comparison DEPENDENCIES/generic/include/boost/units/detail/linear_algebra.hpp @ 16:2665513ce2d3

Add boost headers

+// Boost.Units - A C++ library for zero-overhead dimensional analysis and 
+// unit/quantity manipulation and conversion
+//
+// Copyright (C) 2003-2008 Matthias Christian Schabel
+// Copyright (C) 2008 Steven Watanabe
+//
+// Distributed under 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_UNITS_DETAIL_LINEAR_ALGEBRA_HPP
+#define BOOST_UNITS_DETAIL_LINEAR_ALGEBRA_HPP
+
+#include <boost/units/static_rational.hpp>
+#include <boost/mpl/next.hpp>
+#include <boost/mpl/arithmetic.hpp>
+#include <boost/mpl/and.hpp>
+#include <boost/mpl/assert.hpp>
+
+#include <boost/units/dim.hpp>
+#include <boost/units/dimensionless_type.hpp>
+#include <boost/units/static_rational.hpp>
+#include <boost/units/detail/dimension_list.hpp>
+#include <boost/units/detail/sort.hpp>
+
+namespace boost {
+
+namespace units {
+
+namespace detail {
+
+// typedef list<rational> equation;
+
+template<int N>
+struct eliminate_from_pair_of_equations_impl;
+
+template<class E1, class E2>
+struct eliminate_from_pair_of_equations;
+
+template<int N>
+struct elimination_impl;
+
+template<bool is_zero, bool element_is_last>
+struct elimination_skip_leading_zeros_impl;
+
+template<class Equation, class Vars>
+struct substitute;
+
+template<int N>
+struct substitute_impl;
+
+template<bool is_end>
+struct solve_impl;
+
+template<class T>
+struct solve;
+
+template<int N>
+struct check_extra_equations_impl;
+
+template<int N>
+struct normalize_units_impl;
+
+struct inconsistent {};
+
+// generally useful utilies.
+
+template<int N>
+struct divide_equation {
+    template<class Begin, class Divisor>
+    struct apply {
+        typedef list<typename mpl::divides<typename Begin::item, Divisor>::type, typename divide_equation<N - 1>::template apply<typename Begin::next, Divisor>::type> type;
+    };
+};
+
+template<>
+struct divide_equation<0> {
+    template<class Begin, class Divisor>
+    struct apply {
+        typedef dimensionless_type type;
+    };
+};
+
+// eliminate_from_pair_of_equations takes a pair of
+// equations and eliminates the first variable.
+//
+// equation eliminate_from_pair_of_equations(equation l1, equation l2) {
+//     rational x1 = l1.front();
+//     rational x2 = l2.front();
+//     return(transform(pop_front(l1), pop_front(l2), _1 * x2 - _2 * x1));
+// }
+
+template<int N>
+struct eliminate_from_pair_of_equations_impl {
+    template<class Begin1, class Begin2, class X1, class X2>
+    struct apply {
+        typedef list<
+            typename mpl::minus<
+                typename mpl::times<typename Begin1::item, X2>::type,
+                typename mpl::times<typename Begin2::item, X1>::type
+            >::type,
+            typename eliminate_from_pair_of_equations_impl<N - 1>::template apply<
+                typename Begin1::next,
+                typename Begin2::next,
+                X1,
+                X2
+            >::type
+        > type;
+    };
+};
+
+template<>
+struct eliminate_from_pair_of_equations_impl<0> {
+    template<class Begin1, class Begin2, class X1, class X2>
+    struct apply {
+        typedef dimensionless_type type;
+    };
+};
+
+template<class E1, class E2>
+struct eliminate_from_pair_of_equations {
+    typedef E1 begin1;
+    typedef E2 begin2;
+    typedef typename eliminate_from_pair_of_equations_impl<(E1::size::value - 1)>::template apply<
+        typename begin1::next,
+        typename begin2::next,
+        typename begin1::item,
+        typename begin2::item
+    >::type type;
+};
+
+
+
+// Stage 1.  Determine which dimensions should
+// have dummy base units.  For this purpose
+// row reduce the matrix.
+
+template<int N>
+struct make_zero_vector {
+    typedef list<static_rational<0>, typename make_zero_vector<N - 1>::type> type;
+};
+template<>
+struct make_zero_vector<0> {
+    typedef dimensionless_type type;
+};
+
+template<int Column, int TotalColumns>
+struct create_row_of_identity {
+    typedef list<static_rational<0>, typename create_row_of_identity<Column - 1, TotalColumns - 1>::type> type;
+};
+template<int TotalColumns>
+struct create_row_of_identity<0, TotalColumns> {
+    typedef list<static_rational<1>, typename make_zero_vector<TotalColumns - 1>::type> type;
+};
+template<int Column>
+struct create_row_of_identity<Column, 0> {
+    // error
+};
+
+template<int RemainingRows>
+struct determine_extra_equations_impl;
+
+template<bool first_is_zero, bool is_last>
+struct determine_extra_equations_skip_zeros_impl;
+
+// not the last row and not zero.
+template<>
+struct determine_extra_equations_skip_zeros_impl<false, false> {
+    template<class RowsBegin, int RemainingRows, int CurrentColumn, int TotalColumns, class Result>
+    struct apply {
+        // remove the equation being eliminated against from the set of equations.
+        typedef typename determine_extra_equations_impl<RemainingRows - 1>::template apply<typename RowsBegin::next, typename RowsBegin::item>::type next_equations;
+        // since this column was present, strip it out.
+        typedef Result type;
+    };
+};
+
+// the last row but not zero.
+template<>
+struct determine_extra_equations_skip_zeros_impl<false, true> {
+    template<class RowsBegin, int RemainingRows, int CurrentColumn, int TotalColumns, class Result>
+    struct apply {
+        // remove this equation.
+        typedef dimensionless_type next_equations;
+        // since this column was present, strip it out.
+        typedef Result type;
+    };
+};
+
+
+// the first columns is zero but it is not the last column.
+// continue with the same loop.
+template<>
+struct determine_extra_equations_skip_zeros_impl<true, false> {
+    template<class RowsBegin, int RemainingRows, int CurrentColumn, int TotalColumns, class Result>
+    struct apply {
+        typedef typename RowsBegin::next::item next_row;
+        typedef typename determine_extra_equations_skip_zeros_impl<
+            next_row::item::Numerator == 0,
+            RemainingRows == 2  // the next one will be the last.
+        >::template apply<
+            typename RowsBegin::next,
+            RemainingRows - 1,
+            CurrentColumn,
+            TotalColumns,
+            Result
+        > next;
+        typedef list<typename RowsBegin::item::next, typename next::next_equations> next_equations;
+        typedef typename next::type type;
+    };
+};
+
+// all the elements in this column are zero.
+template<>
+struct determine_extra_equations_skip_zeros_impl<true, true> {
+    template<class RowsBegin, int RemainingRows, int CurrentColumn, int TotalColumns, class Result>
+    struct apply {
+        typedef list<typename RowsBegin::item::next, dimensionless_type> next_equations;
+        typedef list<typename create_row_of_identity<CurrentColumn, TotalColumns>::type, Result> type;
+    };
+};
+
+template<int RemainingRows>
+struct determine_extra_equations_impl {
+    template<class RowsBegin, class EliminateAgainst>
+    struct apply {
+        typedef list<
+            typename eliminate_from_pair_of_equations<typename RowsBegin::item, EliminateAgainst>::type,
+            typename determine_extra_equations_impl<RemainingRows-1>::template apply<typename RowsBegin::next, EliminateAgainst>::type
+        > type;
+    };
+};
+
+template<>
+struct determine_extra_equations_impl<0> {
+    template<class RowsBegin, class EliminateAgainst>
+    struct apply {
+        typedef dimensionless_type type;
+    };
+};
+
+template<int RemainingColumns, bool is_done>
+struct determine_extra_equations {
+    template<class RowsBegin, int TotalColumns, class Result>
+    struct apply {
+        typedef typename RowsBegin::item top_row;
+        typedef typename determine_extra_equations_skip_zeros_impl<
+            top_row::item::Numerator == 0,
+            RowsBegin::item::size::value == 1
+        >::template apply<
+            RowsBegin,
+            RowsBegin::size::value,
+            TotalColumns - RemainingColumns,
+            TotalColumns,
+            Result
+        > column_info;
+        typedef typename determine_extra_equations<
+            RemainingColumns - 1,
+            column_info::next_equations::size::value == 0
+        >::template apply<
+            typename column_info::next_equations,
+            TotalColumns,
+            typename column_info::type
+        >::type type;
+    };
+};
+
+template<int RemainingColumns>
+struct determine_extra_equations<RemainingColumns, true> {
+    template<class RowsBegin, int TotalColumns, class Result>
+    struct apply {
+        typedef typename determine_extra_equations<RemainingColumns - 1, true>::template apply<
+            RowsBegin,
+            TotalColumns,
+            list<typename create_row_of_identity<TotalColumns - RemainingColumns, TotalColumns>::type, Result>
+        >::type type;
+    };
+};
+
+template<>
+struct determine_extra_equations<0, true> {
+    template<class RowsBegin, int TotalColumns, class Result>
+    struct apply {
+        typedef Result type;
+    };
+};
+
+// Stage 2
+// invert the matrix using Gauss-Jordan elimination
+
+
+template<bool is_zero, bool is_last>
+struct invert_strip_leading_zeroes;
+
+template<int N>
+struct invert_handle_after_pivot_row;
+
+// When processing column N, none of the first N rows 
+// can be the pivot column.
+template<int N>
+struct invert_handle_inital_rows {
+    template<class RowsBegin, class IdentityBegin>
+    struct apply {
+        typedef typename invert_handle_inital_rows<N - 1>::template apply<
+            typename RowsBegin::next,
+            typename IdentityBegin::next
+        > next;
+        typedef typename RowsBegin::item current_row;
+        typedef typename IdentityBegin::item current_identity_row;
+        typedef typename next::pivot_row pivot_row;
+        typedef typename next::identity_pivot_row identity_pivot_row;
+        typedef list<
+            typename eliminate_from_pair_of_equations_impl<(current_row::size::value) - 1>::template apply<
+                typename current_row::next,
+                pivot_row,
+                typename current_row::item,
+                static_rational<1>
+            >::type,
+            typename next::new_matrix
+        > new_matrix;
+        typedef list<
+            typename eliminate_from_pair_of_equations_impl<(current_identity_row::size::value)>::template apply<
+                current_identity_row,
+                identity_pivot_row,
+                typename current_row::item,
+                static_rational<1>
+            >::type,
+            typename next::identity_result
+        > identity_result;
+    };
+};
+
+// This handles the switch to searching for a pivot column.
+// The pivot row will be propagated up in the typedefs
+// pivot_row and identity_pivot_row.  It is inserted here.
+template<>
+struct invert_handle_inital_rows<0> {
+    template<class RowsBegin, class IdentityBegin>
+    struct apply {
+        typedef typename RowsBegin::item current_row;
+        typedef typename invert_strip_leading_zeroes<
+            (current_row::item::Numerator == 0),
+            (RowsBegin::size::value == 1)
+        >::template apply<
+            RowsBegin,
+            IdentityBegin
+        > next;
+        // results
+        typedef list<typename next::pivot_row, typename next::new_matrix> new_matrix;
+        typedef list<typename next::identity_pivot_row, typename next::identity_result> identity_result;
+        typedef typename next::pivot_row pivot_row;
+        typedef typename next::identity_pivot_row identity_pivot_row;
+    };
+};
+
+// The first internal element which is not zero.
+template<>
+struct invert_strip_leading_zeroes<false, false> {
+    template<class RowsBegin, class IdentityBegin>
+    struct apply {
+        typedef typename RowsBegin::item current_row;
+        typedef typename current_row::item current_value;
+        typedef typename divide_equation<(current_row::size::value - 1)>::template apply<typename current_row::next, current_value>::type new_equation;
+        typedef typename divide_equation<(IdentityBegin::item::size::value)>::template apply<typename IdentityBegin::item, current_value>::type transformed_identity_equation;
+        typedef typename invert_handle_after_pivot_row<(RowsBegin::size::value - 1)>::template apply<
+            typename RowsBegin::next,
+            typename IdentityBegin::next,
+            new_equation,
+            transformed_identity_equation
+        > next;
+
+        // results
+        // Note that we don't add the pivot row to the
+        // results here, because it needs to propagated up
+        // to the diagonal.
+        typedef typename next::new_matrix new_matrix;
+        typedef typename next::identity_result identity_result;
+        typedef new_equation pivot_row;
+        typedef transformed_identity_equation identity_pivot_row;
+    };
+};
+
+// The one and only non-zero element--at the end
+template<>
+struct invert_strip_leading_zeroes<false, true> {
+    template<class RowsBegin, class IdentityBegin>
+    struct apply {
+        typedef typename RowsBegin::item current_row;
+        typedef typename current_row::item current_value;
+        typedef typename divide_equation<(current_row::size::value - 1)>::template apply<typename current_row::next, current_value>::type new_equation;
+        typedef typename divide_equation<(IdentityBegin::item::size::value)>::template apply<typename IdentityBegin::item, current_value>::type transformed_identity_equation;
+
+        // results
+        // Note that we don't add the pivot row to the
+        // results here, because it needs to propagated up
+        // to the diagonal.
+        typedef dimensionless_type identity_result;
+        typedef dimensionless_type new_matrix;
+        typedef new_equation pivot_row;
+        typedef transformed_identity_equation identity_pivot_row;
+    };
+};
+
+// One of the initial zeroes
+template<>
+struct invert_strip_leading_zeroes<true, false> {
+    template<class RowsBegin, class IdentityBegin>
+    struct apply {
+        typedef typename RowsBegin::item current_row;
+        typedef typename RowsBegin::next::item next_row;
+        typedef typename invert_strip_leading_zeroes<
+            next_row::item::Numerator == 0,
+            RowsBegin::size::value == 2
+        >::template apply<
+            typename RowsBegin::next,
+            typename IdentityBegin::next
+        > next;
+        typedef typename IdentityBegin::item current_identity_row;
+        // these are propagated up.
+        typedef typename next::pivot_row pivot_row;
+        typedef typename next::identity_pivot_row identity_pivot_row;
+        typedef list<
+            typename eliminate_from_pair_of_equations_impl<(current_row::size::value - 1)>::template apply<
+                typename current_row::next,
+                pivot_row,
+                typename current_row::item,
+                static_rational<1>
+            >::type,
+            typename next::new_matrix
+        > new_matrix;
+        typedef list<
+            typename eliminate_from_pair_of_equations_impl<(current_identity_row::size::value)>::template apply<
+                current_identity_row,
+                identity_pivot_row,
+                typename current_row::item,
+                static_rational<1>
+            >::type,
+            typename next::identity_result
+        > identity_result;
+    };
+};
+
+// the last element, and is zero.
+// Should never happen.
+template<>
+struct invert_strip_leading_zeroes<true, true> {
+};
+
+template<int N>
+struct invert_handle_after_pivot_row {
+    template<class RowsBegin, class IdentityBegin, class MatrixPivot, class IdentityPivot>
+    struct apply {
+        typedef typename invert_handle_after_pivot_row<N - 1>::template apply<
+            typename RowsBegin::next,
+            typename IdentityBegin::next,
+            MatrixPivot,
+            IdentityPivot
+        > next;
+        typedef typename RowsBegin::item current_row;
+        typedef typename IdentityBegin::item current_identity_row;
+        typedef MatrixPivot pivot_row;
+        typedef IdentityPivot identity_pivot_row;
+
+        // results
+        typedef list<
+            typename eliminate_from_pair_of_equations_impl<(current_row::size::value - 1)>::template apply<
+                typename current_row::next,
+                pivot_row,
+                typename current_row::item,
+                static_rational<1>
+            >::type,
+            typename next::new_matrix
+        > new_matrix;
+        typedef list<
+            typename eliminate_from_pair_of_equations_impl<(current_identity_row::size::value)>::template apply<
+                current_identity_row,
+                identity_pivot_row,
+                typename current_row::item,
+                static_rational<1>
+            >::type,
+            typename next::identity_result
+        > identity_result;
+    };
+};
+
+template<>
+struct invert_handle_after_pivot_row<0> {
+    template<class RowsBegin, class IdentityBegin, class MatrixPivot, class IdentityPivot>
+    struct apply {
+        typedef dimensionless_type new_matrix;
+        typedef dimensionless_type identity_result;
+    };
+};
+
+template<int N>
+struct invert_impl {
+    template<class RowsBegin, class IdentityBegin>
+    struct apply {
+        typedef typename invert_handle_inital_rows<RowsBegin::size::value - N>::template apply<RowsBegin, IdentityBegin> process_column;
+        typedef typename invert_impl<N - 1>::template apply<
+            typename process_column::new_matrix,
+            typename process_column::identity_result
+        >::type type;
+    };
+};
+
+template<>
+struct invert_impl<0> {
+    template<class RowsBegin, class IdentityBegin>
+    struct apply {
+        typedef IdentityBegin type;
+    };
+};
+
+template<int N>
+struct make_identity {
+    template<int Size>
+    struct apply {
+        typedef list<typename create_row_of_identity<Size - N, Size>::type, typename make_identity<N - 1>::template apply<Size>::type> type;
+    };
+};
+
+template<>
+struct make_identity<0> {
+    template<int Size>
+    struct apply {
+        typedef dimensionless_type type;
+    };
+};
+
+template<class Matrix>
+struct make_square_and_invert {
+    typedef typename Matrix::item top_row;
+    typedef typename determine_extra_equations<(top_row::size::value), false>::template apply<
+        Matrix,                 // RowsBegin
+        top_row::size::value,   // TotalColumns
+        Matrix                  // Result
+    >::type invertible;
+    typedef typename invert_impl<invertible::size::value>::template apply<
+        invertible,
+        typename make_identity<invertible::size::value>::template apply<invertible::size::value>::type
+    >::type type;
+};
+
+
+// find_base_dimensions takes a list of
+// base_units and returns a sorted list
+// of all the base_dimensions they use.
+//
+// list<base_dimension> find_base_dimensions(list<base_unit> l) {
+//     set<base_dimension> dimensions;
+//     for_each(base_unit unit : l) {
+//         for_each(dim d : unit.dimension_type) {
+//             dimensions = insert(dimensions, d.tag_type);
+//         }
+//     }
+//     return(sort(dimensions, _1 > _2, front_inserter(list<base_dimension>())));
+// }
+
+typedef char set_no;
+struct set_yes { set_no dummy[2]; };
+
+template<class T>
+struct wrap {};
+
+struct set_end {
+    static set_no lookup(...);
+    typedef mpl::long_<0> size;
+};
+
+template<class T, class Next>
+struct set : Next {
+    using Next::lookup;
+    static set_yes lookup(wrap<T>*);
+    typedef T item;
+    typedef Next next;
+    typedef typename mpl::next<typename Next::size>::type size;
+};
+
+template<bool has_key>
+struct set_insert;
+
+template<>
+struct set_insert<true> {
+    template<class Set, class T>
+    struct apply {
+        typedef Set type;
+    };
+};
+
+template<>
+struct set_insert<false> {
+    template<class Set, class T>
+    struct apply {
+        typedef set<T, Set> type;
+    };
+};
+
+template<class Set, class T>
+struct has_key {
+    static const long size = sizeof(Set::lookup((wrap<T>*)0));
+    static const bool value = (size == sizeof(set_yes));
+};
+
+template<int N>
+struct find_base_dimensions_impl_impl {
+    template<class Begin, class S>
+    struct apply {
+        typedef typename find_base_dimensions_impl_impl<N-1>::template apply<
+            typename Begin::next,
+            S
+        >::type next;
+
+        typedef typename set_insert<
+            (has_key<next, typename Begin::item::tag_type>::value)
+        >::template apply<
+            next,
+            typename Begin::item::tag_type
+        >::type type;
+    };
+};
+
+template<>
+struct find_base_dimensions_impl_impl<0> {
+    template<class Begin, class S>
+    struct apply {
+        typedef S type;
+    };
+};
+
+template<int N>
+struct find_base_dimensions_impl {
+    template<class Begin>
+    struct apply {
+        typedef typename find_base_dimensions_impl_impl<(Begin::item::dimension_type::size::value)>::template apply<
+            typename Begin::item::dimension_type,
+            typename find_base_dimensions_impl<N-1>::template apply<typename Begin::next>::type
+        >::type type;
+    };
+};
+
+template<>
+struct find_base_dimensions_impl<0> {
+    template<class Begin>
+    struct apply {
+        typedef set_end type;
+    };
+};
+
+template<class T>
+struct find_base_dimensions {
+    typedef typename insertion_sort<
+        typename find_base_dimensions_impl<
+            (T::size::value)
+        >::template apply<T>::type
+    >::type type;
+};
+
+// calculate_base_dimension_coefficients finds
+// the coefficients corresponding to the first
+// base_dimension in each of the dimension_lists.
+// It returns two values.  The first result
+// is a list of the coefficients.  The second
+// is a list with all the incremented iterators.
+// When we encounter a base_dimension that is
+// missing from a dimension_list, we do not
+// increment the iterator and we set the
+// coefficient to zero.
+
+template<bool has_dimension>
+struct calculate_base_dimension_coefficients_func;
+
+template<>
+struct calculate_base_dimension_coefficients_func<true> {
+    template<class T>
+    struct apply {
+        typedef typename T::item::value_type type;
+        typedef typename T::next next;
+    };
+};
+
+template<>
+struct calculate_base_dimension_coefficients_func<false> {
+    template<class T>
+    struct apply {
+        typedef static_rational<0> type;
+        typedef T next;
+    };
+};
+
+// begins_with_dimension returns true iff its first
+// parameter is a valid iterator which yields its
+// second parameter when dereferenced.
+
+template<class Iterator>
+struct begins_with_dimension {
+    template<class Dim>
+    struct apply : 
+        boost::is_same<
+            Dim,
+            typename Iterator::item::tag_type
+        > {};
+};
+
+template<>
+struct begins_with_dimension<dimensionless_type> {
+    template<class Dim>
+    struct apply : mpl::false_ {};
+};
+
+template<int N>
+struct calculate_base_dimension_coefficients_impl {
+    template<class BaseUnitDimensions,class Dim,class T>
+    struct apply {
+        typedef typename calculate_base_dimension_coefficients_func<
+            begins_with_dimension<typename BaseUnitDimensions::item>::template apply<
+                Dim
+            >::value
+        >::template apply<
+            typename BaseUnitDimensions::item
+        > result;
+        typedef typename calculate_base_dimension_coefficients_impl<N-1>::template apply<
+            typename BaseUnitDimensions::next,
+            Dim,
+            list<typename result::type, T>
+        > next_;
+        typedef typename next_::type type;
+        typedef list<typename result::next, typename next_::next> next;
+    };
+};
+
+template<>
+struct calculate_base_dimension_coefficients_impl<0> {
+    template<class Begin, class BaseUnitDimensions, class T>
+    struct apply {
+        typedef T type;
+        typedef dimensionless_type next;
+    };
+};
+
+// add_zeroes pushs N zeroes onto the
+// front of a list.
+//
+// list<rational> add_zeroes(list<rational> l, int N) {
+//     if(N == 0) {
+//         return(l);
+//     } else {
+//         return(push_front(add_zeroes(l, N-1), 0));
+//     }
+// }
+
+template<int N>
+struct add_zeroes_impl {
+    // If you get an error here and your base units are
+    // in fact linearly independent, please report it.
+    BOOST_MPL_ASSERT_MSG((N > 0), base_units_are_probably_not_linearly_independent, (void));
+    template<class T>
+    struct apply {
+        typedef list<
+            static_rational<0>,
+            typename add_zeroes_impl<N-1>::template apply<T>::type
+        > type;
+    };
+};
+
+template<>
+struct add_zeroes_impl<0> {
+    template<class T>
+    struct apply {
+        typedef T type;
+    };
+};
+
+// expand_dimensions finds the exponents of
+// a set of dimensions in a dimension_list.
+// the second parameter is assumed to be
+// a superset of the base_dimensions of
+// the first parameter.
+//
+// list<rational> expand_dimensions(dimension_list, list<base_dimension>);
+
+template<int N>
+struct expand_dimensions {
+    template<class Begin, class DimensionIterator>
+    struct apply {
+        typedef typename calculate_base_dimension_coefficients_func<
+            begins_with_dimension<DimensionIterator>::template apply<typename Begin::item>::value
+        >::template apply<DimensionIterator> result;
+        typedef list<
+            typename result::type,
+            typename expand_dimensions<N-1>::template apply<typename Begin::next, typename result::next>::type
+        > type;
+    };
+};
+
+template<>
+struct expand_dimensions<0> {
+    template<class Begin, class DimensionIterator>
+    struct apply {
+        typedef dimensionless_type type;
+    };
+};
+
+template<int N>
+struct create_unit_matrix {
+    template<class Begin, class Dimensions>
+    struct apply {
+        typedef typename create_unit_matrix<N - 1>::template apply<typename Begin::next, Dimensions>::type next;
+        typedef list<typename expand_dimensions<Dimensions::size::value>::template apply<Dimensions, typename Begin::item::dimension_type>::type, next> type;
+    };
+};
+
+template<>
+struct create_unit_matrix<0> {
+    template<class Begin, class Dimensions>
+    struct apply {
+        typedef dimensionless_type type;
+    };
+};
+
+template<class T>
+struct normalize_units {
+    typedef typename find_base_dimensions<T>::type dimensions;
+    typedef typename create_unit_matrix<(T::size::value)>::template apply<
+        T,
+        dimensions
+    >::type matrix;
+    typedef typename make_square_and_invert<matrix>::type type;
+    static const long extra = (type::size::value) - (T::size::value);
+};
+
+// multiply_add_units computes M x V
+// where M is a matrix and V is a horizontal
+// vector
+//
+// list<rational> multiply_add_units(list<list<rational> >, list<rational>);
+
+template<int N>
+struct multiply_add_units_impl {
+    template<class Begin1, class Begin2 ,class X>
+    struct apply {
+        typedef list<
+            typename mpl::plus<
+                typename mpl::times<
+                    typename Begin2::item,
+                    X
+                >::type,
+                typename Begin1::item
+            >::type,
+            typename multiply_add_units_impl<N-1>::template apply<
+                typename Begin1::next,
+                typename Begin2::next,
+                X
+            >::type
+        > type;
+    };
+};
+
+template<>
+struct multiply_add_units_impl<0> {
+    template<class Begin1, class Begin2 ,class X>
+    struct apply {
+        typedef dimensionless_type type;
+    };
+};
+
+template<int N>
+struct multiply_add_units {
+    template<class Begin1, class Begin2>
+    struct apply {
+        typedef typename multiply_add_units_impl<
+            (Begin2::item::size::value)
+        >::template apply<
+            typename multiply_add_units<N-1>::template apply<
+                typename Begin1::next,
+                typename Begin2::next
+            >::type,
+            typename Begin2::item,
+            typename Begin1::item
+        >::type type;
+    };
+};
+
+template<>
+struct multiply_add_units<1> {
+    template<class Begin1, class Begin2>
+    struct apply {
+        typedef typename add_zeroes_impl<
+            (Begin2::item::size::value)
+        >::template apply<dimensionless_type>::type type1;
+        typedef typename multiply_add_units_impl<
+            (Begin2::item::size::value)
+        >::template apply<
+            type1,
+            typename Begin2::item,
+            typename Begin1::item
+        >::type type;
+    };
+};
+
+
+// strip_zeroes erases the first N elements of a list if
+// they are all zero, otherwise returns inconsistent
+//
+// list strip_zeroes(list l, int N) {
+//     if(N == 0) {
+//         return(l);
+//     } else if(l.front == 0) {
+//         return(strip_zeroes(pop_front(l), N-1));
+//     } else {
+//         return(inconsistent);
+//     }
+// }
+
+template<int N>
+struct strip_zeroes_impl;
+
+template<class T>
+struct strip_zeroes_func {
+    template<class L, int N>
+    struct apply {
+        typedef inconsistent type;
+    };
+};
+
+template<>
+struct strip_zeroes_func<static_rational<0> > {
+    template<class L, int N>
+    struct apply {
+        typedef typename strip_zeroes_impl<N-1>::template apply<typename L::next>::type type;
+    };
+};
+
+template<int N>
+struct strip_zeroes_impl {
+    template<class T>
+    struct apply {
+        typedef typename strip_zeroes_func<typename T::item>::template apply<T, N>::type type;
+    };
+};
+
+template<>
+struct strip_zeroes_impl<0> {
+    template<class T>
+    struct apply {
+        typedef T type;
+    };
+};
+
+// Given a list of base_units, computes the
+// exponents of each base unit for a given
+// dimension.
+//
+// list<rational> calculate_base_unit_exponents(list<base_unit> units, dimension_list dimensions);
+
+template<class T>
+struct is_base_dimension_unit {
+    typedef mpl::false_ type;
+    typedef void base_dimension_type;
+};
+template<class T>
+struct is_base_dimension_unit<list<dim<T, static_rational<1> >, dimensionless_type> > {
+    typedef mpl::true_ type;
+    typedef T base_dimension_type;
+};
+
+template<int N>
+struct is_simple_system_impl {
+    template<class Begin, class Prev>
+    struct apply {
+        typedef is_base_dimension_unit<typename Begin::item::dimension_type> test;
+        typedef mpl::and_<
+            typename test::type,
+            mpl::less<Prev, typename test::base_dimension_type>,
+            typename is_simple_system_impl<N-1>::template apply<
+                typename Begin::next,
+                typename test::base_dimension_type
+            >
+        > type;
+        static const bool value = (type::value);
+    };
+};
+
+template<>
+struct is_simple_system_impl<0> {
+    template<class Begin, class Prev>
+    struct apply : mpl::true_ {
+    };
+};
+
+template<class T>
+struct is_simple_system {
+    typedef T Begin;
+    typedef is_base_dimension_unit<typename Begin::item::dimension_type> test;
+    typedef typename mpl::and_<
+        typename test::type,
+        typename is_simple_system_impl<
+            T::size::value - 1
+        >::template apply<
+            typename Begin::next::type,
+            typename test::base_dimension_type
+        >
+    >::type type;
+    static const bool value = type::value;
+};
+
+template<bool>
+struct calculate_base_unit_exponents_impl;
+
+template<>
+struct calculate_base_unit_exponents_impl<true> {
+    template<class T, class Dimensions>
+    struct apply {
+        typedef typename expand_dimensions<(T::size::value)>::template apply<
+            typename find_base_dimensions<T>::type,
+            Dimensions
+        >::type type;
+    };
+};
+
+template<>
+struct calculate_base_unit_exponents_impl<false> {
+    template<class T, class Dimensions>
+    struct apply {
+        // find the units that correspond to each base dimension
+        typedef normalize_units<T> base_solutions;
+        // pad the dimension with zeroes so it can just be a
+        // list of numbers, making the multiplication easy
+        // e.g. if the arguments are list<pound, foot> and
+        // list<mass,time^-2> then this step will
+        // yield list<0,1,-2>
+        typedef typename expand_dimensions<(base_solutions::dimensions::size::value)>::template apply<
+            typename base_solutions::dimensions,
+            Dimensions
+        >::type dimensions;
+        // take the unit corresponding to each base unit
+        // multiply each of its exponents by the exponent
+        // of the base_dimension in the result and sum.
+        typedef typename multiply_add_units<dimensions::size::value>::template apply<
+            dimensions,
+            typename base_solutions::type
+        >::type units;
+        // Now, verify that the dummy units really
+        // cancel out and remove them.
+        typedef typename strip_zeroes_impl<base_solutions::extra>::template apply<units>::type type;
+    };
+};
+
+template<class T, class Dimensions>
+struct calculate_base_unit_exponents {
+    typedef typename calculate_base_unit_exponents_impl<is_simple_system<T>::value>::template apply<T, Dimensions>::type type;
+};
+
+} // namespace detail
+
+} // namespace units
+
+} // namespace boost
+
+#endif