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- /*
- * Copyright Nick Thompson, 2018
- * 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)
- */
- #include <iostream>
- #include <vector>
- #include <string>
- #include <complex>
- #include <bitset>
- #include <boost/assert.hpp>
- #include <boost/multiprecision/cpp_bin_float.hpp>
- #include <boost/math/constants/constants.hpp>
- #include <boost/math/tools/polynomial.hpp>
- #include <boost/math/tools/roots.hpp>
- #include <boost/math/special_functions/binomial.hpp>
- #include <boost/multiprecision/cpp_complex.hpp>
- #include <boost/multiprecision/complex128.hpp>
- #include <boost/math/quadrature/gauss_kronrod.hpp>
- using std::string;
- using boost::math::tools::polynomial;
- using boost::math::binomial_coefficient;
- using boost::math::tools::schroder_iterate;
- using boost::math::tools::halley_iterate;
- using boost::math::tools::newton_raphson_iterate;
- using boost::math::tools::complex_newton;
- using boost::math::constants::half;
- using boost::math::constants::root_two;
- using boost::math::constants::pi;
- using boost::math::quadrature::gauss_kronrod;
- using boost::multiprecision::cpp_bin_float_100;
- using boost::multiprecision::cpp_complex_100;
- template<class Complex>
- std::vector<std::pair<Complex, Complex>> find_roots(size_t p)
- {
- // Initialize the polynomial; see Mallat, A Wavelet Tour of Signal Processing, equation 7.96
- BOOST_ASSERT(p>0);
- typedef typename Complex::value_type Real;
- std::vector<Complex> coeffs(p);
- for (size_t k = 0; k < coeffs.size(); ++k)
- {
- coeffs[k] = Complex(binomial_coefficient<Real>(p-1+k, k), 0);
- }
- polynomial<Complex> P(std::move(coeffs));
- polynomial<Complex> Pcopy = P;
- polynomial<Complex> Pcopy_prime = P.prime();
- auto orig = [&](Complex z) { return std::make_pair<Complex, Complex>(Pcopy(z), Pcopy_prime(z)); };
- polynomial<Complex> P_prime = P.prime();
- // Polynomial is of degree p-1.
- std::vector<Complex> roots(p-1, {std::numeric_limits<Real>::quiet_NaN(),std::numeric_limits<Real>::quiet_NaN()});
- size_t i = 0;
- while(P.size() > 1)
- {
- Complex guess = {0.0, 1.0};
- std::cout << std::setprecision(std::numeric_limits<Real>::digits10+3);
- auto f = [&](Complex x)->std::pair<Complex, Complex>
- {
- return std::make_pair<Complex, Complex>(P(x), P_prime(x));
- };
- Complex r = complex_newton(f, guess);
- using std::isnan;
- if(isnan(r.real()))
- {
- int i = 50;
- do {
- // Try a different guess
- guess *= Complex(1.0,-1.0);
- r = complex_newton(f, guess);
- std::cout << "New guess: " << guess << ", result? " << r << std::endl;
- } while (isnan(r.real()) && i-- > 0);
- if (isnan(r.real()))
- {
- std::cout << "Polynomial that killed the process: " << P << std::endl;
- throw std::logic_error("Newton iteration did not converge");
- }
- }
- // Refine r with the original function.
- // We only use the polynomial division to ensure we don't get the same root over and over.
- // However, the division induces error which can grow quickly-or slowly! See Numerical Recipes, section 9.5.1.
- r = complex_newton(orig, r);
- if (isnan(r.real()))
- {
- throw std::logic_error("Found a root for the deflated polynomial which is not a root for the original. Indicative of catastrophic numerical error.");
- }
- // Test the root:
- using std::sqrt;
- Real tol = sqrt(sqrt(std::numeric_limits<Real>::epsilon()));
- if (norm(Pcopy(r)) > tol)
- {
- std::cout << "This is a bad root: P" << r << " = " << Pcopy(r) << std::endl;
- std::cout << "Reduced polynomial leading to bad root: " << P << std::endl;
- throw std::logic_error("Donezo.");
- }
- BOOST_ASSERT(i < roots.size());
- roots[i] = r;
- ++i;
- polynomial<Complex> q{-r, {1,0}};
- // This optimization breaks at p = 11. I have no clue why.
- // Unfortunate, because I expect it to be considerably more stable than
- // repeatedly dividing by the complex root.
- /*polynomial<Complex> q;
- if (r.imag() > sqrt(std::numeric_limits<Real>::epsilon()))
- {
- // Then the complex conjugate is also a root:
- using std::conj;
- using std::norm;
- BOOST_ASSERT(i < roots.size());
- roots[i] = conj(r);
- ++i;
- q = polynomial<Complex>({{norm(r), 0}, {-2*r.real(),0}, {1,0}});
- }
- else
- {
- // The imaginary part is numerical noise:
- r.imag() = 0;
- q = polynomial<Complex>({-r, {1,0}});
- }*/
- auto PR = quotient_remainder(P, q);
- // I should validate that the remainder is small, but . . .
- //std::cout << "Remainder = " << PR.second<< std::endl;
- P = PR.first;
- P_prime = P.prime();
- }
- std::vector<std::pair<Complex, Complex>> Qroots(p-1);
- for (size_t i = 0; i < Qroots.size(); ++i)
- {
- Complex y = roots[i];
- Complex z1 = static_cast<Complex>(1) - static_cast<Complex>(2)*y + static_cast<Complex>(2)*sqrt(y*(y-static_cast<Complex>(1)));
- Complex z2 = static_cast<Complex>(1) - static_cast<Complex>(2)*y - static_cast<Complex>(2)*sqrt(y*(y-static_cast<Complex>(1)));
- Qroots[i] = {z1, z2};
- }
- return Qroots;
- }
- template<class Complex>
- std::vector<typename Complex::value_type> daubechies_coefficients(std::vector<std::pair<Complex, Complex>> const & Qroots)
- {
- typedef typename Complex::value_type Real;
- size_t p = Qroots.size() + 1;
- // Choose the minimum abs root; see Mallat, discussion just after equation 7.98
- std::vector<Complex> chosen_roots(p-1);
- for (size_t i = 0; i < p - 1; ++i)
- {
- if(norm(Qroots[i].first) <= 1)
- {
- chosen_roots[i] = Qroots[i].first;
- }
- else
- {
- BOOST_ASSERT(norm(Qroots[i].second) <= 1);
- chosen_roots[i] = Qroots[i].second;
- }
- }
- polynomial<Complex> R{1};
- for (size_t i = 0; i < p-1; ++i)
- {
- Complex ak = chosen_roots[i];
- R *= polynomial<Complex>({-ak/(static_cast<Complex>(1)-ak), static_cast<Complex>(1)/(static_cast<Complex>(1)-ak)});
- }
- polynomial<Complex> a{{half<Real>(), 0}, {half<Real>(),0}};
- polynomial<Complex> poly = root_two<Real>()*pow(a, p)*R;
- std::vector<Complex> result = poly.data();
- // If we reverse, we get the Numerical Recipes and Daubechies convention.
- // If we don't reverse, we get the Pywavelets and Mallat convention.
- // I believe this is because of the sign convention on the DFT, which differs between Daubechies and Mallat.
- // You implement a dot product in Daubechies/NR convention, and a convolution in PyWavelets/Mallat convention.
- // I won't reverse so I can spot check against Pywavelets: http://wavelets.pybytes.com/wavelet/
- //std::reverse(result.begin(), result.end());
- std::vector<Real> h(result.size());
- for (size_t i = 0; i < result.size(); ++i)
- {
- Complex r = result[i];
- BOOST_ASSERT(r.imag() < sqrt(std::numeric_limits<Real>::epsilon()));
- h[i] = r.real();
- }
- // Quick sanity check: We could check all vanishing moments, but that sum is horribly ill-conditioned too!
- Real sum = 0;
- Real scale = 0;
- for (size_t i = 0; i < h.size(); ++i)
- {
- sum += h[i];
- scale += h[i]*h[i];
- }
- BOOST_ASSERT(abs(scale -1) < sqrt(std::numeric_limits<Real>::epsilon()));
- BOOST_ASSERT(abs(sum - root_two<Real>()) < sqrt(std::numeric_limits<Real>::epsilon()));
- return h;
- }
- int main()
- {
- typedef boost::multiprecision::cpp_complex<100> Complex;
- for(size_t p = 1; p < 200; ++p)
- {
- auto roots = find_roots<Complex>(p);
- auto h = daubechies_coefficients(roots);
- std::cout << "h_" << p << "[] = {";
- for (auto& x : h) {
- std::cout << x << ", ";
- }
- std::cout << "} // = h_" << p << "\n\n\n\n";
- }
- }
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