text: //=================================================================================================
  /*!
  //  \file blaze/math/expressions/TSMatDVecMultExpr.h
  //  \brief Header file for the transpose sparse matrix/dense vector multiplication expression
  //
  //  Copyright (C) 2012-2017 Klaus Iglberger - All Rights Reserved
  //
  //  This file is part of the Blaze library. You can redistribute it and/or modify it under
  //  the terms of the New (Revised) BSD License. Redistribution and use in source and binary
  //  forms, with or without modification, are permitted provided that the following conditions
  //  are met:
  //
  //  1. Redistributions of source code must retain the above copyright notice, this list of
  //     conditions and the following disclaimer.
  //  2. Redistributions in binary form must reproduce the above copyright notice, this list
  //     of conditions and the following disclaimer in the documentation and/or other materials
  //     provided with the distribution.
  //  3. Neither the names of the Blaze development group nor the names of its contributors
  //     may be used to endorse or promote products derived from this software without specific
  //     prior written permission.
  //
  //  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY
  //  EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
  //  OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT
  //  SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
  //  INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
  //  TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR
  //  BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
  //  CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
  //  ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH
  //  DAMAGE.
  */
  //=================================================================================================
  
  #ifndef _BLAZE_MATH_EXPRESSIONS_TSMATDVECMULTEXPR_H_
  #define _BLAZE_MATH_EXPRESSIONS_TSMATDVECMULTEXPR_H_
  
  
  //*************************************************************************************************
  // Includes
  //*************************************************************************************************
  
  #include <blaze/math/Aliases.h>
  #include <blaze/math/constraints/ColumnMajorMatrix.h>
  #include <blaze/math/constraints/ColumnVector.h>
  #include <blaze/math/constraints/DenseMatrix.h>
  #include <blaze/math/constraints/DenseVector.h>
  #include <blaze/math/constraints/MatMatMultExpr.h>
  #include <blaze/math/constraints/MatVecMultExpr.h>
  #include <blaze/math/constraints/RequiresEvaluation.h>
  #include <blaze/math/constraints/SparseMatrix.h>
  #include <blaze/math/constraints/Symmetric.h>
  #include <blaze/math/Exception.h>
  #include <blaze/math/expressions/Computation.h>
  #include <blaze/math/expressions/DenseVector.h>
  #include <blaze/math/expressions/Forward.h>
  #include <blaze/math/expressions/MatVecMultExpr.h>
  #include <blaze/math/shims/IsDefault.h>
  #include <blaze/math/shims/Reset.h>
  #include <blaze/math/shims/Serial.h>
  #include <blaze/math/traits/MultTrait.h>
  #include <blaze/math/typetraits/IsAligned.h>
  #include <blaze/math/typetraits/IsComputation.h>
  #include <blaze/math/typetraits/IsDiagonal.h>
  #include <blaze/math/typetraits/IsExpression.h>
  #include <blaze/math/typetraits/IsLower.h>
  #include <blaze/math/typetraits/IsResizable.h>
  #include <blaze/math/typetraits/IsStrictlyLower.h>
  #include <blaze/math/typetraits/IsStrictlyUpper.h>
  #include <blaze/math/typetraits/IsSymmetric.h>
  #include <blaze/math/typetraits/IsUpper.h>
  #include <blaze/math/typetraits/RequiresEvaluation.h>
  #include <blaze/math/typetraits/Size.h>
  #include <blaze/math/views/Check.h>
  #include <blaze/system/Thresholds.h>
  #include <blaze/util/Assert.h>
  #include <blaze/util/DisableIf.h>
  #include <blaze/util/EnableIf.h>
  #include <blaze/util/FunctionTrace.h>
  #include <blaze/util/mpl/If.h>
  #include <blaze/util/Types.h>
  #include <blaze/util/typetraits/RemoveReference.h>
  
  
  namespace blaze {
  
  //=================================================================================================
  //
  //  CLASS TSMATDVECMULTEXPR
  //
  //=================================================================================================
  
  //*************************************************************************************************
  /*!\brief Expression object for transpose sparse matrix-dense vector multiplications.
  // \ingroup dense_vector_expression
  //
  // The TSMatDVecMultExpr class represents the compile time expression for multiplications
  // between column-major sparse matrices and dense vectors.
  */
  template< typename MT    // Type of the left-hand side sparse matrix
          , typename VT >  // Type of the right-hand side dense vector
  class TSMatDVecMultExpr
     : public MatVecMultExpr< DenseVector< TSMatDVecMultExpr<MT,VT>, false > >
     , private Computation
  {
   private:
     //**Type definitions****************************************************************************
     using MRT = ResultType_<MT>;     //!< Result type of the left-hand side sparse matrix expression.
     using VRT = ResultType_<VT>;     //!< Result type of the right-hand side dense vector expression.
     using MCT = CompositeType_<MT>;  //!< Composite type of the left-hand side sparse matrix expression.
     using VCT = CompositeType_<VT>;  //!< Composite type of the right-hand side dense vector expression.
     //**********************************************************************************************
  
     //**********************************************************************************************
     //! Compilation switch for the composite type of the left-hand side sparse matrix expression.
     enum : bool { evaluateMatrix = RequiresEvaluation<MT>::value };
     //**********************************************************************************************
  
     //**********************************************************************************************
     //! Compilation switch for the composite type of the right-hand side dense vector expression.
     enum : bool { evaluateVector = IsComputation<VT>::value || RequiresEvaluation<VT>::value };
     //**********************************************************************************************
  
     //**********************************************************************************************
     /*! \cond BLAZE_INTERNAL */
     //! Helper structure for the explicit application of the SFINAE principle.
     /*! The UseSMPAssign struct is a helper struct for the selection of the parallel evaluation
         strategy. In case either the matrix or the vector operand requires an intermediate
         evaluation, the nested \value will be set to 1, otherwise it will be 0. */
     template< typename T1 >
     struct UseSMPAssign {
        enum : bool { value = ( evaluateMatrix || evaluateVector ) };
     };
     /*! \endcond */
     //**********************************************************************************************
  
   public:
     //**Type definitions****************************************************************************
     using This          = TSMatDVecMultExpr<MT,VT>;    //!< Type of this TSMatDVecMultExpr instance.
     using ResultType    = MultTrait_<MRT,VRT>;         //!< Result type for expression template evaluations.
     using TransposeType = TransposeType_<ResultType>;  //!< Transpose type for expression template evaluations.
     using ElementType   = ElementType_<ResultType>;    //!< Resulting element type.
     using ReturnType    = const ElementType;           //!< Return type for expression template evaluations.
     using CompositeType = const ResultType;            //!< Data type for composite expression templates.
  
     //! Composite type of the left-hand side sparse matrix expression.
     using LeftOperand = If_< IsExpression<MT>, const MT, const MT& >;
  
     //! Composite type of the right-hand side dense vector expression.
     using RightOperand = If_< IsExpression<VT>, const VT, const VT& >;
  
     //! Type for the assignment of the left-hand side sparse matrix operand.
     using LT = IfTrue_< evaluateMatrix, const MRT, MCT >;
  
     //! Type for the assignment of the right-hand side dense vector operand.
     using RT = IfTrue_< evaluateVector, const VRT, VCT >;
     //**********************************************************************************************
  
     //**Compilation flags***************************************************************************
     //! Compilation switch for the expression template evaluation strategy.
     enum : bool { simdEnabled = false };
  
     //! Compilation switch for the expression template assignment strategy.
     enum : bool { smpAssignable = !evaluateMatrix && MT::smpAssignable &&
                                   !evaluateVector && VT::smpAssignable };
     //**********************************************************************************************
  
     //**Constructor*********************************************************************************
     /*!\brief Constructor for the TSMatDVecMultExpr class.
     //
     // \param mat The left-hand side sparse matrix operand of the multiplication expression.
     // \param vec The right-hand side dense vector operand of the multiplication expression.
     */
     explicit inline TSMatDVecMultExpr( const MT& mat, const VT& vec ) noexcept
        : mat_( mat )  // Left-hand side sparse matrix of the multiplication expression
        , vec_( vec )  // Right-hand side dense vector of the multiplication expression
     {
        BLAZE_INTERNAL_ASSERT( mat_.columns() == vec_.size(), "Invalid matrix and vector sizes" );
     }
     //**********************************************************************************************
  
     //**Subscript operator**************************************************************************
     /*!\brief Subscript operator for the direct access to the vector elements.
     //
     // \param index Access index. The index has to be in the range \f$[0..N-1]\f$.
     // \return The resulting value.
     */
     inline ReturnType operator[]( size_t index ) const {
        BLAZE_INTERNAL_ASSERT( index < mat_.rows(), "Invalid vector access index" );
  
        if( IsDiagonal<MT>::value )
        {
           return mat_(index,index) * vec_[index];
        }
        else if( IsLower<MT>::value )
        {
           const size_t n( IsStrictlyLower<MT>::value ? index : index+1UL );
           return subvector( row( mat_, index, unchecked ), 0UL, n, unchecked ) *
                  subvector( vec_, 0UL, n, unchecked );
        }
        else if( IsUpper<MT>::value )
        {
           const size_t begin( IsStrictlyUpper<MT>::value ? index+1UL : index );
           const size_t n    ( mat_.columns() - begin );
           return subvector( row( mat_, index, unchecked ), begin, n, unchecked ) *
                  subvector( vec_, begin, n, unchecked );
        }
        else
        {
           return row( mat_, index, unchecked ) * vec_;
        }
     }
     //**********************************************************************************************
  
     //**At function*********************************************************************************
     /*!\brief Checked access to the vector elements.
     //
     // \param index Access index. The index has to be in the range \f$[0..N-1]\f$.
     // \return The resulting value.
     // \exception std::out_of_range Invalid vector access index.
     */
     inline ReturnType at( size_t index ) const {
        if( index >= mat_.rows() ) {
           BLAZE_THROW_OUT_OF_RANGE( "Invalid vector access index" );
        }
        return (*this)[index];
     }
     //**********************************************************************************************
  
     //**Size function*******************************************************************************
     /*!\brief Returns the current size/dimension of the vector.
     //
     // \return The size of the vector.
     */
     inline size_t size() const noexcept {
        return mat_.rows();
     }
     //**********************************************************************************************
  
     //**Left operand access*************************************************************************
     /*!\brief Returns the left-hand side transpose sparse matrix operand.
     //
     // \return The left-hand side transpose sparse matrix operand.
     */
     inline LeftOperand leftOperand() const noexcept {
        return mat_;
     }
     //**********************************************************************************************
  
     //**Right operand access************************************************************************
     /*!\brief Returns the right-hand side dense vector operand.
     //
     // \return The right-hand side dense vector operand.
     */
     inline RightOperand rightOperand() const noexcept {
        return vec_;
     }
     //**********************************************************************************************
  
     //**********************************************************************************************
     /*!\brief Returns whether the expression can alias with the given address \a alias.
     //
     // \param alias The alias to be checked.
     // \return \a true in case the expression can alias, \a false otherwise.
     */
     template< typename T >
     inline bool canAlias( const T* alias ) const noexcept {
        return ( mat_.isAliased( alias ) || vec_.isAliased( alias ) );
     }
     //**********************************************************************************************
  
     //**********************************************************************************************
     /*!\brief Returns whether the expression is aliased with the given address \a alias.
     //
     // \param alias The alias to be checked.
     // \return \a true in case an alias effect is detected, \a false otherwise.
     */
     template< typename T >
     inline bool isAliased( const T* alias ) const noexcept {
        return ( mat_.isAliased( alias ) || vec_.isAliased( alias ) );
     }
     //**********************************************************************************************
  
     //**********************************************************************************************
     /*!\brief Returns whether the operands of the expression are properly aligned in memory.
     //
     // \return \a true in case the operands are aligned, \a false if not.
     */
     inline bool isAligned() const noexcept {
        return vec_.isAligned();
     }
     //**********************************************************************************************
  
     //**********************************************************************************************
     /*!\brief Returns whether the expression can be used in SMP assignments.
     //
     // \return \a true in case the expression can be used in SMP assignments, \a false if not.
     */
     inline bool canSMPAssign() const noexcept {
        return ( size() > SMP_TSMATDVECMULT_THRESHOLD );
     }
     //**********************************************************************************************
  
   private:
     //**Member variables****************************************************************************
     LeftOperand  mat_;  //!< Left-hand side sparse matrix of the multiplication expression.
     RightOperand vec_;  //!< Right-hand side dense vector of the multiplication expression.
     //**********************************************************************************************
  
     //**Assignment to dense vectors*****************************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Assignment of a transpose sparse matrix-dense vector multiplication to a dense
     //        vector (\f$ \vec{y}=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side dense vector.
     // \param rhs The right-hand side multiplication expression to be assigned.
     // \return void
     //
     // This function implements the performance optimized assignment of a transpose sparse matrix-
     // dense vector multiplication expression to a dense vector.
     */
     template< typename VT2 >  // Type of the target dense vector
     friend inline void assign( DenseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        reset( ~lhs );
  
        if( rhs.mat_.columns() == 0UL ) return;
  
        LT A( serial( rhs.mat_ ) );  // Evaluation of the left-hand side sparse matrix operand
        RT x( serial( rhs.vec_ ) );  // Evaluation of the right-hand side dense vector operand
  
        BLAZE_INTERNAL_ASSERT( A.rows()    == rhs.mat_.rows()   , "Invalid number of rows"    );
        BLAZE_INTERNAL_ASSERT( A.columns() == rhs.mat_.columns(), "Invalid number of columns" );
        BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
        BLAZE_INTERNAL_ASSERT( A.rows()    == (~lhs).size()     , "Invalid vector size"       );
  
        TSMatDVecMultExpr::selectAssignKernel( ~lhs, A, x );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Optimized assignment to dense vectors*******************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Optimized assignment of a transpose sparse matrix-dense vector multiplication
     //        (\f$ \vec{y}=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param y The target left-hand side dense vector.
     // \param A The left-hand side sparse matrix operand.
     // \param x The right-hand side dense vector operand.
     // \return void
     //
     // This function implements the serial assignment kernel for the transpose sparse matrix-
     // dense vector multiplication.
     */
     template< typename VT1    // Type of the left-hand side target vector
             , typename MT1    // Type of the left-hand side matrix operand
             , typename VT2 >  // Type of the right-hand side vector operand
     static inline void selectAssignKernel( VT1& y, const MT1& A, const VT2& x )
     {
        using ConstIterator = ConstIterator_< RemoveReference_<MT1> >;
  
        for( size_t j=0UL; j<A.columns(); ++j )
        {
           ConstIterator element( A.begin(j) );
           const ConstIterator end( A.end(j) );
  
           for( ; element!=end; ++element ) {
              if( IsResizable< ElementType_<VT1> >::value &&
                  isDefault( y[element->index()] ) )
                 y[element->index()] = element->value() * x[j];
              else
                 y[element->index()] += element->value() * x[j];
           }
        }
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Assignment to sparse vectors****************************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Assignment of a transpose sparse matrix-dense vector multiplication to a sparse
     //        vector (\f$ \vec{y}=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side sparse vector.
     // \param rhs The right-hand side multiplication expression to be assigned.
     // \return void
     //
     // This function implements the performance optimized assignment of a transpose sparse matrix-
     // dense vector multiplication expression to a sparse vector.
     */
     template< typename VT2 >  // Type of the target sparse vector
     friend inline void assign( SparseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_BE_COLUMN_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_NOT_REQUIRE_EVALUATION( ResultType );
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        const ResultType tmp( serial( rhs ) );
        assign( ~lhs, tmp );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Addition assignment to dense vectors********************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Addition assignment of a sparse matrix-dense vector multiplication to a dense vector
     //        (\f$ \vec{y}+=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side dense vector.
     // \param rhs The right-hand side multiplication expression to be added.
     // \return void
     //
     // This function implements the performance optimized addition assignment of a sparse matrix-
     // dense vector multiplication expression to a dense vector.
     */
     template< typename VT2 >  // Type of the target dense vector
     friend inline void addAssign( DenseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        if( rhs.mat_.columns() == 0UL ) return;
  
        LT A( serial( rhs.mat_ ) );  // Evaluation of the left-hand side sparse matrix operand
        RT x( serial( rhs.vec_ ) );  // Evaluation of the right-hand side dense vector operand
  
        BLAZE_INTERNAL_ASSERT( A.rows()    == rhs.mat_.rows()   , "Invalid number of rows"    );
        BLAZE_INTERNAL_ASSERT( A.columns() == rhs.mat_.columns(), "Invalid number of columns" );
        BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
        BLAZE_INTERNAL_ASSERT( A.rows()    == (~lhs).size()     , "Invalid vector size"       );
  
        TSMatDVecMultExpr::selectAddAssignKernel( ~lhs, A, x );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Optimized addition assignment to dense vectors**********************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Optimized addition assignment of a transpose sparse matrix-dense vector
     //        multiplication (\f$ \vec{y}+=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param y The target left-hand side dense vector.
     // \param A The left-hand side sparse matrix operand.
     // \param x The right-hand side dense vector operand.
     // \return void
     //
     // This function implements the serial addition assignment kernel for the transpose sparse
     // matrix-dense vector multiplication.
     */
     template< typename VT1    // Type of the left-hand side target vector
             , typename MT1    // Type of the left-hand side matrix operand
             , typename VT2 >  // Type of the right-hand side vector operand
     static inline void selectAddAssignKernel( VT1& y, const MT1& A, const VT2& x )
     {
        using ConstIterator = ConstIterator_< RemoveReference_<MT1> >;
  
        for( size_t j=0UL; j<A.columns(); ++j )
        {
           ConstIterator element( A.begin(j) );
           const ConstIterator end( A.end(j) );
  
           for( ; element!=end; ++element ) {
              y[element->index()] += element->value() * x[j];
           }
        }
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Addition assignment to sparse vectors*******************************************************
     // No special implementation for the addition assignment to sparse vectors.
     //**********************************************************************************************
  
     //**Subtraction assignment to dense vectors*****************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Subtraction assignment of a sparse matrix-dense vector multiplication to a dense
     //        vector (\f$ \vec{y}-=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side dense vector.
     // \param rhs The right-hand side multiplication expression to be subtracted.
     // \return void
     //
     // This function implements the performance optimized subtraction assignment of a sparse matrix-
     // dense vector multiplication expression to a dense vector.
     */
     template< typename VT2 >  // Type of the target dense vector
     friend inline void subAssign( DenseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        if( rhs.mat_.columns() == 0UL ) return;
  
        LT A( serial( rhs.mat_ ) );  // Evaluation of the left-hand side sparse matrix operand
        RT x( serial( rhs.vec_ ) );  // Evaluation of the right-hand side dense vector operand
  
        BLAZE_INTERNAL_ASSERT( A.rows()    == rhs.mat_.rows()   , "Invalid number of rows"    );
        BLAZE_INTERNAL_ASSERT( A.columns() == rhs.mat_.columns(), "Invalid number of columns" );
        BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
        BLAZE_INTERNAL_ASSERT( A.rows()    == (~lhs).size()     , "Invalid vector size"       );
  
        TSMatDVecMultExpr::selectSubAssignKernel( ~lhs, A, x );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Optimized subtraction assignment to dense vectors*******************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Optimized subtraction assignment of a transpose sparse matrix-dense vector
     //        multiplication (\f$ \vec{y}-=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param y The target left-hand side dense vector.
     // \param A The left-hand side sparse matrix operand.
     // \param x The right-hand side dense vector operand.
     // \return void
     //
     // This function implements the serial subtraction assignment kernel for the transpose sparse
     // matrix-dense vector multiplication.
     */
     template< typename VT1    // Type of the left-hand side target vector
             , typename MT1    // Type of the left-hand side matrix operand
             , typename VT2 >  // Type of the right-hand side vector operand
     static inline void selectSubAssignKernel( VT1& y, const MT1& A, const VT2& x )
     {
        using ConstIterator = ConstIterator_< RemoveReference_<MT1> >;
  
        for( size_t j=0UL; j<A.columns(); ++j )
        {
           ConstIterator element( A.begin(j) );
           const ConstIterator end( A.end(j) );
  
           for( ; element!=end; ++element ) {
              y[element->index()] -= element->value() * x[j];
           }
        }
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Subtraction assignment to sparse vectors****************************************************
     // No special implementation for the subtraction assignment to sparse vectors.
     //**********************************************************************************************
  
     //**Multiplication assignment to dense vectors**************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Multiplication assignment of a sparse matrix-dense vector multiplication to a dense
     //        vector (\f$ \vec{y}*=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side dense vector.
     // \param rhs The right-hand side multiplication expression to be multiplied.
     // \return void
     //
     // This function implements the performance optimized multiplication assignment of a sparse
     // matrix-dense vector multiplication expression to a dense vector.
     */
     template< typename VT2 >  // Type of the target dense vector
     friend inline void multAssign( DenseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_BE_COLUMN_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_NOT_REQUIRE_EVALUATION( ResultType );
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        const ResultType tmp( serial( rhs ) );
        multAssign( ~lhs, tmp );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Multiplication assignment to sparse vectors*************************************************
     // No special implementation for the multiplication assignment to sparse vectors.
     //**********************************************************************************************
  
     //**Division assignment to dense vectors********************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Disision assignment of a sparse matrix-dense vector multiplication to a dense vector
     //        (\f$ \vec{y}/=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side dense vector.
     // \param rhs The right-hand side multiplication expression divisor.
     // \return void
     //
     // This function implements the performance optimized division assignment of a sparse matrix-
     // dense vector multiplication expression to a dense vector.
     */
     template< typename VT2 >  // Type of the target dense vector
     friend inline void divAssign( DenseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_BE_COLUMN_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_NOT_REQUIRE_EVALUATION( ResultType );
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        const ResultType tmp( serial( rhs ) );
        divAssign( ~lhs, tmp );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Division assignment to sparse vectors*******************************************************
     // No special implementation for the division assignment to sparse vectors.
     //**********************************************************************************************
  
     //**SMP assignment to dense vectors*************************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief SMP assignment of a transpose sparse matrix-dense vector multiplication to a dense
     //        vector (\f$ \vec{y}=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side dense vector.
     // \param rhs The right-hand side multiplication expression to be assigned.
     // \return void
     //
     // This function implements the performance optimized SMP assignment of a transpose sparse
     // matrix-dense vector multiplication expression to a dense vector. Due to the explicit
     // application of the SFINAE principle, this function can only be selected by the compiler
     // in case the expression specific parallel evaluation strategy is selected.
     */
     template< typename VT2 >  // Type of the target dense vector
     friend inline EnableIf_< UseSMPAssign<VT2> >
        smpAssign( DenseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        reset( ~lhs );
  
        if( rhs.mat_.columns() == 0UL ) return;
  
        LT A( rhs.mat_ );  // Evaluation of the left-hand side sparse matrix operand
        RT x( rhs.vec_ );  // Evaluation of the right-hand side dense vector operand
  
        BLAZE_INTERNAL_ASSERT( A.rows()    == rhs.mat_.rows()   , "Invalid number of rows"    );
        BLAZE_INTERNAL_ASSERT( A.columns() == rhs.mat_.columns(), "Invalid number of columns" );
        BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
        BLAZE_INTERNAL_ASSERT( A.rows()    == (~lhs).size()     , "Invalid vector size"       );
  
        smpAssign( ~lhs, A * x );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**SMP assignment to sparse vectors************************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief SMP assignment of a transpose sparse matrix-dense vector multiplication to a sparse
     //        vector (\f$ \vec{y}=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side sparse vector.
     // \param rhs The right-hand side multiplication expression to be assigned.
     // \return void
     //
     // This function implements the performance optimized SMP assignment of a transpose sparse
     // matrix-dense vector multiplication expression to a sparse vector. Due to the explicit
     // application of the SFINAE principle, this function can only be selected by the compiler
     // in case the expression specific parallel evaluation strategy is selected.
     */
     template< typename VT2 >  // Type of the target sparse vector
     friend inline EnableIf_< UseSMPAssign<VT2> >
        smpAssign( SparseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_BE_COLUMN_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_NOT_REQUIRE_EVALUATION( ResultType );
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        const ResultType tmp( rhs );
        smpAssign( ~lhs, tmp );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**SMP addition assignment to dense vectors****************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief SMP addition assignment of a sparse matrix-dense vector multiplication to a dense
     //        vector (\f$ \vec{y}+=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side dense vector.
     // \param rhs The right-hand side multiplication expression to be added.
     // \return void
     //
     // This function implements the performance optimized SMP addition assignment of a sparse
     // matrix-dense vector multiplication expression to a dense vector. Due to the explicit
     // application of the SFINAE principle, this function can only be selected by the compiler
     // in case the expression specific parallel evaluation strategy is selected.
     */
     template< typename VT2 >  // Type of the target dense vector
     friend inline EnableIf_< UseSMPAssign<VT2> >
        smpAddAssign( DenseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        if( rhs.mat_.columns() == 0UL ) return;
  
        LT A( rhs.mat_ );  // Evaluation of the left-hand side sparse matrix operand
        RT x( rhs.vec_ );  // Evaluation of the right-hand side dense vector operand
  
        BLAZE_INTERNAL_ASSERT( A.rows()    == rhs.mat_.rows()   , "Invalid number of rows"    );
        BLAZE_INTERNAL_ASSERT( A.columns() == rhs.mat_.columns(), "Invalid number of columns" );
        BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
        BLAZE_INTERNAL_ASSERT( A.rows()    == (~lhs).size()     , "Invalid vector size"       );
  
        smpAddAssign( ~lhs, A * x );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**SMP addition assignment to sparse vectors***************************************************
     // No special implementation for the SMP addition assignment to sparse vectors.
     //**********************************************************************************************
  
     //**SMP subtraction assignment to dense vectors*************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief SMP subtraction assignment of a sparse matrix-dense vector multiplication to a dense
     //        vector (\f$ \vec{y}-=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side dense vector.
     // \param rhs The right-hand side multiplication expression to be subtracted.
     // \return void
     //
     // This function implements the performance optimized SMP subtraction assignment of a sparse
     // matrix- dense vector multiplication expression to a dense vector. Due to the explicit
     // application of the SFINAE principle, this function can only be selected by the compiler
     // in case the expression specific parallel evaluation strategy is selected.
     */
     template< typename VT2 >  // Type of the target dense vector
     friend inline EnableIf_< UseSMPAssign<VT2> >
        smpSubAssign( DenseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        if( rhs.mat_.columns() == 0UL ) return;
  
        LT A( rhs.mat_ );  // Evaluation of the left-hand side sparse matrix operand
        RT x( rhs.vec_ );  // Evaluation of the right-hand side dense vector operand
  
        BLAZE_INTERNAL_ASSERT( A.rows()    == rhs.mat_.rows()   , "Invalid number of rows"    );
        BLAZE_INTERNAL_ASSERT( A.columns() == rhs.mat_.columns(), "Invalid number of columns" );
        BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
        BLAZE_INTERNAL_ASSERT( A.rows()    == (~lhs).size()     , "Invalid vector size"       );
  
        smpSubAssign( ~lhs, A * x );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**SMP subtraction assignment to sparse vectors************************************************
     // No special implementation for the SMP subtraction assignment to sparse vectors.
     //**********************************************************************************************
  
     //**SMP multiplication assignment to dense vectors**********************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief SMP multiplication assignment of a sparse matrix-dense vector multiplication to a
     //        dense vector (\f$ \vec{y}*=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side dense vector.
     // \param rhs The right-hand side multiplication expression to be multiplied.
     // \return void
     //
     // This function implements the performance optimized SMP multiplication assignment of a
     // sparse matrix-dense vector multiplication expression to a dense vector. Due to the explicit
     // application of the SFINAE principle, this function can only be selected by the compiler
     // in case the expression specific parallel evaluation strategy is selected.
     */
     template< typename VT2 >  // Type of the target dense vector
     friend inline EnableIf_< UseSMPAssign<VT2> >
        smpMultAssign( DenseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_BE_COLUMN_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_NOT_REQUIRE_EVALUATION( ResultType );
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        const ResultType tmp( rhs );
        smpMultAssign( ~lhs, tmp );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**SMP multiplication assignment to sparse vectors*********************************************
     // No special implementation for the SMP multiplication assignment to sparse vectors.
     //**********************************************************************************************
  
     //**SMP division assignment to dense vectors****************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief SMP division assignment of a sparse matrix-dense vector multiplication to a dense
     //        vector (\f$ \vec{y}/=A*\vec{x} \f$).
     // \ingroup dense_vector
     //
     // \param lhs The target left-hand side dense vector.
     // \param rhs The right-hand side multiplication expression divisor.
     // \return void
     //
     // This function implements the performance optimized SMP division assignment of a sparse
     // matrix-dense vector multiplication expression to a dense vector. Due to the explicit
     // application of the SFINAE principle, this function can only be selected by the compiler
     // in case the expression specific parallel evaluation strategy is selected.
     */
     template< typename VT2 >  // Type of the target dense vector
     friend inline EnableIf_< UseSMPAssign<VT2> >
        smpDivAssign( DenseVector<VT2,false>& lhs, const TSMatDVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_BE_COLUMN_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_NOT_REQUIRE_EVALUATION( ResultType );
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        const ResultType tmp( rhs );
        smpDivAssign( ~lhs, tmp );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**SMP division assignment to sparse vectors***************************************************
     // No special implementation for the SMP division assignment to sparse vectors.
     //**********************************************************************************************
  
     //**Compile time checks*************************************************************************
     /*! \cond BLAZE_INTERNAL */
     BLAZE_CONSTRAINT_MUST_BE_SPARSE_MATRIX_TYPE( MT );
     BLAZE_CONSTRAINT_MUST_BE_COLUMN_MAJOR_MATRIX_TYPE( MT );
     BLAZE_CONSTRAINT_MUST_NOT_BE_SYMMETRIC_MATRIX_TYPE( MT );
     BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE ( VT );
     BLAZE_CONSTRAINT_MUST_BE_COLUMN_VECTOR_TYPE( VT );
     BLAZE_CONSTRAINT_MUST_FORM_VALID_MATVECMULTEXPR( MT, VT );
     /*! \endcond */
     //**********************************************************************************************
  };
  //*************************************************************************************************
  
  
  
  
  //=================================================================================================
  //
  //  GLOBAL BINARY ARITHMETIC OPERATORS
  //
  //=================================================================================================
  
  //*************************************************************************************************
  /*! \cond BLAZE_INTERNAL */
  /*!\brief Backend implementation of the multiplication of a column-major sparse matrix and
  //        a dense vector (\f$ \vec{a}=B*\vec{c} \f$).
  // \ingroup sparse_vector
  //
  // \param mat The left-hand side column-major sparse matrix for the multiplication.
  // \param vec The right-hand side dense vector for the multiplication.
  // \return The resulting vector.
  //
  // This function implements the performance optimized treatment of the multiplication
  // of a column-major sparse matrix and a dense vector. It restructures the expression
  // \f$ \vec{y}=A^T*\vec{x} \f$ to the expression \f$ \vec{y}=A*\vec{x} \f$.
  */
  template< typename MT  // Type of the left-hand side sparse matrix
          , typename VT  // Type of the right-hand side dense vector
          , typename = DisableIf_< IsSymmetric<MT> > >
  inline const TSMatDVecMultExpr<MT,VT>
     tsmatdvecmult( const SparseMatrix<MT,true>& mat, const DenseVector<VT,false>& vec )
  {
     BLAZE_FUNCTION_TRACE;
  
     BLAZE_INTERNAL_ASSERT( (~mat).columns() == (~vec).size(), "Invalid matrix and vector sizes" );
  
     return TSMatDVecMultExpr<MT,VT>( ~mat, ~vec );
  }
  /*! \endcond */
  //*************************************************************************************************
  
  
  //*************************************************************************************************
  /*! \cond BLAZE_INTERNAL */
  /*!\brief Backend implementation of the multiplication of a symmetric column-major sparse matrix
  //        and a dense vector (\f$ \vec{a}=B*\vec{c} \f$).
  // \ingroup sparse_vector
  //
  // \param mat The left-hand side column-major sparse matrix for the multiplication.
  // \param vec The right-hand side dense vector for the multiplication.
  // \return The resulting vector.
  //
  // This function implements the performance optimized treatment of the multiplication of a
  // symmetric column-major sparse matrix and a dense vector. It restructures the expression
  // \f$ \vec{y}=A^T*\vec{x} \f$ to the expression \f$ \vec{y}=A*\vec{x} \f$.
  */
  template< typename MT  // Type of the left-hand side sparse matrix
          , typename VT  // Type of the right-hand side dense vector
          , typename = EnableIf_< IsSymmetric<MT> > >
  inline decltype(auto)
     tsmatdvecmult( const SparseMatrix<MT,true>& mat, const DenseVector<VT,false>& vec )
  {
     BLAZE_FUNCTION_TRACE;
  
     BLAZE_INTERNAL_ASSERT( (~mat).columns() == (~vec).size(), "Invalid matrix and vector sizes" );
  
     return trans( ~mat ) * (~vec);
  }
  /*! \endcond */
  //*************************************************************************************************
  
  
  //*************************************************************************************************
  /*!\brief Multiplication operator for the multiplication of a column-major sparse matrix and a
  //        dense vector (\f$ \vec{y}=A*\vec{x} \f$).
  // \ingroup dense_vector
  //
  // \param mat The left-hand side column-major sparse matrix for the multiplication.
  // \param vec The right-hand side dense vector for the multiplication.
  // \return The resulting vector.
  // \exception std::invalid_argument Matrix and vector sizes do not match.
  //
  // This operator represents the multiplication between a column-major sparse matrix and a dense
  // vector:
  
     \code
     using blaze::columnMajor;
     using blaze::columnVector;
  
     blaze::CompressedMatrix<double,columnMajor> A;
     blaze::DynamicVector<double,columnVector> x, y;
     // ... Resizing and initialization
     y = A * x;
     \endcode
  
  // The operator returns an expression representing a dense vector of the higher-order element
  // type of the two involved element types \a MT::ElementType and \a VT::ElementType. Both the
  // sparse matrix type \a MT and the dense vector type \a VT as well as the two element types
  // \a MT::ElementType and \a VT::ElementType have to be supported by the MultTrait class
  // template.\n
  // In case the current size of the vector \a vec doesn't match the current number of columns
  // of the matrix \a mat, a \a std::invalid_argument is thrown.
  */
  template< typename MT    // Type of the left-hand side sparse matrix
          , typename VT >  // Type of the right-hand side dense vector
  inline decltype(auto)
     operator*( const SparseMatrix<MT,true>& mat, const DenseVector<VT,false>& vec )
  {
     BLAZE_FUNCTION_TRACE;
  
     BLAZE_CONSTRAINT_MUST_NOT_BE_MATMATMULTEXPR_TYPE( MT );
  
     if( (~mat).columns() != (~vec).size() ) {
        BLAZE_THROW_INVALID_ARGUMENT( "Matrix and vector sizes do not match" );
     }
  
     return tsmatdvecmult( ~mat, ~vec );
  }
  //*************************************************************************************************
  
  
  
  
  //=================================================================================================
  //
  //  SIZE SPECIALIZATIONS
  //
  //=================================================================================================
  
  //*************************************************************************************************
  /*! \cond BLAZE_INTERNAL */
  template< typename MT, typename VT >
  struct Size< TSMatDVecMultExpr<MT,VT>, 0UL >
     : public Size<MT,0UL>
  {};
  /*! \endcond */
  //*************************************************************************************************
  
  
  
  
  //=================================================================================================
  //
  //  ISALIGNED SPECIALIZATIONS
  //
  //=================================================================================================
  
  //*************************************************************************************************
  /*! \cond BLAZE_INTERNAL */
  template< typename MT, typename VT >
  struct IsAligned< TSMatDVecMultExpr<MT,VT> >
     : public IsAligned<VT>
  {};
  /*! \endcond */
  //*************************************************************************************************
  
  } // namespace blaze
  
  #endif
  