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- // 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 coefficents 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 `boost::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]
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