    // Copyright (C) 2009  Davis E. King (davis@dlib.net)
    // License: Boost Software License   See LICENSE.txt for the full license.
    // This code was adapted from code from the JAMA part of NIST's TNT library.
    //    See: http://math.nist.gov/tnt/
    #ifndef DLIB_MATRIX_LU_DECOMPOSITION_H
    #define DLIB_MATRIX_LU_DECOMPOSITION_H

    #include "matrix.h"
    #include "matrix_utilities.h"
    #include "matrix_subexp.h"
    #include "matrix_trsm.h"
    #include <algorithm>

    #ifdef DLIB_USE_LAPACK
    #include "lapack/getrf.h"
    #endif

    namespace dlib
    {

        template <
            typename matrix_exp_type
            >
        class lu_decomposition
        {
        public:

            const static long NR = matrix_exp_type::NR;
            const static long NC = matrix_exp_type::NC;
            typedef typename matrix_exp_type::type type;
            typedef typename matrix_exp_type::mem_manager_type mem_manager_type;
            typedef typename matrix_exp_type::layout_type layout_type;

            typedef matrix<type,0,0,mem_manager_type,layout_type>  matrix_type;
            typedef matrix<type,NR,1,mem_manager_type,layout_type> column_vector_type;
            typedef matrix<long,NR,1,mem_manager_type,layout_type> pivot_column_vector_type;

            // You have supplied an invalid type of matrix_exp_type.  You have
            // to use this object with matrices that contain float or double type data.
            COMPILE_TIME_ASSERT((is_same_type<float, type>::value ||
                                 is_same_type<double, type>::value ));

            template <typename EXP>
            lu_decomposition (
                const matrix_exp<EXP> &A
            );

            bool is_square (
            ) const;

            bool is_singular (
            ) const;

            long nr(
            ) const;

            long nc(
            ) const;

            const matrix_type get_l (
            ) const;

            const matrix_type get_u (
            ) const;

            const pivot_column_vector_type& get_pivot (
            ) const;

            type det (
            ) const;

            template <typename EXP>
            const matrix_type solve (
                const matrix_exp<EXP> &B
            ) const;

        private:

            /* Array for internal storage of decomposition.  */
            matrix<type,0,0,mem_manager_type,column_major_layout>  LU;
            long m, n, pivsign;
            pivot_column_vector_type piv;

        };

    // ----------------------------------------------------------------------------------------
    // ----------------------------------------------------------------------------------------
    //                              Public member functions
    // ----------------------------------------------------------------------------------------
    // ----------------------------------------------------------------------------------------

        template <typename matrix_exp_type>
        template <typename EXP>
        lu_decomposition<matrix_exp_type>::
        lu_decomposition (
            const matrix_exp<EXP>& A
        ) :
            LU(A),
            m(A.nr()),
            n(A.nc())
        {
            using namespace std;
            using std::abs;

            COMPILE_TIME_ASSERT((is_same_type<type, typename EXP::type>::value));

            // make sure requires clause is not broken
            DLIB_ASSERT(A.size() > 0,
                "\tlu_decomposition::lu_decomposition(A)"
                << "\n\tInvalid inputs were given to this function"
                << "\n\tA.size(): " << A.size()
                << "\n\tthis:     " << this
                );

    #ifdef DLIB_USE_LAPACK
            matrix<lapack::integer,0,1,mem_manager_type,layout_type> piv_temp;
            lapack::getrf(LU, piv_temp);

            pivsign = 1;

            // Turn the piv_temp vector into a more useful form.  This way we will have the identity
            // rowm(A,piv) == L*U.  The permutation vector that comes out of LAPACK is somewhat
            // different.
            piv = trans(range(0,m-1));
            for (long i = 0; i < piv_temp.size(); ++i)
            {
                // -1 because FORTRAN is indexed starting with 1 instead of 0
                if (piv(piv_temp(i)-1) != piv(i))
                {
                    std::swap(piv(i), piv(piv_temp(i)-1));
                    pivsign = -pivsign;
                }
            }

    #else

            // Use a "left-looking", dot-product, Crout/Doolittle algorithm.

            piv = trans(range(0,m-1));
            pivsign = 1;

            column_vector_type LUcolj(m);

            // Outer loop.
            for (long j = 0; j < n; j++)
            {

                // Make a copy of the j-th column to localize references.
                LUcolj = colm(LU,j);

                // Apply previous transformations.
                for (long i = 0; i < m; i++)
                {
                    // Most of the time is spent in the following dot product.
                    const long kmax = std::min(i,j);
                    type s;
                    if (kmax > 0)
                        s = rowm(LU,i, kmax)*colm(LUcolj,0,kmax);
                    else
                        s = 0;

                    LU(i,j) = LUcolj(i) -= s;
                }

                // Find pivot and exchange if necessary.
                long p = j;
                for (long i = j+1; i < m; i++)
                {
                    if (abs(LUcolj(i)) > abs(LUcolj(p)))
                    {
                        p = i;
                    }
                }
                if (p != j)
                {
                    long k=0;
                    for (k = 0; k < n; k++)
                    {
                        type t = LU(p,k);
                        LU(p,k) = LU(j,k);
                        LU(j,k) = t;
                    }
                    k = piv(p);
                    piv(p) = piv(j);
                    piv(j) = k;
                    pivsign = -pivsign;
                }

                // Compute multipliers.
                if ((j < m) && (LU(j,j) != 0.0))
                {
                    for (long i = j+1; i < m; i++)
                    {
                        LU(i,j) /= LU(j,j);
                    }
                }
            }

    #endif
        }

    // ----------------------------------------------------------------------------------------

        template <typename matrix_exp_type>
        bool lu_decomposition<matrix_exp_type>::
        is_square (
        ) const
        {
            return m == n;
        }

    // ----------------------------------------------------------------------------------------

        template <typename matrix_exp_type>
        long lu_decomposition<matrix_exp_type>::
        nr (
        ) const
        {
            return m;
        }

    // ----------------------------------------------------------------------------------------

        template <typename matrix_exp_type>
        long lu_decomposition<matrix_exp_type>::
        nc (
        ) const
        {
            return n;
        }

    // ----------------------------------------------------------------------------------------

        template <typename matrix_exp_type>
        bool lu_decomposition<matrix_exp_type>::
        is_singular (
        ) const
        {
            /* Is the matrix singular?
              if upper triangular factor U (and hence A) is singular, false otherwise.
            */
            // make sure requires clause is not broken
            DLIB_ASSERT(is_square() == true,
                "\tbool lu_decomposition::is_singular()"
                << "\n\tYou can only use this on square matrices"
                << "\n\tthis: " << this
                );

            type max_val, min_val;
            find_min_and_max (abs(diag(LU)), min_val, max_val);
            type eps = max_val;
            if (eps != 0)
                eps *= std::sqrt(std::numeric_limits<type>::epsilon())/10;
            else
                eps = 1;  // there is no max so just use 1

            return min_val < eps;
        }

    // ----------------------------------------------------------------------------------------

        template <typename matrix_exp_type>
        const typename lu_decomposition<matrix_exp_type>::matrix_type lu_decomposition<matrix_exp_type>::
        get_l (
        ) const
        {
            if (LU.nr() >= LU.nc())
                return lowerm(LU,1.0);
            else
                return lowerm(subm(LU,0,0,m,m), 1.0);
        }

    // ----------------------------------------------------------------------------------------

        template <typename matrix_exp_type>
        const typename lu_decomposition<matrix_exp_type>::matrix_type lu_decomposition<matrix_exp_type>::
        get_u (
        ) const
        {
            if (LU.nr() >= LU.nc())
                return upperm(subm(LU,0,0,n,n));
            else
                return upperm(LU);
        }

    // ----------------------------------------------------------------------------------------

        template <typename matrix_exp_type>
        const typename lu_decomposition<matrix_exp_type>::pivot_column_vector_type& lu_decomposition<matrix_exp_type>::
        get_pivot (
        ) const
        {
            return piv;
        }

    // ----------------------------------------------------------------------------------------

        template <typename matrix_exp_type>
        typename lu_decomposition<matrix_exp_type>::type lu_decomposition<matrix_exp_type>::
        det (
        ) const
        {
            // make sure requires clause is not broken
            DLIB_ASSERT(is_square() == true,
                "\ttype lu_decomposition::det()"
                << "\n\tYou can only use this on square matrices"
                << "\n\tthis: " << this
                );

            // Check if it is singular and if it is just return 0.
            // We want to do this because a prod() operation can easily
            // overcome a single diagonal element that is effectively 0 when
            // LU is a big enough matrix.
            if (is_singular())
                return 0;

            return prod(diag(LU))*static_cast<type>(pivsign);
        }

    // ----------------------------------------------------------------------------------------

        template <typename matrix_exp_type>
        template <typename EXP>
        const typename lu_decomposition<matrix_exp_type>::matrix_type lu_decomposition<matrix_exp_type>::
        solve (
            const matrix_exp<EXP> &B
        ) const
        {
            COMPILE_TIME_ASSERT((is_same_type<type, typename EXP::type>::value));

            // make sure requires clause is not broken
            DLIB_ASSERT(is_square() == true && B.nr() == nr(),
                "\ttype lu_decomposition::solve()"
                << "\n\tInvalid arguments to this function"
                << "\n\tis_square():   " << (is_square()? "true":"false" )
                << "\n\tB.nr():        " << B.nr()
                << "\n\tnr():          " << nr()
                << "\n\tthis:          " << this
                );

            // Copy right hand side with pivoting
            matrix<type,0,0,mem_manager_type,column_major_layout> X(rowm(B, piv));

            using namespace blas_bindings;
            // Solve L*Y = B(piv,:)
            triangular_solver(CblasLeft, CblasLower, CblasNoTrans, CblasUnit, LU, X);
            // Solve U*X = Y;
            triangular_solver(CblasLeft, CblasUpper, CblasNoTrans, CblasNonUnit, LU, X);
            return X;
        }

    // ----------------------------------------------------------------------------------------

    }

    #endif // DLIB_MATRIX_LU_DECOMPOSITION_H