dlib C++ Library - matrix_qr.h

    // 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_QR_DECOMPOSITION_H
    #define DLIB_MATRIX_QR_DECOMPOSITION_H

    #include "matrix.h"
    #include "matrix_utilities.h"
    #include "matrix_subexp.h"

    #ifdef DLIB_USE_LAPACK
    #include "lapack/geqrf.h"
    #include "lapack/ormqr.h"
    #endif

    #include "matrix_trsm.h"

    namespace dlib
    {

        template <
            typename matrix_exp_type
            >
        class qr_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;

            // 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>
            qr_decomposition(
                const matrix_exp<EXP>& A
            );

            bool is_full_rank(
            ) const;

            long nr(
            ) const;

            long nc(
            ) const;

            const matrix_type get_r (
            ) const;

            const matrix_type get_q (
            ) const;

            template <typename T, long R, long C, typename MM, typename L>
            void get_q (
                matrix<T,R,C,MM,L>& Q
            ) const;

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

        private:

    #ifndef DLIB_USE_LAPACK
            template <typename EXP>
            const matrix_type solve_mat (
                const matrix_exp<EXP>& B
            ) const;

            template <typename EXP>
            const matrix_type solve_vect (
                const matrix_exp<EXP>& B
            ) const;
    #endif

            /** Array for internal storage of decomposition.
            @serial internal array storage.
            */
            matrix<type,0,0,mem_manager_type,column_major_layout> QR_;

            /** Row and column dimensions.
            @serial column dimension.
            @serial row dimension.
            */
            long m, n;

            /** Array for internal storage of diagonal of R.
            @serial diagonal of R.
            */
            typedef matrix<type,0,1,mem_manager_type,column_major_layout> column_vector_type;
            column_vector_type tau;
            column_vector_type Rdiag;

        };

    // ----------------------------------------------------------------------------------------
    // ----------------------------------------------------------------------------------------
    //                                      Member functions
    // ----------------------------------------------------------------------------------------
    // ----------------------------------------------------------------------------------------

        template <typename matrix_exp_type>
        template <typename EXP>
        qr_decomposition<matrix_exp_type>::
        qr_decomposition(
            const matrix_exp<EXP>& A
        )
        {
            COMPILE_TIME_ASSERT((is_same_type<type, typename EXP::type>::value));

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

            QR_ = A;
            m = A.nr();
            n = A.nc();

    #ifdef DLIB_USE_LAPACK

            lapack::geqrf(QR_, tau);
            Rdiag = diag(QR_);

    #else
            Rdiag.set_size(n);
            long i=0, j=0, k=0;

            // Main loop.
            for (k = 0; k < n; k++)
            {
                // Compute 2-norm of k-th column without under/overflow.
                type nrm = 0;
                for (i = k; i < m; i++)
                {
                    nrm = hypot(nrm,QR_(i,k));
                }

                if (nrm != 0.0)
                {
                    // Form k-th Householder vector.
                    if (QR_(k,k) < 0)
                    {
                        nrm = -nrm;
                    }
                    for (i = k; i < m; i++)
                    {
                        QR_(i,k) /= nrm;
                    }
                    QR_(k,k) += 1.0;

                    // Apply transformation to remaining columns.
                    for (j = k+1; j < n; j++)
                    {
                        type s = 0.0;
                        for (i = k; i < m; i++)
                        {
                            s += QR_(i,k)*QR_(i,j);
                        }
                        s = -s/QR_(k,k);
                        for (i = k; i < m; i++)
                        {
                            QR_(i,j) += s*QR_(i,k);
                        }
                    }
                }
                Rdiag(k) = -nrm;
            }
    #endif
        }

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

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

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

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

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

        template <typename matrix_exp_type>
        bool qr_decomposition<matrix_exp_type>::
        is_full_rank(
        ) const
        {
            type eps = max(abs(Rdiag));
            if (eps != 0)
                eps *= std::sqrt(std::numeric_limits<type>::epsilon())/100;
            else
                eps = 1;  // there is no max so just use 1

            // check if any of the elements of Rdiag are effectively 0
            return min(abs(Rdiag)) > eps;
        }

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

        template <typename matrix_exp_type>
        const typename qr_decomposition<matrix_exp_type>::matrix_type qr_decomposition<matrix_exp_type>::
        get_r(
        ) const
        {
            matrix_type R(n,n);
            for (long i = 0; i < n; i++)
            {
                for (long j = 0; j < n; j++)
                {
                    if (i < j)
                    {
                        R(i,j) = QR_(i,j);
                    }
                    else if (i == j)
                    {
                        R(i,j) = Rdiag(i);
                    }
                    else
                    {
                        R(i,j) = 0.0;
                    }
                }
            }
            return R;
        }

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

        template <typename matrix_exp_type>
        const typename qr_decomposition<matrix_exp_type>::matrix_type qr_decomposition<matrix_exp_type>::
        get_q(
        ) const
        {
            matrix_type Q;
            get_q(Q);
            return Q;
        }

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

        template <typename matrix_exp_type>
        template <typename T, long R, long C, typename MM, typename L>
        void qr_decomposition<matrix_exp_type>::
        get_q(
            matrix<T,R,C,MM,L>& X
        ) const
        {
    #ifdef DLIB_USE_LAPACK
            // Take only the first n columns of an identity matrix.  This way
            // X ends up being an m by n matrix.
            X = colm(identity_matrix<type>(m), range(0,n-1));

            // Compute Y = Q*X
            lapack::ormqr('L','N', QR_, tau, X);

    #else
            long i=0, j=0, k=0;

            X.set_size(m,n);
            for (k = n-1; k >= 0; k--)
            {
                for (i = 0; i < m; i++)
                {
                    X(i,k) = 0.0;
                }
                X(k,k) = 1.0;
                for (j = k; j < n; j++)
                {
                    if (QR_(k,k) != 0)
                    {
                        type s = 0.0;
                        for (i = k; i < m; i++)
                        {
                            s += QR_(i,k)*X(i,j);
                        }
                        s = -s/QR_(k,k);
                        for (i = k; i < m; i++)
                        {
                            X(i,j) += s*QR_(i,k);
                        }
                    }
                }
            }
    #endif
        }

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

        template <typename matrix_exp_type>
        template <typename EXP>
        const typename qr_decomposition<matrix_exp_type>::matrix_type qr_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(B.nr() == nr(),
                "\tconst matrix_type qr_decomposition::solve(B)"
                << "\n\tInvalid inputs were given to this function"
                << "\n\tB.nr():         " << B.nr()
                << "\n\tnr():           " << nr()
                << "\n\tthis:           " << this
                );

    #ifdef DLIB_USE_LAPACK

            using namespace blas_bindings;
            matrix<type,0,0,mem_manager_type,column_major_layout> X(B);
            // Compute Y = transpose(Q)*B
            lapack::ormqr('L','T',QR_, tau, X);
            // Solve R*X = Y;
            triangular_solver(CblasLeft, CblasUpper, CblasNoTrans, CblasNonUnit, QR_, X, n);

            /* return n x nx portion of X */
            return subm(X,0,0,n,B.nc());

    #else
            // just call the right version of the solve function
            if (B.nc() == 1)
                return solve_vect(B);
            else
                return solve_mat(B);
    #endif
        }

    // ----------------------------------------------------------------------------------------
    // ----------------------------------------------------------------------------------------
    //                           Private member functions
    // ----------------------------------------------------------------------------------------
    // ----------------------------------------------------------------------------------------

    #ifndef DLIB_USE_LAPACK

        template <typename matrix_exp_type>
        template <typename EXP>
        const typename qr_decomposition<matrix_exp_type>::matrix_type qr_decomposition<matrix_exp_type>::
        solve_vect(
            const matrix_exp<EXP>& B
        ) const
        {

            column_vector_type x(B);

            // Compute Y = transpose(Q)*B
            for (long k = 0; k < n; k++)
            {
                type s = 0.0;
                for (long i = k; i < m; i++)
                {
                    s += QR_(i,k)*x(i);
                }
                s = -s/QR_(k,k);
                for (long i = k; i < m; i++)
                {
                    x(i) += s*QR_(i,k);
                }
            }
            // Solve R*X = Y;
            for (long k = n-1; k >= 0; k--)
            {
                x(k) /= Rdiag(k);
                for (long i = 0; i < k; i++)
                {
                    x(i) -= x(k)*QR_(i,k);
                }
            }

            /* return n x 1 portion of x */
            return colm(x,0,n);
        }

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

        template <typename matrix_exp_type>
        template <typename EXP>
        const typename qr_decomposition<matrix_exp_type>::matrix_type qr_decomposition<matrix_exp_type>::
        solve_mat(
            const matrix_exp<EXP>& B
        ) const
        {
            const long nx = B.nc();
            matrix_type X(B);
            long i=0, j=0, k=0;

            // Compute Y = transpose(Q)*B
            for (k = 0; k < n; k++)
            {
                for (j = 0; j < nx; j++)
                {
                    type s = 0.0;
                    for (i = k; i < m; i++)
                    {
                        s += QR_(i,k)*X(i,j);
                    }
                    s = -s/QR_(k,k);
                    for (i = k; i < m; i++)
                    {
                        X(i,j) += s*QR_(i,k);
                    }
                }
            }
            // Solve R*X = Y;
            for (k = n-1; k >= 0; k--)
            {
                for (j = 0; j < nx; j++)
                {
                    X(k,j) /= Rdiag(k);
                }
                for (i = 0; i < k; i++)
                {
                    for (j = 0; j < nx; j++)
                    {
                        X(i,j) -= X(k,j)*QR_(i,k);
                    }
                }
            }

            /* return n x nx portion of X */
            return subm(X,0,0,n,nx);
        }

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

    #endif // DLIB_USE_LAPACK not defined

    }

    #endif // DLIB_MATRIX_QR_DECOMPOSITION_H