// (C) Copyright John Maddock 2006. // 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) #define BOOST_UBLAS_TYPE_CHECK_EPSILON (type_traits<real_type>::type_sqrt (boost::math::tools::epsilon <real_type>())) #define BOOST_UBLAS_TYPE_CHECK_MIN (type_traits<real_type>::type_sqrt ( boost::math::tools::min_value<real_type>())) #define BOOST_UBLAS_NDEBUG #include <boost/math/bindings/rr.hpp> namespace std{ using boost::math::ntl::pow; } // workaround for spirit parser. #include <boost/math/tools/remez.hpp> #include <boost/math/tools/test.hpp> #include <boost/math/special_functions/binomial.hpp> #include <boost/spirit/core.hpp> #include <boost/spirit/actor.hpp> #include <boost/lexical_cast.hpp> #include <iostream> #include <iomanip> #include <string> #include <boost/test/included/test_exec_monitor.hpp> // for test_main extern boost::math::ntl::RR f(const boost::math::ntl::RR& x, int variant); extern void show_extra( const boost::math::tools::polynomial<boost::math::ntl::RR>& n, const boost::math::tools::polynomial<boost::math::ntl::RR>& d, const boost::math::ntl::RR& x_offset, const boost::math::ntl::RR& y_offset, int variant); using namespace boost::spirit; boost::math::ntl::RR a(0), b(1); // range to optimise over bool rel_error(true); bool pin(false); int orderN(3); int orderD(1); int target_precision = boost::math::tools::digits<long double>(); int working_precision = target_precision * 2; bool started(false); int variant(0); int skew(0); int brake(50); boost::math::ntl::RR x_offset(0), y_offset(0), x_scale(1); bool auto_offset_y; boost::shared_ptr<boost::math::tools::remez_minimax<boost::math::ntl::RR> > p_remez; boost::math::ntl::RR the_function(const boost::math::ntl::RR& val) { return f(x_scale * (val + x_offset), variant) + y_offset; } void step_some(unsigned count) { try{ NTL::RR::SetPrecision(working_precision); if(!started) { // // If we have an automatic y-offset calculate it now: // if(auto_offset_y) { boost::math::ntl::RR fa, fb, fm; fa = f(x_scale * (a + x_offset), variant); fb = f(x_scale * (b + x_offset), variant); fm = f(x_scale * ((a+b)/2 + x_offset), variant); y_offset = -(fa + fb + fm) / 3; NTL::RR::SetOutputPrecision(5); std::cout << "Setting auto-y-offset to " << y_offset << std::endl; } // // Truncate offsets to float precision: // x_offset = NTL::RoundToPrecision(x_offset.value(), 20); y_offset = NTL::RoundToPrecision(y_offset.value(), 20); // // Construct new Remez state machine: // p_remez.reset(new boost::math::tools::remez_minimax<boost::math::ntl::RR>( &the_function, orderN, orderD, a, b, pin, rel_error, skew, working_precision)); std::cout << "Max error in interpolated form: " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast<double>(p_remez->max_error()) << std::endl; // // Signal that we've started: // started = true; } unsigned i; for(i = 0; i < count; ++i) { std::cout << "Stepping..." << std::endl; p_remez->set_brake(brake); boost::math::ntl::RR r = p_remez->iterate(); NTL::RR::SetOutputPrecision(3); std::cout << "Maximum Deviation Found: " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast<double>(p_remez->max_error()) << std::endl << "Expected Error Term: " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast<double>(p_remez->error_term()) << std::endl << "Maximum Relative Change in Control Points: " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast<double>(r) << std::endl; } } catch(const std::exception& e) { std::cout << "Step failed with exception: " << e.what() << std::endl; } } void step(const char*, const char*) { step_some(1); } void show(const char*, const char*) { NTL::RR::SetPrecision(working_precision); if(started) { boost::math::tools::polynomial<boost::math::ntl::RR> n = p_remez->numerator(); boost::math::tools::polynomial<boost::math::ntl::RR> d = p_remez->denominator(); std::vector<boost::math::ntl::RR> cn = n.chebyshev(); std::vector<boost::math::ntl::RR> cd = d.chebyshev(); int prec = 2 + (target_precision * 3010LL)/10000; std::cout << std::scientific << std::setprecision(prec); NTL::RR::SetOutputPrecision(prec); boost::numeric::ublas::vector<boost::math::ntl::RR> v = p_remez->zero_points(); std::cout << " Zeros = {\n"; unsigned i; for(i = 0; i < v.size(); ++i) { std::cout << " " << v[i] << std::endl; } std::cout << " }\n"; v = p_remez->chebyshev_points(); std::cout << " Chebeshev Control Points = {\n"; for(i = 0; i < v.size(); ++i) { std::cout << " " << v[i] << std::endl; } std::cout << " }\n"; std::cout << "X offset: " << x_offset << std::endl; std::cout << "X scale: " << x_scale << std::endl; std::cout << "Y offset: " << y_offset << std::endl; std::cout << "P = {"; for(i = 0; i < n.size(); ++i) { std::cout << " " << n[i] << "L," << std::endl; } std::cout << " }\n"; std::cout << "Q = {"; for(i = 0; i < d.size(); ++i) { std::cout << " " << d[i] << "L," << std::endl; } std::cout << " }\n"; std::cout << "CP = {"; for(i = 0; i < cn.size(); ++i) { std::cout << " " << cn[i] << "L," << std::endl; } std::cout << " }\n"; std::cout << "CQ = {"; for(i = 0; i < cd.size(); ++i) { std::cout << " " << cd[i] << "L," << std::endl; } std::cout << " }\n"; show_extra(n, d, x_offset, y_offset, variant); } else { std::cerr << "Nothing to display" << std::endl; } } void do_graph(unsigned points) { NTL::RR::SetPrecision(working_precision); boost::math::ntl::RR step = (b - a) / (points - 1); boost::math::ntl::RR x = a; while(points > 1) { NTL::RR::SetOutputPrecision(10); std::cout << std::setprecision(10) << std::setw(30) << std::left << boost::lexical_cast<std::string>(x) << the_function(x) << std::endl; --points; x += step; } std::cout << std::setprecision(10) << std::setw(30) << std::left << boost::lexical_cast<std::string>(b) << the_function(b) << std::endl; } void graph(const char*, const char*) { do_graph(3); } template <class T> void do_test(T, const char* name) { boost::math::ntl::RR::SetPrecision(working_precision); if(started) { // // We want to test the approximation at fixed precision: // either float, double or long double. Begin by getting the // polynomials: // boost::math::tools::polynomial<T> n, d; boost::math::tools::polynomial<boost::math::ntl::RR> nr, dr; nr = p_remez->numerator(); dr = p_remez->denominator(); n = nr; d = dr; std::vector<boost::math::ntl::RR> cn1, cd1; cn1 = nr.chebyshev(); cd1 = dr.chebyshev(); std::vector<T> cn, cd; for(unsigned i = 0; i < cn1.size(); ++i) { cn.push_back(boost::math::tools::real_cast<T>(cn1[i])); } for(unsigned i = 0; i < cd1.size(); ++i) { cd.push_back(boost::math::tools::real_cast<T>(cd1[i])); } // // We'll test at the Chebeshev control points which is where // (in theory) the largest deviation should occur. For good // measure we'll test at the zeros as well: // boost::numeric::ublas::vector<boost::math::ntl::RR> zeros(p_remez->zero_points()), cheb(p_remez->chebyshev_points()); boost::math::ntl::RR max_error(0), cheb_max_error(0); // // Do the tests at the zeros: // std::cout << "Starting tests at " << name << " precision...\n"; std::cout << "Absissa Error (Poly) Error (Cheb)\n"; for(unsigned i = 0; i < zeros.size(); ++i) { boost::math::ntl::RR true_result = the_function(zeros[i]); T absissa = boost::math::tools::real_cast<T>(zeros[i]); boost::math::ntl::RR test_result = n.evaluate(absissa) / d.evaluate(absissa); boost::math::ntl::RR cheb_result = boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa); boost::math::ntl::RR err, cheb_err; if(rel_error) { err = boost::math::tools::relative_error(test_result, true_result); cheb_err = boost::math::tools::relative_error(cheb_result, true_result); } else { err = fabs(test_result - true_result); cheb_err = fabs(cheb_result - true_result); } if(err > max_error) max_error = err; if(cheb_err > cheb_max_error) cheb_max_error = cheb_err; std::cout << std::setprecision(6) << std::setw(15) << std::left << absissa << std::setw(15) << std::left << boost::math::tools::real_cast<T>(err) << boost::math::tools::real_cast<T>(cheb_err) << std::endl; } // // Do the tests at the Chebeshev control points: // for(unsigned i = 0; i < cheb.size(); ++i) { boost::math::ntl::RR true_result = the_function(cheb[i]); T absissa = boost::math::tools::real_cast<T>(cheb[i]); boost::math::ntl::RR test_result = n.evaluate(absissa) / d.evaluate(absissa); boost::math::ntl::RR cheb_result = boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa); boost::math::ntl::RR err, cheb_err; if(rel_error) { err = boost::math::tools::relative_error(test_result, true_result); cheb_err = boost::math::tools::relative_error(cheb_result, true_result); } else { err = fabs(test_result - true_result); cheb_err = fabs(cheb_result - true_result); } if(err > max_error) max_error = err; std::cout << std::setprecision(6) << std::setw(15) << std::left << absissa << std::setw(15) << std::left << boost::math::tools::real_cast<T>(err) << boost::math::tools::real_cast<T>(cheb_err) << std::endl; } std::string msg = "Max Error found at "; msg += name; msg += " precision = "; msg.append(62 - 17 - msg.size(), ' '); std::cout << msg << std::setprecision(6) << "Poly: " << std::setw(20) << std::left << boost::math::tools::real_cast<T>(max_error) << "Cheb: " << boost::math::tools::real_cast<T>(cheb_max_error) << std::endl; } else { std::cout << "Nothing to test: try converging an approximation first!!!" << std::endl; } } void test_float(const char*, const char*) { do_test(float(0), "float"); } void test_double(const char*, const char*) { do_test(double(0), "double"); } void test_long(const char*, const char*) { do_test((long double)(0), "long double"); } void test_all(const char*, const char*) { do_test(float(0), "float"); do_test(double(0), "double"); do_test((long double)(0), "long double"); } template <class T> void do_test_n(T, const char* name, unsigned count) { boost::math::ntl::RR::SetPrecision(working_precision); if(started) { // // We want to test the approximation at fixed precision: // either float, double or long double. Begin by getting the // polynomials: // boost::math::tools::polynomial<T> n, d; boost::math::tools::polynomial<boost::math::ntl::RR> nr, dr; nr = p_remez->numerator(); dr = p_remez->denominator(); n = nr; d = dr; std::vector<boost::math::ntl::RR> cn1, cd1; cn1 = nr.chebyshev(); cd1 = dr.chebyshev(); std::vector<T> cn, cd; for(unsigned i = 0; i < cn1.size(); ++i) { cn.push_back(boost::math::tools::real_cast<T>(cn1[i])); } for(unsigned i = 0; i < cd1.size(); ++i) { cd.push_back(boost::math::tools::real_cast<T>(cd1[i])); } boost::math::ntl::RR max_error(0), max_cheb_error(0); boost::math::ntl::RR step = (b - a) / count; // // Do the tests at the zeros: // std::cout << "Starting tests at " << name << " precision...\n"; std::cout << "Absissa Error (poly) Error (Cheb)\n"; for(boost::math::ntl::RR x = a; x <= b; x += step) { boost::math::ntl::RR true_result = the_function(x); T absissa = boost::math::tools::real_cast<T>(x); boost::math::ntl::RR test_result = n.evaluate(absissa) / d.evaluate(absissa); boost::math::ntl::RR cheb_result = boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa); boost::math::ntl::RR err, cheb_err; if(rel_error) { err = boost::math::tools::relative_error(test_result, true_result); cheb_err = boost::math::tools::relative_error(cheb_result, true_result); } else { err = fabs(test_result - true_result); cheb_err = fabs(cheb_result - true_result); } if(err > max_error) max_error = err; if(cheb_err > max_cheb_error) max_cheb_error = cheb_err; std::cout << std::setprecision(6) << std::setw(15) << std::left << boost::math::tools::real_cast<double>(absissa) << (test_result < true_result ? "-" : "") << std::setw(20) << std::left << boost::math::tools::real_cast<double>(err) << boost::math::tools::real_cast<double>(cheb_err) << std::endl; } std::string msg = "Max Error found at "; msg += name; msg += " precision = "; //msg.append(62 - 17 - msg.size(), ' '); std::cout << msg << "Poly: " << std::setprecision(6) //<< std::setw(15) << std::left << boost::math::tools::real_cast<T>(max_error) << " Cheb: " << boost::math::tools::real_cast<T>(max_cheb_error) << std::endl; } else { std::cout << "Nothing to test: try converging an approximation first!!!" << std::endl; } } void test_n(unsigned n) { do_test_n(boost::math::ntl::RR(), "boost::math::ntl::RR", n); } void test_float_n(unsigned n) { do_test_n(float(0), "float", n); } void test_double_n(unsigned n) { do_test_n(double(0), "double", n); } void test_long_n(unsigned n) { do_test_n((long double)(0), "long double", n); } void rotate(const char*, const char*) { if(p_remez) { p_remez->rotate(); } else { std::cerr << "Nothing to rotate" << std::endl; } } void rescale(const char*, const char*) { if(p_remez) { p_remez->rescale(a, b); } else { std::cerr << "Nothing to rescale" << std::endl; } } void graph_poly(const char*, const char*) { int i = 50; boost::math::ntl::RR::SetPrecision(working_precision); if(started) { // // We want to test the approximation at fixed precision: // either float, double or long double. Begin by getting the // polynomials: // boost::math::tools::polynomial<boost::math::ntl::RR> n, d; n = p_remez->numerator(); d = p_remez->denominator(); boost::math::ntl::RR max_error(0); boost::math::ntl::RR step = (b - a) / i; std::cout << "Evaluating Numerator...\n"; boost::math::ntl::RR val; for(val = a; val <= b; val += step) std::cout << n.evaluate(val) << std::endl; std::cout << "Evaluating Denominator...\n"; for(val = a; val <= b; val += step) std::cout << d.evaluate(val) << std::endl; } else { std::cout << "Nothing to test: try converging an approximation first!!!" << std::endl; } } int test_main(int, char* []) { std::string line; real_parser<long double/*boost::math::ntl::RR*/ > const rr_p; while(std::getline(std::cin, line)) { if(parse(line.c_str(), str_p("quit"), space_p).full) return 0; if(false == parse(line.c_str(), ( str_p("range")[assign_a(started, false)] && real_p[assign_a(a)] && real_p[assign_a(b)] || str_p("relative")[assign_a(started, false)][assign_a(rel_error, true)] || str_p("absolute")[assign_a(started, false)][assign_a(rel_error, false)] || str_p("pin")[assign_a(started, false)] && str_p("true")[assign_a(pin, true)] || str_p("pin")[assign_a(started, false)] && str_p("false")[assign_a(pin, false)] || str_p("pin")[assign_a(started, false)] && str_p("1")[assign_a(pin, true)] || str_p("pin")[assign_a(started, false)] && str_p("0")[assign_a(pin, false)] || str_p("pin")[assign_a(started, false)][assign_a(pin, true)] || str_p("order")[assign_a(started, false)] && uint_p[assign_a(orderN)] && uint_p[assign_a(orderD)] || str_p("order")[assign_a(started, false)] && uint_p[assign_a(orderN)] || str_p("target-precision") && uint_p[assign_a(target_precision)] || str_p("working-precision")[assign_a(started, false)] && uint_p[assign_a(working_precision)] || str_p("variant")[assign_a(started, false)] && int_p[assign_a(variant)] || str_p("skew")[assign_a(started, false)] && int_p[assign_a(skew)] || str_p("brake") && int_p[assign_a(brake)] || str_p("step") && int_p[&step_some] || str_p("step")[&step] || str_p("poly")[&graph_poly] || str_p("info")[&show] || str_p("graph") && uint_p[&do_graph] || str_p("graph")[&graph] || str_p("x-offset") && real_p[assign_a(x_offset)] || str_p("x-scale") && real_p[assign_a(x_scale)] || str_p("y-offset") && str_p("auto")[assign_a(auto_offset_y, true)] || str_p("y-offset") && real_p[assign_a(y_offset)][assign_a(auto_offset_y, false)] || str_p("test") && str_p("float") && uint_p[&test_float_n] || str_p("test") && str_p("float")[&test_float] || str_p("test") && str_p("double") && uint_p[&test_double_n] || str_p("test") && str_p("double")[&test_double] || str_p("test") && str_p("long") && uint_p[&test_long_n] || str_p("test") && str_p("long")[&test_long] || str_p("test") && str_p("all")[&test_all] || str_p("test") && uint_p[&test_n] || str_p("rotate")[&rotate] || str_p("rescale") && real_p[assign_a(a)] && real_p[assign_a(b)] && epsilon_p[&rescale] ), space_p).full) { std::cout << "Unable to parse directive: \"" << line << "\"" << std::endl; } else { std::cout << "Variant = " << variant << std::endl; std::cout << "range = [" << a << "," << b << "]" << std::endl; std::cout << "Relative Error = " << rel_error << std::endl; std::cout << "Pin to Origin = " << pin << std::endl; std::cout << "Order (Num/Denom) = " << orderN << "/" << orderD << std::endl; std::cout << "Target Precision = " << target_precision << std::endl; std::cout << "Working Precision = " << working_precision << std::endl; std::cout << "Skew = " << skew << std::endl; std::cout << "Brake = " << brake << std::endl; std::cout << "X Offset = " << x_offset << std::endl; std::cout << "X scale = " << x_scale << std::endl; std::cout << "Y Offset = "; if(auto_offset_y) std::cout << "Auto ("; std::cout << y_offset; if(auto_offset_y) std::cout << ")"; std::cout << std::endl; } } return 0; }