libs/math/example/fft_sines_table.cpp
// Use, modification and distribution are subject to 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) // Copyright Paul A. Bristow 2013. // Copyright Christopher Kormanyos 2012, 2013. // Copyright John Maddock 2013. // This file is written to be included from a Quickbook .qbk document. // It can be compiled by the C++ compiler, and run. Any output can // also be added here as comment or included or pasted in elsewhere. // Caution: this file contains Quickbook markup as well as code // and comments: don't change any of the special comment markups! #ifdef _MSC_VER # pragma warning (disable : 4996) // -D_SCL_SECURE_NO_WARNINGS. #endif //[fft_sines_table_example_1 /*`[h5 Using Boost.Multiprecision to generate a high-precision array of sine coefficients for use with FFT.] The Boost.Multiprecision library can be used for computations requiring precision exceeding that of standard built-in types such as `float`, `double` and `long double`. For extended-precision calculations, Boost.Multiprecision supplies a template data type called `cpp_bin_float`. The number of decimal digits of precision is fixed at compile-time via a template parameter. One often needs to compute tables of numbers in mathematical software. To avoid the [@https://en.wikipedia.org/wiki/Rounding#Table-maker's_dilemma Table-maker's dilemma] it is necessary to use a higher precision type to compute the table values so that they have the nearest representable bit-pattern for the type, say `double`, of the table value. This example is a program `fft_since_table.cpp` that writes a header file `sines.hpp` containing an array of sine coefficients for use with a Fast Fourier Transform (FFT), that can be included by the FFT program. To use Boost.Multiprecision's high-precision floating-point types and constants, we need some includes: */ #include <boost/math/constants/constants.hpp> // using boost::math::constants::pi; #include <boost/multiprecision/cpp_bin_float.hpp> // for // using boost::multiprecision::cpp_bin_float and // using boost::multiprecision::cpp_bin_float_50; // using boost::multiprecision::cpp_bin_float_quad; #include <boost/array.hpp> // or <array> for std::array #include <iostream> #include <limits> #include <vector> #include <algorithm> #include <iomanip> #include <iterator> #include <fstream> /*`First, this example defines a prolog text string which is a C++ comment with the program licence, copyright etc. (You would of course, tailor this to your needs, including *your* copyright claim). This will appear at the top of the written header file `sines.hpp`. */ //] [fft_sines_table_example_1] static const char* prolog = { "// Use, modification and distribution are subject to the\n" "// Boost Software License, Version 1.0.\n" "// (See accompanying file LICENSE_1_0.txt\n" "// or copy at ""http://www.boost.org/LICENSE_1_0.txt)\n\n" "// Copyright A N Other, 2019.\n\n" }; //[fft_sines_table_example_2 using boost::multiprecision::cpp_bin_float_50; using boost::math::constants::pi; //] [fft_sines_table_example_2] // VS 2010 (wrongly) requires these at file scope, not local scope in `main`. // This program also requires `-std=c++11` option to compile using Clang and GCC. int main() { //[fft_sines_table_example_3 /*`A fast Fourier transform (FFT), for example, may use a table of the values of sin(([pi]/2[super n]) in its implementation details. In order to maximize the precision in the FFT implementation, the precision of the tabulated trigonometric values should exceed that of the built-in floating-point type used in the FFT. The sample below computes a table of the values of sin([pi]/2[super n]) in the range 1 <= n <= 31. This program makes use of, among other program elements, the data type `boost::multiprecision::cpp_bin_float_50` for a precision of 50 decimal digits from Boost.Multiprecision, the value of constant [pi] retrieved from Boost.Math, guaranteed to be initialized with the very last bit of precision for the type, here `cpp_bin_float_50`, and a C++11 lambda function combined with `std::for_each()`. */ /*`define the number of values (32) in the array of sines. */ std::size_t size = 32U; //cpp_bin_float_50 p = pi<cpp_bin_float_50>(); cpp_bin_float_50 p = boost::math::constants::pi<cpp_bin_float_50>(); std::vector <cpp_bin_float_50> sin_values (size); unsigned n = 1U; // Generate the sine values. std::for_each ( sin_values.begin (), sin_values.end (), [&n](cpp_bin_float_50& y) { y = sin( pi<cpp_bin_float_50>() / pow(cpp_bin_float_50 (2), n)); ++n; } ); /*`Define the floating-point type for the generated file, either built-in `double, `float, or `long double`, or a user defined type like `cpp_bin_float_50`. */ std::string fp_type = "double"; std::cout << "Generating an `std::array` or `std::array` for floating-point type: " << fp_type << ". " << std::endl; /*`By default, output would only show the standard 6 decimal digits, so set precision to show enough significant digits for the chosen floating-point type. For `cpp_bin_float_50` is 50. (50 decimal digits should be ample for most applications). */ std::streamsize precision = std::numeric_limits<cpp_bin_float_50>::digits10; std::cout << "Sines table precision is " << precision << " decimal digits. " << std::endl; /*`Of course, one could also choose a lower precision for the table values, for example, `std::streamsize precision = std::numeric_limits<cpp_bin_float_quad>::max_digits10;` 128-bit 'quad' precision of 36 decimal digits would be sufficient for the most precise current `long double` implementations using 128-bit. In general, it should be a couple of decimal digits more (guard digits) than `std::numeric_limits<RealType>::max_digits10` for the target system floating-point type. (If the implementation does not provide `max_digits10`, the the Kahan formula `std::numeric_limits<RealType>::digits * 3010/10000 + 2` can be used instead). The compiler will read these values as decimal digits strings and use the nearest representation for the floating-point type. Now output all the sine table, to a file of your chosen name. */ const char sines_name[] = "sines.hpp"; // Assuming in same directory as .exe std::ofstream fout(sines_name, std::ios_base::out); // Creates if no file exists, // & uses default overwrite/ ios::replace. if (fout.is_open() == false) { // failed to open OK! std::cout << "Open file " << sines_name << " failed!" << std::endl; return EXIT_FAILURE; } else { // Write prolog etc as a C++ comment. std::cout << "Open file " << sines_name << " for output OK." << std::endl; fout << prolog << "// Table of " << sin_values.size() << " values with " << precision << " decimal digits precision,\n" "// generated by program fft_sines_table.cpp.\n" << std::endl; fout << "#include <array> // std::array" << std::endl; // Write the table of sines as a C++ array. fout << "\nstatic const std::array<double, " << size << "> sines =\n" "{{\n"; // 2nd { needed for some old GCC compiler versions. fout.precision(precision); for (unsigned int i = 0U; ;) { fout << " " << sin_values[i]; if (i == sin_values.size()-1) { // next is last value. fout << "\n}}; // array sines\n"; // 2nd } needed for some old GCC compiler versions. break; } else { fout << ",\n"; i++; } } // for fout.close(); std::cout << "Closed file " << sines_name << " for output." << std::endl; } //`The output file generated can be seen at [@../../example/sines.hpp] //] [/fft_sines_table_example_3] return EXIT_SUCCESS; } // int main() /* //[fft_sines_table_example_output The printed table is: 1 0.70710678118654752440084436210484903928483593768847 0.38268343236508977172845998403039886676134456248563 0.19509032201612826784828486847702224092769161775195 0.098017140329560601994195563888641845861136673167501 0.049067674327418014254954976942682658314745363025753 0.024541228522912288031734529459282925065466119239451 0.012271538285719926079408261951003212140372319591769 0.0061358846491544753596402345903725809170578863173913 0.003067956762965976270145365490919842518944610213452 0.0015339801862847656123036971502640790799548645752374 0.00076699031874270452693856835794857664314091945206328 0.00038349518757139558907246168118138126339502603496474 0.00019174759731070330743990956198900093346887403385916 9.5873799095977345870517210976476351187065612851145e-05 4.7936899603066884549003990494658872746866687685767e-05 2.3968449808418218729186577165021820094761474895673e-05 1.1984224905069706421521561596988984804731977538387e-05 5.9921124526424278428797118088908617299871778780951e-06 2.9960562263346607504548128083570598118251878683408e-06 1.4980281131690112288542788461553611206917585861527e-06 7.4901405658471572113049856673065563715595930217207e-07 3.7450702829238412390316917908463317739740476297248e-07 1.8725351414619534486882457659356361712045272098287e-07 9.3626757073098082799067286680885620193236507169473e-08 4.681337853654909269511551813854009695950362701667e-08 2.3406689268274552759505493419034844037886207223779e-08 1.1703344634137277181246213503238103798093456639976e-08 5.8516723170686386908097901008341396943900085051757e-09 2.9258361585343193579282304690689559020175857150074e-09 1.4629180792671596805295321618659637103742615227834e-09 */ //] [/fft_sines_table_example_output] //[fft_sines_table_example_check /*` The output can be copied as text and readily integrated into a given source code. Alternatively, the output can be written to a text or even be used within a self-written automatic code generator as this example. A computer algebra system can be used to verify the results obtained from Boost.Math and Boost.Multiprecision. For example, the __Mathematica computer algebra system can obtain a similar table with the command: Table[N[Sin[Pi / (2^n)], 50], {n, 1, 31, 1}] The __WolframAlpha computational knowledge engine can also be used to generate this table. The same command can be pasted into the compute box. */ //] [/fft_sines_table_example_check]