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// (C) Copyright Nick Thompson 2019. // 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) #ifndef BOOST_MATH_TOOLS_CONDITION_NUMBERS_HPP #define BOOST_MATH_TOOLS_CONDITION_NUMBERS_HPP #include #include #include #include namespace boost { namespace math { namespace tools { template class summation_condition_number { public: summation_condition_number(Real const x = 0) { using std::abs; m_l1 = abs(x); m_sum = x; m_c = 0; } void operator+=(Real const & x) { using std::abs; // No need to Kahan the l1 calc; it's well conditioned: m_l1 += abs(x); BOOST_MATH_IF_CONSTEXPR (kahan) { Real y = x - m_c; Real t = m_sum + y; m_c = (t-m_sum) -y; m_sum = t; } else { m_sum += x; } } inline void operator-=(Real const & x) { this->operator+=(-x); } // Is operator*= relevant? Presumably everything gets rescaled, // (m_sum -> k*m_sum, m_l1->k*m_l1, m_c->k*m_c), // but is this sensible? More important is it useful? // In addition, it might change the condition number. Real operator()() const { using std::abs; if (m_sum == Real(0) && m_l1 != Real(0)) { return std::numeric_limits::infinity(); } return m_l1/abs(m_sum); } Real sum() const { // Higham, 1993, "The Accuracy of Floating Point Summation": // "In [17] and [18], Kahan describes a variation of compensated summation in which the final sum is also corrected // thus s=s+e is appended to the algorithm above)." return m_sum + m_c; } Real l1_norm() const { return m_l1; } private: Real m_l1; Real m_sum; Real m_c; }; template Real evaluation_condition_number(F const & f, Real const & x) { using std::abs; using std::isnan; using std::sqrt; using boost::math::differentiation::finite_difference_derivative; Real fx = f(x); if (isnan(fx)) { return std::numeric_limits::quiet_NaN(); } bool caught_exception = false; Real fp; #ifndef BOOST_MATH_NO_EXCEPTIONS try { #endif fp = finite_difference_derivative(f, x); #ifndef BOOST_MATH_NO_EXCEPTIONS } catch(...) { caught_exception = true; } #endif if (isnan(fp) || caught_exception) { // Check if the right derivative exists: fp = finite_difference_derivative(f, x); if (isnan(fp)) { // Check if a left derivative exists: const Real eps = (std::numeric_limits::epsilon)(); Real h = - 2 * sqrt(eps); h = boost::math::differentiation::detail::make_xph_representable(x, h); Real yh = f(x + h); Real y0 = f(x); Real diff = yh - y0; fp = diff / h; if (isnan(fp)) { return std::numeric_limits::quiet_NaN(); } } } if (fx == 0) { if (x==0 || fp==0) { return std::numeric_limits::quiet_NaN(); } return std::numeric_limits::infinity(); } return abs(x*fp/fx); } }}} // Namespaces #endif