19 //-T(0.0000581136075044138168L) // j20 N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913 // -T(2853534237182741069uLL) / 49102686267859224000000uLL // j20 // error C2177: constant too big, // so must use BOOST_MATH_BIG_CONSTANT(T, ) format in hope of using suffix Q for quad or decimal digits string for others. //-T(0.000058113607504413816772205464778828177256611844221913L), // j20 N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000058113607504413816772205464778828177256611844221913) // j20 - last used by Fukushima // More terms don't seem to give any improvement (worse in fact) and are not use for many z values. //BOOST_MATH_BIG_CONSTANT(T, +0.000039076684867439051635395583044527492132109160553593), // j21 //BOOST_MATH_BIG_CONSTANT(T, -0.000026338064747231098738584082718649443078703982217219), // j22 //BOOST_MATH_BIG_CONSTANT(T, +0.000017790345805079585400736282075184540383274460464169), // j23 //BOOST_MATH_BIG_CONSTANT(T, -0.000012040352739559976942274116578992585158113153190354), // j24 //BOOST_MATH_BIG_CONSTANT(T, +8.1635319824966121713827512573558687050675701559448E-6), // j25 //BOOST_MATH_BIG_CONSTANT(T, -5.5442032085673591366657251660804575198155559225316E-6) // j26 // -T(5.5442032085673591366657251660804575198155559225316E-6L) // j26 // 21 to 26 Added for long double. }; // static const T q[] /* // Temporary copy of original double values for comparison; these are reproduced well. static const T q[] = { -1L, // j0 +1L, // j1 -0.333333333333333333L, // 1/3 j2 +0.152777777777777778L, // 11/72 j3 -0.0796296296296296296L, // 43/540 +0.0445023148148148148L, -0.0259847148736037625L, +0.0156356325323339212L, -0.00961689202429943171L, +0.00601454325295611786L, -0.00381129803489199923L, +0.00244087799114398267L, -0.00157693034468678425L, +0.00102626332050760715L, -0.000672061631156136204L, +0.000442473061814620910L, -0.000292677224729627445L, +0.000194387276054539318L, -0.000129574266852748819L, +0.0000866503580520812717L, -0.0000581136075044138168L // j20 }; */ // Decide how many series terms to use, increasing as z approaches the singularity, // balancing run-time versus computational noise from round-off. // In practice, we truncate the series expansion at a certain order. // If the order is too large, not only does the amount of computation increase, // but also the round-off errors accumulate. // See Fukushima equation 35, page 85 for logic of choice of number of series terms. BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs. const T absp = abs(p); #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS { int terms = 20; // Default to using all terms. if (absp < 0.01159) { // Very near singularity. terms = 6; } else if (absp < 0.0766) { // Near singularity. terms = 10; } std::streamsize saved_precision = std::cout.precision(3); std::cout << "abs(p) = " << absp << ", terms = " << terms << std::endl; std::cout.precision(saved_precision); } #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS if (absp < 0.01159) { // Only 6 near-singularity series terms are useful. return -1 + p * (1 + p * (q[2] + p * (q[3] + p * (q[4] + p * (q[5] + p * q[6] ))))); } else if (absp < 0.0766) // Use 10 near-singularity series terms. { // Use 10 near-singularity series terms. return -1 + p * (1 + p * (q[2] + p * (q[3] + p * (q[4] + p * (q[5] + p * (q[6] + p * (q[7] + p * (q[8] + p * (q[9] + p * q[10] ))))))))); } else { // Use all 20 near-singularity series terms. return -1 + p * (1 + p * (q[2] + p * (q[3] + p * (q[4] + p * (q[5] + p * (q[6] + p * (q[7] + p * (q[8] + p * (q[9] + p * (q[10] + p * (q[11] + p * (q[12] + p * (q[13] + p * (q[14] + p * (q[15] + p * (q[16] + p * (q[17] + p * (q[18] + p * (q[19] + p * q[20] // Last Fukushima term. ))))))))))))))))))); // + // more terms for more precise T: long double ... //// but makes almost no difference, so don't use more terms? // p*q[21] + // p*q[22] + // p*q[23] + // p*q[24] + // p*q[25] // ))))))))))))))))))); } } // template<typename T = double> T lambert_w_singularity_series(const T p) ///////////////////////////////////////////////////////////////////////////////////////////// //! \brief Series expansion used near zero (abs(z) < 0.05). //! \details //! Coefficients of the inverted series expansion of the Lambert W function around z = 0. //! Tosio Fukushima always uses all 17 terms of a Taylor series computed using Wolfram with //! InverseSeries[Series[z Exp[z],{z,0,17}]] //! Tosio Fukushima / Journal of Computational and Applied Mathematics 244 (2013) page 86. //! Decimal values of specifications for built-in floating-point types below //! are 21 digits precision == max_digits10 for long double. //! Care! Some coefficients might overflow some fixed_point types. //! This version is intended to allow use by user-defined types //! like Boost.Multiprecision quad and cpp_dec_float types. //! The three specializations below for built-in float, double //! (and perhaps long double) will be chosen in preference for these types. //! This version uses rationals computed by Wolfram as far as possible, //! limited by maximum size of uLL integers. //! For higher term, uses decimal digit strings computed by Wolfram up to the maximum possible using uLL rationals, //! and then higher coefficients are computed as necessary using function lambert_w0_small_z_series_term //! until the precision required by the policy is achieved. //! InverseSeries[Series[z Exp[z],{z,0,34}]] also computed. // Series evaluation for LambertW(z) as z -> 0. // See http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/ // http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/MainEq1.L.gif //! \brief lambert_w0_small_z uses a tag_type to select a variant depending on the size of the type. //! The Lambert W is computed by lambert_w0_small_z for small z. //! The cutoff for z smallness determined by Tosio Fukushima by trial and error is (abs(z) < 0.05), //! but the optimum might be a function of the size of the type of z. //! \details //! The tag_type selection is based on the value @c std::numeric_limits<T>::max_digits10. //! This allows distinguishing between long double types that commonly vary between 64 and 80-bits, //! and also compilers that have a float type using 64 bits and/or long double using 128-bits. //! It assumes that max_digits10 is defined correctly or this might fail to make the correct selection. //! causing very small differences in computing lambert_w that would be very difficult to detect and diagnose. //! Cannot switch on @c std::numeric_limits<>::max() because comparison values may overflow the compiler limit. //! Cannot switch on @c std::numeric_limits<long double>::max_exponent10() //! because both 80 and 128 bit floating-point implementations use 11 bits for the exponent. //! So must rely on @c std::numeric_limits<long double>::max_digits10. //! Specialization of float zero series expansion used for small z (abs(z) < 0.05). //! Specializations of lambert_w0_small_z for built-in types. //! These specializations should be chosen in preference to T version. //! For example: lambert_w0_small_z(0.001F) should use the float version. //! (Parameter Policy is not used by built-in types when all terms are used during an inline computation, //! but for the tag_type selection to work, they all must include Policy in their signature. // Forward declaration of variants of lambert_w0_small_z. template <typename T, typename Policy> T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 0> const&); // for float (32-bit) type. template <typename T, typename Policy> T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 1> const&); // for double (64-bit) type. template <typename T, typename Policy> T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 2> const&); // for long double (double extended 80-bit) type. template <typename T, typename Policy> T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 3> const&); // for long double (128-bit) type. template <typename T, typename Policy> T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 4> const&); // for float128 quadmath Q type. template <typename T, typename Policy> T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 5> const&); // Generic multiprecision T. // Set tag_type depending on max_digits10. template <typename T, typename Policy> T lambert_w0_small_z(T x, const Policy& pol) { //std::numeric_limits<T>::max_digits10 == 36 ? 3 : // 128-bit long double. using tag_type = std::integral_constant<int, std::numeric_limits<T>::is_specialized == 0 ? 5 : #ifndef BOOST_NO_CXX11_NUMERIC_LIMITS std::numeric_limits<T>::max_digits10 <= 9 ? 0 : // for float 32-bit. std::numeric_limits<T>::max_digits10 <= 17 ? 1 : // for double 64-bit. std::numeric_limits<T>::max_digits10 <= 22 ? 2 : // for 80-bit double extended. std::numeric_limits<T>::max_digits10 < 37 ? 4 // for both 128-bit long double (3) and 128-bit quad suffix Q type (4). #else std::numeric_limits<T>::radix != 2 ? 5 : std::numeric_limits<T>::digits <= 24 ? 0 : // for float 32-bit. std::numeric_limits<T>::digits <= 53 ? 1 : // for double 64-bit. std::numeric_limits<T>::digits <= 64 ? 2 : // for 80-bit double extended. std::numeric_limits<T>::digits <= 113 ? 4 // for both 128-bit long double (3) and 128-bit quad suffix Q type (4). #endif : 5>; // All Generic multiprecision types. // std::cout << "\ntag type = " << tag_type << std::endl; // error C2275: 'tag_type': illegal use of this type as an expression. return lambert_w0_small_z(x, pol, tag_type()); } // template <typename T> T lambert_w0_small_z(T x) //! Specialization of float (32-bit) series expansion used for small z (abs(z) < 0.05). // Only 9 Coefficients are computed to 21 decimal digits precision, ample for 32-bit float used by most platforms. // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], // as proposed by Tosio Fukushima and implemented by Darko Veberic. template <typename T, typename Policy> T lambert_w0_small_z(T z, const Policy&, std::integral_constant<int, 0> const&) { #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. std::cout << "\ntag_type 0 float lambert_w0_small_z called with z = " << z << " using " << 9 << " terms of precision " << std::numeric_limits<float>::max_digits10 << " decimal digits. " << std::endl; #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES T result = z * (1 - // j1 z^1 term = 1 z * (1 - // j2 z^2 term = -1 z * (static_cast<float>(3uLL) / 2uLL - // 3/2 // j3 z^3 term = 1.5. z * (2.6666666666666666667F - // 8/3 // j4 z * (5.2083333333333333333F - // -125/24 // j5 z * (10.8F - // j6 z * (23.343055555555555556F - // j7 z * (52.012698412698412698F - // j8 z * 118.62522321428571429F)))))))); // j9 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::cout << "return w = " << result << std::endl; std::cout.precision(prec); // Restore. #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES return result; } // template <typename T> T lambert_w0_small_z(T x, std::integral_constant<int, 0> const&) //! Specialization of double (64-bit double) series expansion used for small z (abs(z) < 0.05). // 17 Coefficients are computed to 21 decimal digits precision suitable for 64-bit double used by most platforms. // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], as proposed by Tosio Fukushima and implemented by Veberic. template <typename T, typename Policy> T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 1> const&) { #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. std::cout << "\ntag_type 1 double lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision, " << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl; #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES T result = z * (1. - // j1 z^1 z * (1. - // j2 z^2 z * (1.5 - // 3/2 // j3 z^3 z * (2.6666666666666666667 - // 8/3 // j4 z * (5.2083333333333333333 - // -125/24 // j5 z * (10.8 - // j6 z * (23.343055555555555556 - // j7 z * (52.012698412698412698 - // j8 z * (118.62522321428571429 - // j9 z * (275.57319223985890653 - // j10 z * (649.78717234347442681 - // j11 z * (1551.1605194805194805 - // j12 z * (3741.4497029592385495 - // j13 z * (9104.5002411580189358 - // j14 z * (22324.308512706601434 - // j15 z * (55103.621972903835338 - // j16 z * 136808.86090394293563)))))))))))))))); // j17 z^17 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::cout << "return w = " << result << std::endl; std::cout.precision(prec); // Restore. #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES return result; } // T lambert_w0_small_z(const T z, std::integral_constant<int, 1> const&) //! Specialization of long double (80-bit double extended) series expansion used for small z (abs(z) < 0.05). // 21 Coefficients are computed to 21 decimal digits precision suitable for 80-bit long double used by some // platforms including GCC and Clang when generating for Intel X86 floating-point processors with 80-bit operations enabled (the default). // (This is NOT used by Microsoft Visual Studio where double and long always both use only 64-bit type. // Nor used for 128-bit float128.) template <typename T, typename Policy> T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 2> const&) { #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. std::cout << "\ntag_type 2 long double (80-bit double extended) lambert_w0_small_z called with z = " << z << " using " << 21 << " terms of precision, " << std::numeric_limits<long double>::max_digits10 << " decimal digits. " << std::endl; #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES // T result = // z * (1.L - // j1 z^1 // z * (1.L - // j2 z^2 // z * (1.5L - // 3/2 // j3 // z * (2.6666666666666666667L - // 8/3 // j4 // z * (5.2083333333333333333L - // -125/24 // j5 // z * (10.800000000000000000L - // j6 // z * (23.343055555555555556L - // j7 // z * (52.012698412698412698L - // j8 // z * (118.62522321428571429L - // j9 // z * (275.57319223985890653L - // j10 // z * (649.78717234347442681L - // j11 // z * (1551.1605194805194805L - // j12 // z * (3741.4497029592385495L - // j13 // z * (9104.5002411580189358L - // j14 // z * (22324.308512706601434L - // j15 // z * (55103.621972903835338L - // j16 // z * (136808.86090394293563L - // j17 z^17 last term used by Fukushima double. // z * (341422.050665838363317L - // z^18 // z * (855992.9659966075514633L - // z^19 // z * (2.154990206091088289321e6L - // z^20 // z * 5.4455529223144624316423e6L // z^21 // )))))))))))))))))))); // T result = z * (1.L - // z j1 z * (1.L - // z^2 z * (1.500000000000000000000000000000000L - // z^3 z * (2.666666666666666666666666666666666L - // z ^ 4 z * (5.208333333333333333333333333333333L - // z ^ 5 z * (10.80000000000000000000000000000000L - // z ^ 6 z * (23.34305555555555555555555555555555L - // z ^ 7 z * (52.01269841269841269841269841269841L - // z ^ 8 z * (118.6252232142857142857142857142857L - // z ^ 9 z * (275.5731922398589065255731922398589L - // z ^ 10 z * (649.7871723434744268077601410934744L - // z ^ 11 z * (1551.160519480519480519480519480519L - // z ^ 12 z * (3741.449702959238549516327294105071L - //z ^ 13 z * (9104.500241158018935796713574491352L - // z ^ 14 z * (22324.308512706601434280005708577137L - // z ^ 15 z * (55103.621972903835337697771560205422L - // z ^ 16 z * (136808.86090394293563342215789305736L - // z ^ 17 z * (341422.05066583836331735491399356945L - // z^18 z * (855992.9659966075514633630250633224L - // z^19 z * (2.154990206091088289321708745358647e6L // z^20 distance -5 without term 20 )))))))))))))))))))); #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::cout << "return w = " << result << std::endl; std::cout.precision(precision); // Restore. #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES return result; } // long double lambert_w0_small_z(const T z, std::integral_constant<int, 1> const&) //! Specialization of 128-bit long double series expansion used for small z (abs(z) < 0.05). // 34 Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], // and are suffixed by L as they are assumed of type long double. // (This is NOT used for 128-bit quad boost::multiprecision::float128 type which required a suffix Q // nor multiprecision type cpp_bin_float_quad that can only be initialised at full precision of the type // constructed with a decimal digit string like "2.6666666666666666666666666666666666666666666666667".) template <typename T, typename Policy> T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 3> const&) { #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. std::cout << "\ntag_type 3 long double (128-bit) lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision, " << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl; #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES T result = z * (1.L - // j1 z * (1.L - // j2 z * (1.5L - // 3/2 // j3 z * (2.6666666666666666666666666666666666L - // 8/3 // j4 z * (5.2052083333333333333333333333333333L - // -125/24 // j5 z * (10.800000000000000000000000000000000L - // j6 z * (23.343055555555555555555555555555555L - // j7 z * (52.0126984126984126984126984126984126L - // j8 z * (118.625223214285714285714285714285714L - // j9 z * (275.57319223985890652557319223985890L - // * z ^ 10 - // j10 z * (649.78717234347442680776014109347442680776014109347L - // j11 z * (1551.1605194805194805194805194805194805194805194805L - // j12 z * (3741.4497029592385495163272941050718828496606274384L - // j13 z * (9104.5002411580189357967135744913522691300469078247L - // j14 z * (22324.308512706601434280005708577137148565719994291L - // j15 z * (55103.621972903835337697771560205422639285073147507L - // j16 z * 136808.86090394293563342215789305736395683485630576L // j17 )))))))))))))))); #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::cout << "return w = " << result << std::endl; std::cout.precision(precision); // Restore. #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES return result; } // T lambert_w0_small_z(const T z, std::integral_constant<int, 3> const&) //! Specialization of 128-bit quad series expansion used for small z (abs(z) < 0.05). // 34 Taylor series coefficients used were computed by Wolfram to 50 decimal digits using instruction // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], // and are suffixed by Q as they are assumed of type quad. // This could be used for 128-bit quad (which requires a suffix Q for full precision). // But experiments with GCC 7.2.0 show that while this gives full 128-bit precision // when the -f-ext-numeric-literals option is in force and the libquadmath library available, // over the range -0.049 to +0.049, // it is slightly slower than getting a double approximation followed by a single Halley step. #ifdef BOOST_HAS_FLOAT128 template <typename T, typename Policy> T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 4> const&) { #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. std::cout << "\ntag_type 4 128-bit quad float128 lambert_w0_small_z called with z = " << z << " using " << 34 << " terms of precision, " << std::numeric_limits<float128>::max_digits10 << " max decimal digits." << std::endl; #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES T result = z * (1.Q - // z j1 z * (1.Q - // z^2 z * (1.500000000000000000000000000000000Q - // z^3 z * (2.666666666666666666666666666666666Q - // z ^ 4 z * (5.208333333333333333333333333333333Q - // z ^ 5 z * (10.80000000000000000000000000000000Q - // z ^ 6 z * (23.34305555555555555555555555555555Q - // z ^ 7 z * (52.01269841269841269841269841269841Q - // z ^ 8 z * (118.6252232142857142857142857142857Q - // z ^ 9 z * (275.5731922398589065255731922398589Q - // z ^ 10 z * (649.7871723434744268077601410934744Q - // z ^ 11 z * (1551.160519480519480519480519480519Q - // z ^ 12 z * (3741.449702959238549516327294105071Q - //z ^ 13 z * (9104.500241158018935796713574491352Q - // z ^ 14 z * (22324.308512706601434280005708577137Q - // z ^ 15 z * (55103.621972903835337697771560205422Q - // z ^ 16 z * (136808.86090394293563342215789305736Q - // z ^ 17 z * (341422.05066583836331735491399356945Q - // z^18 z * (855992.9659966075514633630250633224Q - // z^19 z * (2.154990206091088289321708745358647e6Q - // 20 z * (5.445552922314462431642316420035073e6Q - // 21 z * (1.380733000216662949061923813184508e7Q - // 22 z * (3.511704498513923292853869855945334e7Q - // 23 z * (8.956800256102797693072819557780090e7Q - // 24 z * (2.290416846187949813964782641734774e8Q - // 25 z * (5.871035041171798492020292225245235e8Q - // 26 z * (1.508256053857792919641317138812957e9Q - // 27 z * (3.882630161293188940385873468413841e9Q - // 28 z * (1.001394313665482968013913601565723e10Q - // 29 z * (2.587356736265760638992878359024929e10Q - // 30 z * (6.696209709358073856946120522333454e10Q - // 31 z * (1.735711659599198077777078238043644e11Q - // 32 z * (4.505680465642353886756098108484670e11Q - // 33 z * (1.171223178256487391904047636564823e12Q //z^34 )))))))))))))))))))))))))))))))))); #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::cout << "return w = " << result << std::endl; std::cout.precision(precision); // Restore. #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES return result; } // T lambert_w0_small_z(const T z, std::integral_constant<int, 4> const&) float128 #else template <typename T, typename Policy> inline T lambert_w0_small_z(const T z, const Policy& pol, std::integral_constant<int, 4> const&) { return lambert_w0_small_z(z, pol, std::integral_constant<int, 5>()); } #endif // BOOST_HAS_FLOAT128 //! Series functor to compute series term using pow and factorial. //! \details Functor is called after evaluating polynomial with the coefficients as rationals below. template <typename T> struct lambert_w0_small_z_series_term { using result_type = T; //! \param _z Lambert W argument z. //! \param -term -pow<18>(z) / 6402373705728000uLL //! \param _k number of terms == initially 18 // Note *after* evaluating N terms, its internal state has k = N and term = (-1)^N z^N. lambert_w0_small_z_series_term(T _z, T _term, int _k) : k(_k), z(_z), term(_term) { } T operator()() { // Called by sum_series until needs precision set by factor (policy::get_epsilon). using std::pow; ++k; term *= -z / k; //T t = pow(z, k) * pow(T(k), -1 + k) / factorial<T>(k); // (z^k * k(k-1)^k) / k! T result = term * pow(T(k), -1 + k); // term * k^(k-1) // std::cout << " k = " << k << ", term = " << term << ", result = " << result << std::endl; return result; // } private: int k; T z; T term; }; // template <typename T> struct lambert_w0_small_z_series_term //! Generic variant for T a User-defined types like Boost.Multiprecision. template <typename T, typename Policy> inline T lambert_w0_small_z(T z, const Policy& pol, std::integral_constant<int, 5> const&) { #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save. std::cout << "Generic lambert_w0_small_z called with z = " << z << " using as many terms needed for precision." << std::endl; std::cout << "Argument z is of type " << typeid(T).name() << std::endl; #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES // First several terms of the series are tabulated and evaluated as a polynomial: // this will save us a bunch of expensive calls to pow. // Then our series functor is initialized "as if" it had already reached term 18, // enough evaluation of built-in 64-bit double and float (and 80-bit long double?) types. // Coefficients should be stored such that the coefficients for the x^i terms are in poly[i]. static const T coeff[] = { 0, // z^0 Care: zeroth term needed by tools::evaluate_polynomial, but not in the Wolfram equation, so indexes are one different! 1, // z^1 term. -1, // z^2 term static_cast<T>(3uLL) / 2uLL, // z^3 term. -static_cast<T>(8uLL) / 3uLL, // z^4 static_cast<T>(125uLL) / 24uLL, // z^5 -static_cast<T>(54uLL) / 5uLL, // z^6 static_cast<T>(16807uLL) / 720uLL, // z^7 -static_cast<T>(16384uLL) / 315uLL, // z^8 static_cast<T>(531441uLL) / 4480uLL, // z^9 -static_cast<T>(156250uLL) / 567uLL, // z^10 static_cast<T>(2357947691uLL) / 3628800uLL, // z^11 -static_cast<T>(2985984uLL) / 1925uLL, // z^12 static_cast<T>(1792160394037uLL) / 479001600uLL, // z^13 -static_cast<T>(7909306972uLL) / 868725uLL, // z^14 static_cast<T>(320361328125uLL) / 14350336uLL, // z^15 -static_cast<T>(35184372088832uLL) / 638512875uLL, // z^16 static_cast<T>(2862423051509815793uLL) / 20922789888000uLL, // z^17 term -static_cast<T>(5083731656658uLL) / 14889875uLL, // z^18 term. = 136808.86090394293563342215789305735851647769682393 // z^18 is biggest that can be computed as rational using the largest possible uLL integers, // so higher terms cannot be potentially compiler-computed as uLL rationals. // Wolfram (5083731656658 z ^ 18) / 14889875 or // -341422.05066583836331735491399356945575432970390954 z^18 // See note below calling the functor to compute another term, // sufficient for 80-bit long double precision. // Wolfram -341422.05066583836331735491399356945575432970390954 z^19 term. // (5480386857784802185939 z^19)/6402373705728000 // But now this variant is not used to compute long double // as specializations are provided above. }; // static const T coeff[] /* Table of 19 computed coefficients: #0 0 #1 1 #2 -1 #3 1.5 #4 -2.6666666666666666666666666666666665382713370408509 #5 5.2083333333333333333333333333333330765426740817019 #6 -10.800000000000000000000000000000000616297582203915 #7 23.343055555555555555555555555555555076212991619177 #8 -52.012698412698412698412698412698412659282693193402 #9 118.62522321428571428571428571428571146835390992496 #10 -275.57319223985890652557319223985891400375196748314 #11 649.7871723434744268077601410934743969785223845882 #12 -1551.1605194805194805194805194805194947599566007429 #13 3741.4497029592385495163272941050719510009019331763 #14 -9104.5002411580189357967135744913524243896052869184 #15 22324.308512706601434280005708577137322392070452582 #16 -55103.621972903835337697771560205423203318720697224 #17 136808.86090394293563342215789305735851647769682393 136808.86090394293563342215789305735851647769682393 == Exactly same as Wolfram computed value. #18 -341422.05066583836331735491399356947486381600607416 341422.05066583836331735491399356945575432970390954 z^19 Wolfram value differs at 36 decimal digit, as expected. */ using boost::math::policies::get_epsilon; // for type T. using boost::math::tools::sum_series; using boost::math::tools::evaluate_polynomial; // http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/roots/rational.html // std::streamsize prec = std::cout.precision(std::numeric_limits <T>::max_digits10); T result = evaluate_polynomial(coeff, z); // template <std::size_t N, typename T, typename V> // V evaluate_polynomial(const T(&poly)[N], const V& val); // Size of coeff found from N //std::cout << "evaluate_polynomial(coeff, z); == " << result << std::endl; //std::cout << "result = " << result << std::endl; // It's an artefact of the way I wrote the functor: *after* evaluating N // terms, its internal state has k = N and term = (-1)^N z^N. So after // evaluating 18 terms, we initialize the functor to the term we've just // evaluated, and then when it's called, it increments itself to the next term. // So 18!is 6402373705728000, which is where that comes from. // The 19th coefficient of the polynomial is actually, 19 ^ 18 / 19!= // 104127350297911241532841 / 121645100408832000 which after removing GCDs // reduces down to Wolfram rational 5480386857784802185939 / 6402373705728000. // Wolfram z^19 term +(5480386857784802185939 z^19) /6402373705728000 // +855992.96599660755146336302506332246623424823099755 z^19 //! Evaluate Functor. lambert_w0_small_z_series_term<T> s(z, -pow<18>(z) / 6402373705728000uLL, 18); // Temporary to list the coefficients. //std::cout << " Table of coefficients" << std::endl; //std::streamsize saved_precision = std::cout.precision(50); //for (size_t i = 0; i != 19; i++) //{ // std::cout << "#" << i << " " << coeff[i] << std::endl; //} //std::cout.precision(saved_precision); std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); // Max iterations from policy. #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES std::cout << "max iter from policy = " << max_iter << std::endl; // // max iter from policy = 1000000 is default. #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES result = sum_series(s, get_epsilon<T, Policy>(), max_iter, result); // result == evaluate_polynomial. //sum_series(Functor& func, int bits, std::uintmax_t& max_terms, const U& init_value) // std::cout << "sum_series(s, get_epsilon<T, Policy>(), max_iter, result); = " << result << std::endl; //T epsilon = get_epsilon<T, Policy>(); //std::cout << "epsilon from policy = " << epsilon << std::endl; // epsilon from policy = 1.93e-34 for T == quad // 5.35e-51 for t = cpp_bin_float_50 // std::cout << " get eps = " << get_epsilon<T, Policy>() << std::endl; // quad eps = 1.93e-34, bin_float_50 eps = 5.35e-51 policies::check_series_iterations<T>("boost::math::lambert_w0_small_z<%1%>(%1%)", max_iter, pol); #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS std::cout << "z = " << z << " needed " << max_iter << " iterations." << std::endl; std::cout.precision(prec); // Restore. #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS return result; } // template <typename T, typename Policy> inline T lambert_w0_small_z_series(T z, const Policy& pol) // Approximate lambert_w0 (used for z values that are outside range of lookup table or rational functions) // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162. template <typename T> inline T lambert_w0_approx(T z) { BOOST_MATH_STD_USING T lz = log(z); T llz = log(lz); T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162. return w; // std::cout << "w max " << max_w << std::endl; // double 703.227 } ////////////////////////////////////////////////////////////////////////////////////////// //! \brief Lambert_w0 implementations for float, double and higher precisions. //! 3rd parameter used to select which version is used. //! /details Rational polynomials are provided for several range of argument z. //! For very small values of z, and for z very near the branch singularity at -e^-1 (~= -0.367879), //! two other series functions are used. //! float precision polynomials are used for 32-bit (usually float) precision (for speed) //! double precision polynomials are used for 64-bit (usually double) precision. //! For higher precisions, a 64-bit double approximation is computed first, //! and then refined using Halley iterations. template <typename T> inline T do_get_near_singularity_param(T z) { BOOST_MATH_STD_USING const T p2 = 2 * (boost::math::constants::e<T>() * z + 1); const T p = sqrt(p2); return p; } template <typename T, typename Policy> inline T get_near_singularity_param(T z, const Policy) { using value_type = typename policies::evaluation<T, Policy>::type; return static_cast<T>(do_get_near_singularity_param(static_cast<value_type>(z))); } // Forward declarations: //template <typename T, typename Policy> T lambert_w0_small_z(T z, const Policy& pol); //template <typename T, typename Policy> //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 0>&); // 32 bit usually float. //template <typename T, typename Policy> //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 1>&); // 64 bit usually double. //template <typename T, typename Policy> //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 2>&); // 80-bit long double. template <typename T> T lambert_w_positive_rational_float(T z) { BOOST_MATH_STD_USING if (z < 2) { if (z < 0.5) { // 0.05 < z < 0.5 // Maximum Deviation Found: 2.993e-08 // Expected Error Term : 2.993e-08 // Maximum Relative Change in Control Points : 7.555e-04 Y offset : -8.196592331e-01 static const T Y = 8.196592331e-01f; static const T P[] = { 1.803388345e-01f, -4.820256838e-01f, -1.068349741e+00f, -3.506624319e-02f, }; static const T Q[] = { 1.000000000e+00f, 2.871703469e+00f, 1.690949264e+00f, }; return z * (Y + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z)); } else { // 0.5 < z < 2 // Max error in interpolated form: 1.018e-08 static const T Y = 5.503368378e-01f; static const T P[] = { 4.493332766e-01f, 2.543432707e-01f, -4.808788799e-01f, -1.244425316e-01f, }; static const T Q[] = { 1.000000000e+00f, 2.780661241e+00f, 1.830840318e+00f, 2.407221031e-01f, }; return z * (Y + boost::math::tools::evaluate_rational(P, Q, z)); } } else if (z < 6) { // 2 < z < 6 // Max error in interpolated form: 2.944e-08 static const T Y = 1.162393570e+00f; static const T P[] = { -1.144183394e+00f, -4.712732855e-01f, 1.563162512e-01f, 1.434010911e-02f, }; static const T Q[] = { 1.000000000e+00f, 1.192626340e+00f, 2.295580708e-01f, 5.477869455e-03f, }; return Y + boost::math::tools::evaluate_rational(P, Q, z); } else if (z < 18) { // 6 < z < 18 // Max error in interpolated form: 5.893e-08 static const T Y = 1.809371948e+00f; static const T P[] = { -1.689291769e+00f, -3.337812742e-01f, 3.151434873e-02f, 1.134178734e-03f, }; static const T Q[] = { 1.000000000e+00f, 5.716915685e-01f, 4.489521292e-02f, 4.076716763e-04f, }; return Y + boost::math::tools::evaluate_rational(P, Q, z); } else if (z < 9897.12905874) // 2.8 < log(z) < 9.2 { // Max error in interpolated form: 1.771e-08 static const T Y = -1.402973175e+00f; static const T P[] = { 1.966174312e+00f, 2.350864728e-01f, -5.098074353e-02f, -1.054818339e-02f, }; static const T Q[] = { 1.000000000e+00f, 4.388208264e-01f, 8.316639634e-02f, 3.397187918e-03f, -1.321489743e-05f, }; T log_w = log(z); return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w); } else if (z < 7.896296e+13) // 9.2 < log(z) <= 32 { // Max error in interpolated form: 5.821e-08 static const T Y = -2.735729218e+00f; static const T P[] = { 3.424903470e+00f, 7.525631787e-02f, -1.427309584e-02f, -1.435974178e-05f, }; static const T Q[] = { 1.000000000e+00f, 2.514005579e-01f, 6.118994652e-03f, -1.357889535e-05f, 7.312865624e-08f, }; T log_w = log(z); return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w); } else // 32 < log(z) < 100 { // Max error in interpolated form: 1.491e-08 static const T Y = -4.012863159e+00f; static const T P[] = { 4.431629226e+00f, 2.756690487e-01f, -2.992956930e-03f, -4.912259384e-05f, }; static const T Q[] = { 1.000000000e+00f, 2.015434591e-01f, 4.949426142e-03f, 1.609659944e-05f, -5.111523436e-09f, }; T log_w = log(z); return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w); } } template <typename T, typename Policy> T lambert_w_negative_rational_float(T z, const Policy& pol) { BOOST_MATH_STD_USING if (z > -0.27) { if (z < -0.051) { // -0.27 < z < -0.051 // Max error in interpolated form: 5.080e-08 static const T Y = 1.255809784e+00f; static const T P[] = { -2.558083412e-01f, -2.306524098e+00f, -5.630887033e+00f, -3.803974556e+00f, }; static const T Q[] = { 1.000000000e+00f, 5.107680783e+00f, 7.914062868e+00f, 3.501498501e+00f, }; return z * (Y + boost::math::tools::evaluate_rational(P, Q, z)); } else { // Very small z so use a series function. return lambert_w0_small_z(z, pol); } } else if (z > -0.3578794411714423215955237701) { // Very close to branch singularity. // Max error in interpolated form: 5.269e-08 static const T Y = 1.220928431e-01f; static const T P[] = { -1.221787446e-01f, -6.816155875e+00f, 7.144582035e+01f, 1.128444390e+03f, }; static const T Q[] = { 1.000000000e+00f, 6.480326790e+01f, 1.869145243e+02f, -1.361804274e+03f, 1.117826726e+03f, }; T d = z + 0.367879441171442321595523770161460867445811f; return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d)); } else { // z is very close (within 0.01) of the singularity at e^-1. return lambert_w_singularity_series(get_near_singularity_param(z, pol)); } } //! Lambert_w0 @b 'float' implementation, selected when T is 32-bit precision. template <typename T, typename Policy> inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 1>&) { static const char* function = "boost::math::lambert_w0<%1%>"; // For error messages. BOOST_MATH_STD_USING // Aid ADL of std functions. if ((boost::math::isnan)(z)) { return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol); } if ((boost::math::isinf)(z)) { return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol); } if (z >= 0.05) // Fukushima switch point. // if (z >= 0.045) // 34 terms makes 128-bit 'exact' below 0.045. { // Normal ranges using several rational polynomials. return lambert_w_positive_rational_float(z); } else if (z <= -0.3678794411714423215955237701614608674458111310f) { if (z < -0.3678794411714423215955237701614608674458111310f) return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol); return -1; } else // z < 0.05 { return lambert_w_negative_rational_float(z, pol); } } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 1>&) for 32-bit usually float. template <typename T> T lambert_w_positive_rational_double(T z) { BOOST_MATH_STD_USING if (z < 2) { if (z < 0.5) { // Max error in interpolated form: 2.255e-17 static const T offset = 8.19659233093261719e-01; static const T P[] = { 1.80340766906685177e-01, 3.28178241493119307e-01, -2.19153620687139706e+00, -7.24750929074563990e+00, -7.28395876262524204e+00, -2.57417169492512916e+00, -2.31606948888704503e-01 }; static const T Q[] = { 1.00000000000000000e+00, 7.36482529307436604e+00, 2.03686007856430677e+01, 2.62864592096657307e+01, 1.59742041380858333e+01, 4.03760534788374589e+00, 2.91327346750475362e-01 }; return z * (offset + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z)); } else { // Max error in interpolated form: 3.806e-18 static const T offset = 5.50335884094238281e-01; static const T P[] = { 4.49664083944098322e-01, 1.90417666196776909e+00, 1.99951368798255994e+00, -6.91217310299270265e-01, -1.88533935998617058e+00, -7.96743968047750836e-01, -1.02891726031055254e-01, -3.09156013592636568e-03 }; static const T Q[] = { 1.00000000000000000e+00, 6.45854489419584014e+00, 1.54739232422116048e+01, 1.72606164253337843e+01, 9.29427055609544096e+00, 2.29040824649748117e+00, 2.21610620995418981e-01, 5.70597669908194213e-03 }; return z * (offset + boost::math::tools::evaluate_rational(P, Q, z)); } } else if (z < 6) { // 2 < z < 6 // Max error in interpolated form: 1.216e-17 static const T Y = 1.16239356994628906e+00; static const T P[] = { -1.16230494982099475e+00, -3.38528144432561136e+00, -2.55653717293161565e+00, -3.06755172989214189e-01, 1.73149743765268289e-01, 3.76906042860014206e-02, 1.84552217624706666e-03, 1.69434126904822116e-05, }; static const T Q[] = { 1.00000000000000000e+00, 3.77187616711220819e+00, 4.58799960260143701e+00, 2.24101228462292447e+00, 4.54794195426212385e-01, 3.60761772095963982e-02, 9.25176499518388571e-04, 4.43611344705509378e-06, }; return Y + boost::math::tools::evaluate_rational(P, Q, z); } else if (z < 18) { // 6 < z < 18 // Max error in interpolated form: 1.985e-19 static const T offset = 1.80937194824218750e+00; static const T P[] = { -1.80690935424793635e+00, -3.66995929380314602e+00, -1.93842957940149781e+00, -2.94269984375794040e-01, 1.81224710627677778e-03, 2.48166798603547447e-03, 1.15806592415397245e-04, 1.43105573216815533e-06, 3.47281483428369604e-09 }; static const T Q[] = { 1.00000000000000000e+00, 2.57319080723908597e+00, 1.96724528442680658e+00, 5.84501352882650722e-01, 7.37152837939206240e-02, 3.97368430940416778e-03, 8.54941838187085088e-05, 6.05713225608426678e-07, 8.17517283816615732e-10 }; return offset + boost::math::tools::evaluate_rational(P, Q, z); } else if (z < 9897.12905874) // 2.8 < log(z) < 9.2 { // Max error in interpolated form: 1.195e-18 static const T Y = -1.40297317504882812e+00; static const T P[] = { 1.97011826279311924e+00, 1.05639945701546704e+00, 3.33434529073196304e-01, 3.34619153200386816e-02, -5.36238353781326675e-03, -2.43901294871308604e-03, -2.13762095619085404e-04, -4.85531936495542274e-06, -2.02473518491905386e-08, }; static const T Q[] = { 1.00000000000000000e+00, 8.60107275833921618e-01, 4.10420467985504373e-01, 1.18444884081994841e-01, 2.16966505556021046e-02, 2.24529766630769097e-03, 9.82045090226437614e-05, 1.36363515125489502e-06, 3.44200749053237945e-09, }; T log_w = log(z); return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w); } else if (z < 7.896296e+13) // 9.2 < log(z) <= 32 { // Max error in interpolated form: 6.529e-18 static const T Y = -2.73572921752929688e+00; static const T P[] = { 3.30547638424076217e+00, 1.64050071277550167e+00, 4.57149576470736039e-01, 4.03821227745424840e-02, -4.99664976882514362e-04, -1.28527893803052956e-04, -2.95470325373338738e-06, -1.76662025550202762e-08, -1.98721972463709290e-11, }; static const T Q[] = { 1.00000000000000000e+00, 6.91472559412458759e-01, 2.48154578891676774e-01, 4.60893578284335263e-02, 3.60207838982301946e-03, 1.13001153242430471e-04, 1.33690948263488455e-06, 4.97253225968548872e-09, 3.39460723731970550e-12, }; T log_w = log(z); return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w); } else if (z < 2.6881171e+43) // 32 < log(z) < 100 { // Max error in interpolated form: 2.015e-18 static const T Y = -4.01286315917968750e+00; static const T P[] = { 5.07714858354309672e+00, -3.32994414518701458e+00, -8.61170416909864451e-01, -4.01139705309486142e-02, -1.85374201771834585e-04, 1.08824145844270666e-05, 1.17216905810452396e-07, 2.97998248101385990e-10, 1.42294856434176682e-13, }; static const T Q[] = { 1.00000000000000000e+00, -4.85840770639861485e-01, -3.18714850604827580e-01, -3.20966129264610534e-02, -1.06276178044267895e-03, -1.33597828642644955e-05, -6.27900905346219472e-08, -9.35271498075378319e-11, -2.60648331090076845e-14, }; T log_w = log(z); return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w); } else // 100 < log(z) < 710 { // Max error in interpolated form: 5.277e-18 static const T Y = -5.70115661621093750e+00; static const T P[] = { 6.42275660145116698e+00, 1.33047964073367945e+00, 6.72008923401652816e-02, 1.16444069958125895e-03, 7.06966760237470501e-06, 5.48974896149039165e-09, -7.00379652018853621e-11, -1.89247635913659556e-13, -1.55898770790170598e-16, -4.06109208815303157e-20, -2.21552699006496737e-24, }; static const T Q[] = { 1.00000000000000000e+00, 3.34498588416632854e-01, 2.51519862456384983e-02, 6.81223810622416254e-04, 7.94450897106903537e-06, 4.30675039872881342e-08, 1.10667669458467617e-10, 1.31012240694192289e-13, 6.53282047177727125e-17, 1.11775518708172009e-20, 3.78250395617836059e-25, }; T log_w = log(z); return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w); } } template <typename T, typename Policy> T lambert_w_negative_rational_double(T z, const Policy& pol) { BOOST_MATH_STD_USING if (z > -0.1) { if (z < -0.051) { // -0.1 < z < -0.051 // Maximum Deviation Found: 4.402e-22 // Expected Error Term : 4.240e-22 // Maximum Relative Change in Control Points : 4.115e-03 static const T Y = 1.08633995056152344e+00; static const T P[] = { -8.63399505615014331e-02, -1.64303871814816464e+00, -7.71247913918273738e+00, -1.41014495545382454e+01, -1.02269079949257616e+01, -2.17236002836306691e+00, }; static const T Q[] = { 1.00000000000000000e+00, 7.44775406945739243e+00, 2.04392643087266541e+01, 2.51001961077774193e+01, 1.31256080849023319e+01, 2.11640324843601588e+00, }; return z * (Y + boost::math::tools::evaluate_rational(P, Q, z)); } else { // Very small z > 0.051: return lambert_w0_small_z(z, pol); } } else if (z > -0.2) { // -0.2 < z < -0.1 // Maximum Deviation Found: 2.898e-20 // Expected Error Term : 2.873e-20 // Maximum Relative Change in Control Points : 3.779e-04 static const T Y = 1.20359611511230469e+00; static const T P[] = { -2.03596115108465635e-01, -2.95029082937201859e+00, -1.54287922188671648e+01, -3.81185809571116965e+01, -4.66384358235575985e+01, -2.59282069989642468e+01, -4.70140451266553279e+00, }; static const T Q[] = { 1.00000000000000000e+00, 9.57921436074599929e+00, 3.60988119290234377e+01, 6.73977699505546007e+01, 6.41104992068148823e+01, 2.82060127225153607e+01, 4.10677610657724330e+00, }; return z * (Y + boost::math::tools::evaluate_rational(P, Q, z)); } else if (z > -0.3178794411714423215955237) { // Max error in interpolated form: 6.996e-18 static const T Y = 3.49680423736572266e-01; static const T P[] = { -3.49729841718749014e-01, -6.28207407760709028e+01, -2.57226178029669171e+03, -2.50271008623093747e+04, 1.11949239154711388e+05, 1.85684566607844318e+06, 4.80802490427638643e+06, 2.76624752134636406e+06, }; static const T Q[] = { 1.00000000000000000e+00, 1.82717661215113000e+02, 8.00121119810280100e+03, 1.06073266717010129e+05, 3.22848993926057721e+05, -8.05684814514171256e+05, -2.59223192927265737e+06, -5.61719645211570871e+05, 6.27765369292636844e+04, }; T d = z + 0.367879441171442321595523770161460867445811; return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d)); } else if (z > -0.3578794411714423215955237701) { // Max error in interpolated form: 1.404e-17 static const T Y = 5.00126481056213379e-02; static const T P[] = { -5.00173570682372162e-02, -4.44242461870072044e+01, -9.51185533619946042e+03, -5.88605699015429386e+05, -1.90760843597427751e+06, 5.79797663818311404e+08, 1.11383352508459134e+10, 5.67791253678716467e+10, 6.32694500716584572e+10, }; static const T Q[] = { 1.00000000000000000e+00, 9.08910517489981551e+02, 2.10170163753340133e+05, 1.67858612416470327e+07, 4.90435561733227953e+08, 4.54978142622939917e+09, 2.87716585708739168e+09, -4.59414247951143131e+10, -1.72845216404874299e+10, }; T d = z + 0.36787944117144232159552377016146086744581113103176804; return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d)); } else { // z is very close (within 0.01) of the singularity at -e^-1, // so use a series expansion from R. M. Corless et al. const T p2 = 2 * (boost::math::constants::e<T>() * z + 1); const T p = sqrt(p2); return lambert_w_detail::lambert_w_singularity_series(p); } } //! Lambert_w0 @b 'double' implementation, selected when T is 64-bit precision. template <typename T, typename Policy> inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 2>&) { static const char* function = "boost::math::lambert_w0<%1%>"; BOOST_MATH_STD_USING // Aid ADL of std functions. // Detect unusual case of 32-bit double with a wider/64-bit long double static_assert(std::numeric_limits<double>::digits >= 53, "Our double precision coefficients will be truncated, " "please file a bug report with details of your platform's floating point types " "- or possibly edit the coefficients to have " "an appropriate size-suffix for 64-bit floats on your platform - L?"); if ((boost::math::isnan)(z)) { return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol); } if ((boost::math::isinf)(z)) { return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol); } if (z >= 0.05) { return lambert_w_positive_rational_double(z); } else if (z <= -0.36787944117144232159552377016146086744581113103176804) // Precision is max_digits10(cpp_bin_float_50). { if (z < -0.36787944117144232159552377016146086744581113103176804) { return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol); } return -1; } else { return lambert_w_negative_rational_double(z, pol); } } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 2>&) 64-bit precision, usually double. //! lambert_W0 implementation for extended precision types including //! long double (80-bit and 128-bit), ??? //! quad float128, Boost.Multiprecision types like cpp_bin_float_quad, cpp_bin_float_50... template <typename T, typename Policy> inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 0>&) { static const char* function = "boost::math::lambert_w0<%1%>"; BOOST_MATH_STD_USING // Aid ADL of std functions. // Filter out special cases first: if ((boost::math::isnan)(z)) { return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol); } if (fabs(z) <= 0.05f) { // Very small z: return lambert_w0_small_z(z, pol); } if (z > (std::numeric_limits<double>::max)()) { if ((boost::math::isinf)(z)) { return policies::raise_overflow_error<T>(function, 0, pol); // Or might return infinity if available else max_value, // but other Boost.Math special functions raise overflow. } // z is larger than the largest double, so cannot use the polynomial to get an approximation, // so use the asymptotic approximation and Halley iterate: T w = lambert_w0_approx(z); // Make an inline function as also used elsewhere. //T lz = log(z); //T llz = log(lz); //T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162. return lambert_w_halley_iterate(w, z); } if (z < -0.3578794411714423215955237701) { // Very close to branch point so rational polynomials are not usable. if (z <= -boost::math::constants::exp_minus_one<T>()) { if (z == -boost::math::constants::exp_minus_one<T>()) { // Exactly at the branch point singularity. return -1; } return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol); } // z is very close (within 0.01) of the branch singularity at -e^-1 // so use a series approximation proposed by Corless et al. const T p2 = 2 * (boost::math::constants::e<T>() * z + 1); const T p = sqrt(p2); T w = lambert_w_detail::lambert_w_singularity_series(p); return lambert_w_halley_iterate(w, z); } // Phew! If we get here we are in the normal range of the function, // so get a double precision approximation first, then iterate to full precision of T. // We define a tag_type that is: // true_type if there are so many digits precision wanted that iteration is necessary. // false_type if a single Halley step is sufficient. using precision_type = typename policies::precision<T, Policy>::type; using tag_type = std::integral_constant<bool, (precision_type::value == 0) || (precision_type::value > 113) ? true // Unknown at compile-time, variable/arbitrary, or more than float128 or cpp_bin_quad 128-bit precision. : false // float, double, float128, cpp_bin_quad 128-bit, so single Halley step. >; // For speed, we also cast z to type double when that is possible // if (std::is_constructible<double, T>() == true). T w = lambert_w0_imp(maybe_reduce_to_double(z, std::is_constructible<double, T>()), pol, std::integral_constant<int, 2>()); return lambert_w_maybe_halley_iterate(w, z, tag_type()); } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 0>&) all extended precision types. // Lambert w-1 implementation // ============================================================================================== //! Lambert W for W-1 branch, -max(z) < z <= -1/e. // TODO is -max(z) allowed? template<typename T, typename Policy> T lambert_wm1_imp(const T z, const Policy& pol) { // Catch providing an integer value as parameter x to lambert_w, for example, lambert_w(1). // Need to ensure it is a floating-point type (of the desired type, float 1.F, double 1., or long double 1.L), // or static_casted integer, for example: static_cast<float>(1) or static_cast<cpp_dec_float_50>(1). // Want to allow fixed_point types too, so do not just test for floating-point. // Integral types should be promoted to double by user Lambert w functions. // If integral type provided to user function lambert_w0 or lambert_wm1, // then should already have been promoted to double. static_assert(!std::is_integral<T>::value, "Must be floating-point or fixed type (not integer type), for example: lambert_wm1(1.), not lambert_wm1(1)!"); BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs. const char* function = "boost::math::lambert_wm1<RealType>(<RealType>)"; // Used for error messages. // Check for edge and corner cases first: if ((boost::math::isnan)(z)) { return policies::raise_domain_error(function, "Argument z is NaN!", z, pol); } // isnan if ((boost::math::isinf)(z)) { return policies::raise_domain_error(function, "Argument z is infinite!", z, pol); } // isinf if (z == static_cast<T>(0)) { // z is exactly zero so return -std::numeric_limits<T>::infinity(); if (std::numeric_limits<T>::has_infinity) { return -std::numeric_limits<T>::infinity(); } else { return -tools::max_value<T>(); } } if (std::numeric_limits<T>::has_denorm) { // All real types except arbitrary precision. if (!(boost::math::isnormal)(z)) { // Almost zero - might also just return infinity like z == 0 or max_value? return policies::raise_overflow_error(function, "Argument z = %1% is denormalized! (must be z > (std::numeric_limits<RealType>::min)() or z == 0)", z, pol); } } if (z > static_cast<T>(0)) { // return policies::raise_domain_error(function, "Argument z = %1% is out of range (z <= 0) for Lambert W-1 branch! (Try Lambert W0 branch?)", z, pol); } if (z > -boost::math::tools::min_value<T>()) { // z is denormalized, so cannot be computed. // -std::numeric_limits<T>::min() is smallest for type T, // for example, for double: lambert_wm1(-2.2250738585072014e-308) = -714.96865723796634 return policies::raise_overflow_error(function, "Argument z = %1% is too small (z < -std::numeric_limits<T>::min so denormalized) for Lambert W-1 branch!", z, pol); } if (z == -boost::math::constants::exp_minus_one<T>()) // == singularity/branch point z = -exp(-1) = -3.6787944. { // At singularity, so return exactly -1. return -static_cast<T>(1); } // z is too negative for the W-1 (or W0) branch. if (z < -boost::math::constants::exp_minus_one<T>()) // > singularity/branch point z = -exp(-1) = -3.6787944. { return policies::raise_domain_error(function, "Argument z = %1% is out of range (z < -exp(-1) = -3.6787944... <= 0) for Lambert W-1 (or W0) branch!", z, pol); } if (z < static_cast<T>(-0.35)) { // Close to singularity/branch point z = -0.3678794411714423215955237701614608727 but on W-1 branch. const T p2 = 2 * (boost::math::constants::e<T>() * z + 1); if (p2 == 0) { // At the singularity at branch point. return -1; } if (p2 > 0) { T w_series = lambert_w_singularity_series(T(-sqrt(p2))); if (boost::math::tools::digits<T>() > 53) { // Multiprecision, so try a Halley refinement. w_series = lambert_w_detail::lambert_w_halley_iterate(w_series, z); #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); std::cout << "Lambert W-1 Halley updated to " << w_series << std::endl; std::cout.precision(saved_precision); #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN } return w_series; } // Should not get here. return policies::raise_domain_error(function, "Argument z = %1% is out of range for Lambert W-1 branch. (Should not get here - please report!)", z, pol); } // if (z < -0.35) using lambert_w_lookup::wm1es; using lambert_w_lookup::wm1zs; using lambert_w_lookup::noof_wm1zs; // size == 64 // std::cout <<" Wm1zs[63] (== G[64]) = " << " " << wm1zs[63] << std::endl; // Wm1zs[63] (== G[64]) = -1.0264389699511283e-26 // Check that z argument value is not smaller than lookup_table G[64] // std::cout << "(z > wm1zs[63]) = " << std::boolalpha << (z > wm1zs[63]) << std::endl; if (z >= wm1zs[63]) // wm1zs[63] = -1.0264389699511282259046957018510946438e-26L W = 64.00000000000000000 { // z >= -1.0264389699511303e-26 (but z != 0 and z >= std::numeric_limits<T>::min() and so NOT denormalized). // Some info on Lambert W-1 values for extreme values of z. // std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // std::cout << "-std::numeric_limits<float>::min() = " << -(std::numeric_limits<float>::min)() << std::endl; // std::cout << "-std::numeric_limits<double>::min() = " << -(std::numeric_limits<double>::min)() << std::endl; // -std::numeric_limits<float>::min() = -1.1754943508222875e-38 // -std::numeric_limits<double>::min() = -2.2250738585072014e-308 // N[productlog(-1, -1.1754943508222875 * 10^-38 ), 50] = -91.856775324595479509567756730093823993834155027858 // N[productlog(-1, -2.2250738585072014e-308 * 10^-308 ), 50] = -1424.8544521230553853558132180518404363617968042942 // N[productlog(-1, -1.4325445274604020119111357113179868158* 10^-27), 37] = -65.99999999999999999999999999999999955 // R.M.Corless, G.H.Gonnet, D.E.G.Hare, D.J.Jeffrey, and D.E.Knuth, // On the Lambert W function, Adv.Comput.Math., vol. 5, pp. 329, 1996. // Francois Chapeau-Blondeau and Abdelilah Monir // Numerical Evaluation of the Lambert W Function // IEEE Transactions On Signal Processing, VOL. 50, NO. 9, Sep 2002 // https://pdfs.semanticscholar.org/7a5a/76a9369586dd0dd34dda156d8f2779d1fd59.pdf // Estimate Lambert W using ln(-z) ... // This is roughly the power of ten * ln(10) ~= 2.3. n ~= 10^n // and improve by adding a second term -ln(ln(-z)) T guess; // bisect lowest possible Gk[=64] (for lookup_t type) T lz = log(-z); T llz = log(-lz); guess = lz - llz + (llz / lz); // Chapeau-Blondeau equation 20, page 2162. #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); std::cout << "z = " << z << ", guess = " << guess << ", ln(-z) = " << lz << ", ln(-ln(-z) = " << llz << ", llz/lz = " << (llz / lz) << std::endl; // z = -1.0000000000000001e-30, guess = -73.312782616731482, ln(-z) = -69.077552789821368, ln(-ln(-z) = 4.2352298269101114, llz/lz = -0.061311231447304194 // z = -9.9999999999999999e-91, guess = -212.56650048504233, ln(-z) = -207.23265836946410, ln(-ln(-z) = 5.3338421155782205, llz/lz = -0.025738424423764311 // >z = -2.2250738585072014e-308, guess = -714.95942238244606, ln(-z) = -708.39641853226408, ln(-ln(-z) = 6.5630038501819854, llz/lz = -0.0092645920821846622 int d10 = policies::digits_base10<T, Policy>(); // policy template parameter digits10 int d2 = policies::digits<T, Policy>(); // digits base 2 from policy. std::cout << "digits10 = " << d10 << ", digits2 = " << d2 // For example: digits10 = 1, digits2 = 5 << std::endl; std::cout.precision(saved_precision); #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY if (policies::digits<T, Policy>() < 12) { // 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. return guess; } T result = lambert_w_detail::lambert_w_halley_iterate(guess, z); return result; // Was Fukushima // G[k=64] == g[63] == -1.02643897e-26 //return policies::raise_domain_error(function, // "Argument z = %1% is too small (< -1.02643897e-26) ! (Should not occur, please report.", // z, pol); } // Z too small so use approximation and Halley. // Else Use a lookup table to find the nearest integer part of Lambert W-1 as starting point for Bisection. if (boost::math::tools::digits<T>() > 53) { // T is more precise than 64-bit double (or long double, or ?), // so compute an approximate value using only one Schroeder refinement, // (avoiding any double-precision Halley refinement from policy double_digits2<50> 53 - 3 = 50 // because are next going to use Halley refinement at full/high precision using this as an approximation). using boost::math::policies::precision; using boost::math::policies::digits10; using boost::math::policies::digits2; using boost::math::policies::policy; // Compute a 50-bit precision approximate W0 in a double (no Halley refinement). T double_approx(static_cast<T>(lambert_wm1_imp(must_reduce_to_double(z, std::is_constructible<double, T>()), policy<digits2<50>>()))); #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); std::cout << "Lambert_wm1 Argument Type " << typeid(T).name() << " approximation double = " << double_approx << std::endl; std::cout.precision(saved_precision); #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1 // Perform additional Halley refinement(s) to ensure that // get a near as possible to correct result (usually +/- one epsilon). T result = lambert_w_halley_iterate(double_approx, z); #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); std::cout << "Result " << typeid(T).name() << " precision Halley refinement = " << result << std::endl; std::cout.precision(saved_precision); #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1 return result; } // digits > 53 - higher precision than double. else // T is double or less precision. { // Use a lookup table to find the nearest integer part of Lambert W as starting point for Bisection. using namespace boost::math::lambert_w_detail::lambert_w_lookup; // Bracketing sequence n = (2, 4, 8, 16, 32, 64) for W-1 branch. (0 is -infinity) // Since z is probably quite small, start with lowest n (=2). int n = 2; if (wm1zs[n - 1] > z) { goto bisect; } for (int j = 1; j <= 5; ++j) { n *= 2; if (wm1zs[n - 1] > z) { goto overshot; } } // else z < g[63] == -1.0264389699511303e-26, so Lambert W-1 integer part > 64. // This should not now occur (should be caught by test and code above) so should be a logic_error? return policies::raise_domain_error(function, "Argument z = %1% is too small (< -1.026439e-26) (logic error - please report!)", z, pol); overshot: { int nh = n / 2; for (int j = 1; j <= 5; ++j) { nh /= 2; // halve step size. if (nh <= 0) { break; // goto bisect; } if (wm1zs[n - nh - 1] > z) { n -= nh; } } } bisect: --n; // g[n] now holds lambert W of floor integer n and g[n+1] the ceil part; // these are used as initial values for bisection. #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); std::cout << "Result lookup W-1(" << z << ") bisection between wm1zs[" << n - 1 << "] = " << wm1zs[n - 1] << " and wm1zs[" << n << "] = " << wm1zs[n] << ", bisect mean = " << (wm1zs[n - 1] + wm1zs[n]) / 2 << std::endl; std::cout.precision(saved_precision); #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP // Compute bisections is the number of bisections computed from n, // such that a single application of the fifth-order Schroeder update formula // after the bisections is enough to evaluate Lambert W-1 with (near?) 53-bit accuracy. // Fukushima established these by trial and error? int bisections = 11; // Assume maximum number of bisections will be needed (most common case). if (n >= 8) { bisections = 8; } else if (n >= 3) { bisections = 9; } else if (n >= 2) { bisections = 10; } // Bracketing, Fukushima section 2.3, page 82: // (Avoiding using exponential function for speed). // Only use @c lookup_t precision, default double, for bisection (again for speed), // and use later Halley refinement for higher precisions. using lambert_w_lookup::halves; using lambert_w_lookup::sqrtwm1s; using calc_type = typename std::conditional<std::is_constructible<lookup_t, T>::value, lookup_t, T>::type; calc_type w = -static_cast<calc_type>(n); // Equation 25, calc_type y = static_cast<calc_type>(z * wm1es[n - 1]); // Equation 26, // Perform the bisections fractional bisections for necessary precision. for (int j = 0; j < bisections; ++j) { // Equation 27. calc_type wj = w - halves[j]; // Subtract 1/2, 1/4, 1/8 ... calc_type yj = y * sqrtwm1s[j]; // Multiply by sqrt(1/e), ... if (wj < yj) { w = wj; y = yj; } } // for j return static_cast<T>(schroeder_update(w, y)); // Schroeder 5th order method refinement. // else // Perform additional Halley refinement(s) to ensure that // // get a near as possible to correct result (usually +/- epsilon). // { // // result = lambert_w_halley_iterate(result, z); // result = lambert_w_halley_step(result, z); // Just one Halley step should be enough. //#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_HALLEY // std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // std::cout << "Halley refinement estimate = " << result << std::endl; // std::cout.precision(saved_precision); //#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W1_HALLEY // return result; // Halley // } // Schroeder or Schroeder and Halley. } } // template<typename T = double> T lambert_wm1_imp(const T z) } // namespace lambert_w_detail ///////////////////////////// User Lambert w functions. ////////////////////////////// //! Lambert W0 using User-defined policy. template <typename T, typename Policy> inline typename boost::math::tools::promote_args<T>::type lambert_w0(T z, const Policy& pol) { // Promote integer or expression template arguments to double, // without doing any other internal promotion like float to double. using result_type = typename tools::promote_args<T>::type; // Work out what precision has been selected, // based on the Policy and the number type. using precision_type = typename policies::precision<result_type, Policy>::type; // and then select the correct implementation based on that precision (not the type T): using tag_type = std::integral_constant<int, (precision_type::value == 0) || (precision_type::value > 53) ? 0 // either variable precision (0), or greater than 64-bit precision. : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision. : 2 // 64-bit (probably double) precision. >; return lambert_w_detail::lambert_w0_imp(result_type(z), pol, tag_type()); // } // lambert_w0(T z, const Policy& pol) //! Lambert W0 using default policy. template <typename T> inline typename tools::promote_args<T>::type lambert_w0(T z) { // Promote integer or expression template arguments to double, // without doing any other internal promotion like float to double. using result_type = typename tools::promote_args<T>::type; // Work out what precision has been selected, based on the Policy and the number type. // For the default policy version, we want the *default policy* precision for T. using precision_type = typename policies::precision<result_type, policies::policy<>>::type; // and then select the correct implementation based on that (not the type T): using tag_type = std::integral_constant<int, (precision_type::value == 0) || (precision_type::value > 53) ? 0 // either variable precision (0), or greater than 64-bit precision. : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision. : 2 // 64-bit (probably double) precision. >; return lambert_w_detail::lambert_w0_imp(result_type(z), policies::policy<>(), tag_type()); } // lambert_w0(T z) using default policy. //! W-1 branch (-max(z) < z <= -1/e). //! Lambert W-1 using User-defined policy. template <typename T, typename Policy> inline typename tools::promote_args<T>::type lambert_wm1(T z, const Policy& pol) { // Promote integer or expression template arguments to double, // without doing any other internal promotion like float to double. using result_type = typename tools::promote_args<T>::type; return lambert_w_detail::lambert_wm1_imp(result_type(z), pol); // } //! Lambert W-1 using default policy. template <typename T> inline typename tools::promote_args<T>::type lambert_wm1(T z) { using result_type = typename tools::promote_args<T>::type; return lambert_w_detail::lambert_wm1_imp(result_type(z), policies::policy<>()); } // lambert_wm1(T z) // First derivative of Lambert W0 and W-1. template <typename T, typename Policy> inline typename tools::promote_args<T>::type lambert_w0_prime(T z, const Policy& pol) { using result_type = typename tools::promote_args<T>::type; using std::numeric_limits; if (z == 0) { return static_cast<result_type>(1); } // This is the sensible choice if we regard the Lambert-W function as complex analytic. // Of course on the real line, it's just undefined. if (z == - boost::math::constants::exp_minus_one<result_type>()) { return numeric_limits<result_type>::has_infinity ? numeric_limits<result_type>::infinity() : boost::math::tools::max_value<result_type>(); } // if z < -1/e, we'll let lambert_w0 do the error handling: result_type w = lambert_w0(result_type(z), pol); // If w ~ -1, then presumably this can get inaccurate. // Is there an accurate way to evaluate 1 + W(-1/e + eps)? // Yes: This is discussed in the Princeton Companion to Applied Mathematics, // 'The Lambert-W function', Section 1.3: Series and Generating Functions. // 1 + W(-1/e + x) ~ sqrt(2ex). // Nick is not convinced this formula is more accurate than the naive one. // However, for z != -1/e, we never get rounded to w = -1 in any precision I've tested (up to cpp_bin_float_100). return w / (z * (1 + w)); } // lambert_w0_prime(T z) template <typename T> inline typename tools::promote_args<T>::type lambert_w0_prime(T z) { return lambert_w0_prime(z, policies::policy<>()); } template <typename T, typename Policy> inline typename tools::promote_args<T>::type lambert_wm1_prime(T z, const Policy& pol) { using std::numeric_limits; using result_type = typename tools::promote_args<T>::type; //if (z == 0) //{ // return static_cast<result_type>(1); //} //if (z == - boost::math::constants::exp_minus_one<result_type>()) if (z == 0 || z == - boost::math::constants::exp_minus_one<result_type>()) { return numeric_limits<result_type>::has_infinity ? -numeric_limits<result_type>::infinity() : -boost::math::tools::max_value<result_type>(); } result_type w = lambert_wm1(z, pol); return w/(z*(1+w)); } // lambert_wm1_prime(T z) template <typename T> inline typename tools::promote_args<T>::type lambert_wm1_prime(T z) { return lambert_wm1_prime(z, policies::policy<>()); } }} //boost::math namespaces #endif // #ifdef BOOST_MATH_SF_LAMBERT_W_HPP
