// (C) Copyright Nick Thompson 2018.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_TOOLS_NORMS_HPP
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#define BOOST_MATH_TOOLS_NORMS_HPP
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#include <algorithm>
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#include <iterator>
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#include <boost/assert.hpp>
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#include <boost/math/tools/complex.hpp>
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namespace boost::math::tools {
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// Mallat, "A Wavelet Tour of Signal Processing", equation 2.60:
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template<class ForwardIterator>
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auto total_variation(ForwardIterator first, ForwardIterator last)
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{
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using T = typename std::iterator_traits<ForwardIterator>::value_type;
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using std::abs;
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BOOST_ASSERT_MSG(first != last && std::next(first) != last, "At least two samples are required to compute the total variation.");
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auto it = first;
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if constexpr (std::is_unsigned<T>::value)
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{
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T tmp = *it;
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double tv = 0;
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while (++it != last)
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{
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if (*it > tmp)
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{
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tv += *it - tmp;
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}
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else
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{
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tv += tmp - *it;
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}
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tmp = *it;
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}
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return tv;
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}
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else if constexpr (std::is_integral<T>::value)
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{
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double tv = 0;
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double tmp = *it;
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while(++it != last)
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{
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double tmp2 = *it;
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tv += abs(tmp2 - tmp);
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tmp = *it;
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}
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return tv;
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}
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else
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{
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T tmp = *it;
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T tv = 0;
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while (++it != last)
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{
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tv += abs(*it - tmp);
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tmp = *it;
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}
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return tv;
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}
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}
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template<class Container>
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inline auto total_variation(Container const & v)
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{
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return total_variation(v.cbegin(), v.cend());
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}
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template<class ForwardIterator>
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auto sup_norm(ForwardIterator first, ForwardIterator last)
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{
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BOOST_ASSERT_MSG(first != last, "At least one value is required to compute the sup norm.");
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using T = typename std::iterator_traits<ForwardIterator>::value_type;
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using std::abs;
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if constexpr (boost::math::tools::is_complex_type<T>::value)
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{
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auto it = std::max_element(first, last, [](T a, T b) { return abs(b) > abs(a); });
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return abs(*it);
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}
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else if constexpr (std::is_unsigned<T>::value)
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{
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return *std::max_element(first, last);
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}
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else
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{
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auto pair = std::minmax_element(first, last);
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if (abs(*pair.first) > abs(*pair.second))
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{
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return abs(*pair.first);
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}
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else
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{
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return abs(*pair.second);
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}
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}
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}
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template<class Container>
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inline auto sup_norm(Container const & v)
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{
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return sup_norm(v.cbegin(), v.cend());
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}
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template<class ForwardIterator>
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auto l1_norm(ForwardIterator first, ForwardIterator last)
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{
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using T = typename std::iterator_traits<ForwardIterator>::value_type;
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using std::abs;
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if constexpr (std::is_unsigned<T>::value)
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{
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double l1 = 0;
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for (auto it = first; it != last; ++it)
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{
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l1 += *it;
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}
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return l1;
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}
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else if constexpr (std::is_integral<T>::value)
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{
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double l1 = 0;
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for (auto it = first; it != last; ++it)
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{
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double tmp = *it;
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l1 += abs(tmp);
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}
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return l1;
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}
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else
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{
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decltype(abs(*first)) l1 = 0;
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for (auto it = first; it != last; ++it)
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{
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l1 += abs(*it);
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}
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return l1;
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}
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}
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template<class Container>
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inline auto l1_norm(Container const & v)
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{
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return l1_norm(v.cbegin(), v.cend());
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}
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template<class ForwardIterator>
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auto l2_norm(ForwardIterator first, ForwardIterator last)
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{
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using T = typename std::iterator_traits<ForwardIterator>::value_type;
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using std::abs;
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using std::norm;
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using std::sqrt;
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using std::is_floating_point;
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if constexpr (boost::math::tools::is_complex_type<T>::value)
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{
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typedef typename T::value_type Real;
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Real l2 = 0;
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for (auto it = first; it != last; ++it)
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{
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l2 += norm(*it);
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}
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Real result = sqrt(l2);
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if (!isfinite(result))
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{
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Real a = sup_norm(first, last);
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l2 = 0;
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for (auto it = first; it != last; ++it)
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{
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l2 += norm(*it/a);
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}
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return a*sqrt(l2);
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}
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return result;
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}
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else if constexpr (is_floating_point<T>::value ||
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std::numeric_limits<T>::max_exponent)
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{
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T l2 = 0;
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for (auto it = first; it != last; ++it)
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{
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l2 += (*it)*(*it);
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}
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T result = sqrt(l2);
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// Higham, Accuracy and Stability of Numerical Algorithms,
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// Problem 27.5 presents a different algorithm to deal with overflow.
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// The algorithm used here takes 3 passes *if* there is overflow.
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// Higham's algorithm is 1 pass, but more requires operations than the no overflow case.
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// I'm operating under the assumption that overflow is rare since the dynamic range of floating point numbers is huge.
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if (!isfinite(result))
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{
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T a = sup_norm(first, last);
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l2 = 0;
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for (auto it = first; it != last; ++it)
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{
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T tmp = *it/a;
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l2 += tmp*tmp;
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}
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return a*sqrt(l2);
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}
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return result;
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}
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else
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{
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double l2 = 0;
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for (auto it = first; it != last; ++it)
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{
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double tmp = *it;
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l2 += tmp*tmp;
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}
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return sqrt(l2);
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}
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}
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template<class Container>
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inline auto l2_norm(Container const & v)
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{
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return l2_norm(v.cbegin(), v.cend());
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}
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template<class ForwardIterator>
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size_t l0_pseudo_norm(ForwardIterator first, ForwardIterator last)
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{
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using RealOrComplex = typename std::iterator_traits<ForwardIterator>::value_type;
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size_t count = 0;
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for (auto it = first; it != last; ++it)
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{
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if (*it != RealOrComplex(0))
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{
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++count;
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}
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}
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return count;
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}
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template<class Container>
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inline size_t l0_pseudo_norm(Container const & v)
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{
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return l0_pseudo_norm(v.cbegin(), v.cend());
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}
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template<class ForwardIterator>
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size_t hamming_distance(ForwardIterator first1, ForwardIterator last1, ForwardIterator first2)
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{
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size_t count = 0;
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auto it1 = first1;
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auto it2 = first2;
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while (it1 != last1)
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{
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if (*it1++ != *it2++)
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{
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++count;
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}
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}
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return count;
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}
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template<class Container>
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inline size_t hamming_distance(Container const & v, Container const & w)
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{
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return hamming_distance(v.cbegin(), v.cend(), w.cbegin());
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}
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template<class ForwardIterator>
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auto lp_norm(ForwardIterator first, ForwardIterator last, unsigned p)
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{
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using std::abs;
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using std::pow;
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using std::is_floating_point;
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using std::isfinite;
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using RealOrComplex = typename std::iterator_traits<ForwardIterator>::value_type;
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if constexpr (boost::math::tools::is_complex_type<RealOrComplex>::value)
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{
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using std::norm;
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using Real = typename RealOrComplex::value_type;
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Real lp = 0;
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for (auto it = first; it != last; ++it)
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{
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lp += pow(abs(*it), p);
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}
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auto result = pow(lp, Real(1)/Real(p));
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if (!isfinite(result))
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{
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auto a = boost::math::tools::sup_norm(first, last);
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Real lp = 0;
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for (auto it = first; it != last; ++it)
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{
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lp += pow(abs(*it)/a, p);
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}
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result = a*pow(lp, Real(1)/Real(p));
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}
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return result;
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}
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else if constexpr (is_floating_point<RealOrComplex>::value || std::numeric_limits<RealOrComplex>::max_exponent)
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{
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BOOST_ASSERT_MSG(p >= 0, "For p < 0, the lp norm is not a norm");
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RealOrComplex lp = 0;
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for (auto it = first; it != last; ++it)
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{
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lp += pow(abs(*it), p);
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}
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RealOrComplex result = pow(lp, RealOrComplex(1)/RealOrComplex(p));
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if (!isfinite(result))
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{
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RealOrComplex a = boost::math::tools::sup_norm(first, last);
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lp = 0;
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for (auto it = first; it != last; ++it)
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{
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lp += pow(abs(*it)/a, p);
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}
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result = a*pow(lp, RealOrComplex(1)/RealOrComplex(p));
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}
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return result;
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}
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else
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{
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double lp = 0;
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for (auto it = first; it != last; ++it)
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{
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double tmp = *it;
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lp += pow(abs(tmp), p);
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}
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double result = pow(lp, 1.0/double(p));
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if (!isfinite(result))
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{
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double a = boost::math::tools::sup_norm(first, last);
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lp = 0;
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for (auto it = first; it != last; ++it)
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{
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double tmp = *it;
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lp += pow(abs(tmp)/a, p);
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}
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result = a*pow(lp, double(1)/double(p));
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}
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return result;
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}
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}
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template<class Container>
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inline auto lp_norm(Container const & v, unsigned p)
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{
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return lp_norm(v.cbegin(), v.cend(), p);
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}
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template<class ForwardIterator>
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auto lp_distance(ForwardIterator first1, ForwardIterator last1, ForwardIterator first2, unsigned p)
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{
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using std::pow;
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using std::abs;
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using std::is_floating_point;
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using std::isfinite;
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using RealOrComplex = typename std::iterator_traits<ForwardIterator>::value_type;
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auto it1 = first1;
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auto it2 = first2;
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if constexpr (boost::math::tools::is_complex_type<RealOrComplex>::value)
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{
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using Real = typename RealOrComplex::value_type;
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using std::norm;
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Real dist = 0;
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while(it1 != last1)
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{
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auto tmp = *it1++ - *it2++;
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dist += pow(abs(tmp), p);
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}
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return pow(dist, Real(1)/Real(p));
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}
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else if constexpr (is_floating_point<RealOrComplex>::value || std::numeric_limits<RealOrComplex>::max_exponent)
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{
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RealOrComplex dist = 0;
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while(it1 != last1)
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{
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auto tmp = *it1++ - *it2++;
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dist += pow(abs(tmp), p);
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}
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return pow(dist, RealOrComplex(1)/RealOrComplex(p));
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}
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else
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{
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double dist = 0;
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while(it1 != last1)
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{
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double tmp1 = *it1++;
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double tmp2 = *it2++;
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// Naively you'd expect the integer subtraction to be faster,
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// but this can overflow or wraparound:
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//double tmp = *it1++ - *it2++;
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dist += pow(abs(tmp1 - tmp2), p);
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}
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return pow(dist, 1.0/double(p));
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}
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}
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template<class Container>
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inline auto lp_distance(Container const & v, Container const & w, unsigned p)
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{
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return lp_distance(v.cbegin(), v.cend(), w.cbegin(), p);
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}
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template<class ForwardIterator>
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auto l1_distance(ForwardIterator first1, ForwardIterator last1, ForwardIterator first2)
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{
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using std::abs;
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using std::is_floating_point;
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using std::isfinite;
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using T = typename std::iterator_traits<ForwardIterator>::value_type;
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auto it1 = first1;
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auto it2 = first2;
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if constexpr (boost::math::tools::is_complex_type<T>::value)
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{
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using Real = typename T::value_type;
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Real sum = 0;
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while (it1 != last1) {
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sum += abs(*it1++ - *it2++);
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}
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return sum;
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}
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else if constexpr (is_floating_point<T>::value || std::numeric_limits<T>::max_exponent)
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{
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T sum = 0;
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while (it1 != last1)
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{
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sum += abs(*it1++ - *it2++);
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}
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return sum;
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}
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else if constexpr (std::is_unsigned<T>::value)
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{
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double sum = 0;
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while(it1 != last1)
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{
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T x1 = *it1++;
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T x2 = *it2++;
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if (x1 > x2)
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{
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sum += (x1 - x2);
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}
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else
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{
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sum += (x2 - x1);
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}
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}
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return sum;
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}
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else if constexpr (std::is_integral<T>::value)
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{
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double sum = 0;
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while(it1 != last1)
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{
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double x1 = *it1++;
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double x2 = *it2++;
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sum += abs(x1-x2);
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}
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return sum;
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}
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else
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{
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BOOST_ASSERT_MSG(false, "Could not recognize type.");
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}
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}
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template<class Container>
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auto l1_distance(Container const & v, Container const & w)
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{
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using std::size;
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BOOST_ASSERT_MSG(size(v) == size(w),
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"L1 distance requires both containers to have the same number of elements");
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return l1_distance(v.cbegin(), v.cend(), w.begin());
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}
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template<class ForwardIterator>
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auto l2_distance(ForwardIterator first1, ForwardIterator last1, ForwardIterator first2)
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{
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using std::abs;
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using std::norm;
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using std::sqrt;
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using std::is_floating_point;
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using std::isfinite;
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using T = typename std::iterator_traits<ForwardIterator>::value_type;
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auto it1 = first1;
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auto it2 = first2;
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if constexpr (boost::math::tools::is_complex_type<T>::value)
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{
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using Real = typename T::value_type;
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Real sum = 0;
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while (it1 != last1) {
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sum += norm(*it1++ - *it2++);
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}
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return sqrt(sum);
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}
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else if constexpr (is_floating_point<T>::value || std::numeric_limits<T>::max_exponent)
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{
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T sum = 0;
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while (it1 != last1)
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{
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T tmp = *it1++ - *it2++;
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sum += tmp*tmp;
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}
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return sqrt(sum);
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}
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else if constexpr (std::is_unsigned<T>::value)
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{
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double sum = 0;
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while(it1 != last1)
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{
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T x1 = *it1++;
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T x2 = *it2++;
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if (x1 > x2)
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{
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double tmp = x1-x2;
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sum += tmp*tmp;
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}
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else
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{
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double tmp = x2 - x1;
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sum += tmp*tmp;
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}
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}
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return sqrt(sum);
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}
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else
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{
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double sum = 0;
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while(it1 != last1)
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{
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double x1 = *it1++;
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double x2 = *it2++;
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double tmp = x1-x2;
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sum += tmp*tmp;
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}
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return sqrt(sum);
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}
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}
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template<class Container>
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auto l2_distance(Container const & v, Container const & w)
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{
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using std::size;
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BOOST_ASSERT_MSG(size(v) == size(w),
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"L2 distance requires both containers to have the same number of elements");
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return l2_distance(v.cbegin(), v.cend(), w.begin());
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}
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template<class ForwardIterator>
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auto sup_distance(ForwardIterator first1, ForwardIterator last1, ForwardIterator first2)
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{
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using std::abs;
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using std::norm;
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using std::sqrt;
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using std::is_floating_point;
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using std::isfinite;
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using T = typename std::iterator_traits<ForwardIterator>::value_type;
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auto it1 = first1;
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auto it2 = first2;
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if constexpr (boost::math::tools::is_complex_type<T>::value)
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{
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using Real = typename T::value_type;
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Real sup_sq = 0;
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while (it1 != last1) {
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Real tmp = norm(*it1++ - *it2++);
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if (tmp > sup_sq) {
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sup_sq = tmp;
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}
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}
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return sqrt(sup_sq);
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}
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else if constexpr (is_floating_point<T>::value || std::numeric_limits<T>::max_exponent)
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{
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T sup = 0;
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while (it1 != last1)
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{
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T tmp = *it1++ - *it2++;
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if (sup < abs(tmp))
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{
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sup = abs(tmp);
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}
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}
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return sup;
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}
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else // integral values:
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{
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double sup = 0;
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while(it1 != last1)
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{
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T x1 = *it1++;
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T x2 = *it2++;
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double tmp;
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if (x1 > x2)
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{
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tmp = x1-x2;
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}
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else
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{
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tmp = x2 - x1;
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}
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if (sup < tmp) {
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sup = tmp;
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}
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}
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return sup;
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}
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}
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template<class Container>
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auto sup_distance(Container const & v, Container const & w)
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{
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using std::size;
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BOOST_ASSERT_MSG(size(v) == size(w),
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"sup distance requires both containers to have the same number of elements");
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return sup_distance(v.cbegin(), v.cend(), w.begin());
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}
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}
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#endif
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