text: //=================================================================================================
  /*!
  //  \file blaze/math/expressions/TSMatSVecMultExpr.h
  //  \brief Header file for the transpose sparse matrix/sparse vector multiplication expression
  //
  //  Copyright (C) 2013 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_TSMATSVECMULTEXPR_H_
  #define _BLAZE_MATH_EXPRESSIONS_TSMATSVECMULTEXPR_H_
  
  
  //*************************************************************************************************
  // Includes
  //*************************************************************************************************
  
  #include <vector>
  #include <blaze/math/constraints/ColumnMajorMatrix.h>
  #include <blaze/math/constraints/ColumnVector.h>
  #include <blaze/math/constraints/MatVecMultExpr.h>
  #include <blaze/math/constraints/SparseMatrix.h>
  #include <blaze/math/constraints/SparseVector.h>
  #include <blaze/math/dense/DynamicVector.h>
  #include <blaze/math/expressions/Computation.h>
  #include <blaze/math/expressions/Forward.h>
  #include <blaze/math/expressions/MatVecMultExpr.h>
  #include <blaze/math/expressions/SparseVector.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/IsComputation.h>
  #include <blaze/math/typetraits/IsExpression.h>
  #include <blaze/math/typetraits/IsMatMatMultExpr.h>
  #include <blaze/math/typetraits/IsResizable.h>
  #include <blaze/math/typetraits/RequiresEvaluation.h>
  #include <blaze/math/typetraits/Rows.h>
  #include <blaze/math/typetraits/Size.h>
  #include <blaze/system/Thresholds.h>
  #include <blaze/util/Assert.h>
  #include <blaze/util/Byte.h>
  #include <blaze/util/constraints/Reference.h>
  #include <blaze/util/DisableIf.h>
  #include <blaze/util/EnableIf.h>
  #include <blaze/util/Exception.h>
  #include <blaze/util/logging/FunctionTrace.h>
  #include <blaze/util/SelectType.h>
  #include <blaze/util/Types.h>
  #include <blaze/util/typetraits/RemoveReference.h>
  
  
  namespace blaze {
  
  //=================================================================================================
  //
  //  CLASS SMATDVECMULTEXPR
  //
  //=================================================================================================
  
  //*************************************************************************************************
  /*!\brief Expression object for sparse matrix-sparse vector multiplications.
  // \ingroup sparse_vector_expression
  //
  // The TSMatSVecMultExpr class represents the compile time expression for multiplications
  // between column-major sparse matrices and sparse vectors.
  */
  template< typename MT    // Type of the left-hand side sparse matrix
          , typename VT >  // Type of the right-hand side sparse vector
  class TSMatSVecMultExpr : public SparseVector< TSMatSVecMultExpr<MT,VT>, false >
                          , private MatVecMultExpr
                          , private Computation
  {
   private:
     //**Type definitions****************************************************************************
     typedef typename MT::ResultType     MRT;  //!< Result type of the left-hand side sparse matrix expression.
     typedef typename VT::ResultType     VRT;  //!< Result type of the right-hand side sparse vector expression.
     typedef typename MT::CompositeType  MCT;  //!< Composite type of the left-hand side sparse matrix expression.
     typedef typename VT::CompositeType  VCT;  //!< Composite type of the right-hand side sparse vector expression.
     //**********************************************************************************************
  
     //**********************************************************************************************
     //! Compilation switch for the composite type of the left-hand side sparse matrix expression.
     enum { evaluateMatrix = RequiresEvaluation<MT>::value };
     //**********************************************************************************************
  
     //**********************************************************************************************
     //! Compilation switch for the composite type of the right-hand side sparse vector expression.
     enum { evaluateVector = RequiresEvaluation<VT>::value || IsComputation<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 { value = ( evaluateMatrix || evaluateVector ) };
     };
     /*! \endcond */
     //**********************************************************************************************
  
   public:
     //**Type definitions****************************************************************************
     typedef TSMatSVecMultExpr<MT,VT>            This;           //!< Type of this TSMatSVecMultExpr instance.
     typedef typename MultTrait<MRT,VRT>::Type   ResultType;     //!< Result type for expression template evaluations.
     typedef typename ResultType::TransposeType  TransposeType;  //!< Transpose type for expression template evaluations.
     typedef typename ResultType::ElementType    ElementType;    //!< Resulting element type.
     typedef const ElementType                   ReturnType;     //!< Return type for expression template evaluations.
  
     //! Data type for composite expression templates.
     typedef const ResultType  CompositeType;
  
     //! Composite type of the left-hand side sparse matrix expression.
     typedef typename SelectType< IsExpression<MT>::value, const MT, const MT& >::Type  LeftOperand;
  
     //! Composite type of the right-hand side sparse vector expression.
     typedef typename SelectType< IsExpression<VT>::value, const VT, const VT& >::Type  RightOperand;
  
     //! Type for the assignment of the left-hand side sparse matrix operand.
     typedef typename SelectType< evaluateMatrix, const MRT, MCT >::Type  LT;
  
     //! Type for the assignment of the right-hand side sparse vector operand.
     typedef typename SelectType< evaluateVector, const VRT, VCT >::Type  RT;
     //**********************************************************************************************
  
     //**Compilation flags***************************************************************************
     //! Compilation switch for the expression template assignment strategy.
     enum { smpAssignable = !evaluateMatrix && MT::smpAssignable &&
                            !evaluateVector && VT::smpAssignable };
     //**********************************************************************************************
  
     //**Constructor*********************************************************************************
     /*!\brief Constructor for the TSMatSVecMultExpr class.
     //
     // \param mat The left-hand side sparse matrix operand of the multiplication expression.
     // \param vec The right-hand side sparse vector operand of the multiplication expression.
     */
     explicit inline TSMatSVecMultExpr( const MT& mat, const VT& vec )
        : mat_( mat )  // Left-hand side sparse matrix of the multiplication expression
        , vec_( vec )  // Right-hand side sparse 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" );
  
        typedef typename RemoveReference<MCT>::Type::ConstIterator  MatrixIterator;
        typedef typename RemoveReference<VCT>::Type::ConstIterator  VectorIterator;
  
        MCT A( mat_ );  // Evaluation of the left-hand side sparse matrix operand
        VCT x( vec_ );  // Evaluation of the right-hand side sparse vector operand
  
        BLAZE_INTERNAL_ASSERT( A.rows()    == mat_.rows()   , "Invalid number of rows"    );
        BLAZE_INTERNAL_ASSERT( A.columns() == mat_.columns(), "Invalid number of columns" );
        BLAZE_INTERNAL_ASSERT( x.size()    == vec_.size()   , "Invalid vector size"       );
  
        ElementType res;
  
        const VectorIterator vend( x.end() );
        VectorIterator velem( x.begin() );
  
        if( vec_.size() > 0UL && velem != vend ) {
           res = A( index, velem->index() ) * velem->value();
           ++velem;
           for( ; velem!=vend; ++velem )
              res += A( index, velem->index() ) * velem->value();
        }
        else {
           reset( res );
        }
  
        return res;
     }
     //**********************************************************************************************
  
     //**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 {
        return mat_.rows();
     }
     //**********************************************************************************************
  
     //**NonZeros function***************************************************************************
     /*!\brief Returns an estimation for the number of non-zero elements in the sparse vector.
     //
     // \return The estimate for the number of non-zero elements in the sparse vector.
     */
     inline size_t nonZeros() const {
        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 {
        return mat_;
     }
     //**********************************************************************************************
  
     //**Right operand function**********************************************************************
     /*!\brief Returns the right-hand side sparse vector operand.
     //
     // \return The right-hand side sparse vector operand.
     */
     inline RightOperand rightOperand() const {
        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 {
        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 {
        return ( mat_.isAliased( alias ) || vec_.isAliased( alias ) );
     }
     //**********************************************************************************************
  
     //**********************************************************************************************
     /*!\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 {
        return ( size() > SMP_SMATSVECMULT_THRESHOLD );
     }
     //**********************************************************************************************
  
   private:
     //**Member variables****************************************************************************
     LeftOperand  mat_;  //!< Left-hand side sparse matrix of the multiplication expression.
     RightOperand vec_;  //!< Right-hand side sparse vector of the multiplication expression.
     //**********************************************************************************************
  
     //**Assignment to dense vectors*****************************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Assignment of a transpose sparse matrix-sparse vector multiplication to a dense
     //        vector.
     // \ingroup sparse_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-
     // sparse vector multiplication expression to a dense vector.
     */
     template< typename VT1 >  // Type of the target dense vector
     friend inline void assign( DenseVector<VT1,false>& lhs, const TSMatSVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        // Resetting the left-hand side target dense vector
        reset( ~lhs );
  
        // Evaluation of the right-hand side sparse vector operand
        RT x( serial( rhs.vec_ ) );
        if( x.nonZeros() == 0UL ) return;
  
        // Evaluation of the left-hand side sparse matrix operand
        LT A( serial( rhs.mat_ ) );
  
        // Checking the evaluated operators
        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"       );
  
        // Performing the transpose sparse matrix-sparse vector multiplication
        TSMatSVecMultExpr::selectAssignKernel( ~lhs, A, x );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Default assignment to dense vectors*********************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Default assignment of a transpose sparse matrix-sparse vector multiplication
     //        (\f$ \vec{y}=A*\vec{x} \f$).
     // \ingroup sparse_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 sparse vector operand.
     // \return void
     //
     // This function implements the default assignment kernel for the transpose sparse matrix-
     // sparse 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 )
     {
        typedef typename RemoveReference<MT1>::Type::ConstIterator  MatrixIterator;
        typedef typename RemoveReference<VT2>::Type::ConstIterator  VectorIterator;
  
        const VectorIterator vend ( x.end() );
        VectorIterator       velem( x.begin() );
  
        for( ; velem!=vend; ++velem )
        {
           const MatrixIterator mend ( A.end  ( velem->index() ) );
           MatrixIterator       melem( A.begin( velem->index() ) );
  
           for( ; melem!=mend; ++melem ) {
              if( IsResizable<typename VT1::ElementType>::value &&
                  isDefault( y[melem->index()] ) )
                 y[melem->index()] = melem->value() * velem->value();
              else
                 y[melem->index()] += melem->value() * velem->value();
           }
        }
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Assignment to sparse vectors****************************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Assignment of a transpose sparse matrix-sparse vector multiplication to a sparse vector.
     // \ingroup sparse_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-
     // sparse vector multiplication expression to a sparse vector.
     */
     template< typename VT1 >  // Type of the target sparse vector
     friend inline void assign( SparseVector<VT1,false>& lhs, const TSMatSVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        typedef typename RemoveReference<LT>::Type::ConstIterator  MatrixIterator;
        typedef typename RemoveReference<RT>::Type::ConstIterator  VectorIterator;
  
        RT x( serial( rhs.vec_ ) );  // Evaluation of the right-hand side sparse vector operand
        if( x.nonZeros() == 0UL ) return;
  
        LT A( serial( rhs.mat_ ) );  // Evaluation of the left-hand side sparse matrix 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"       );
  
        DynamicVector<ElementType> tmp( (~lhs).size() );
        std::vector<byte> indices( (~lhs).size(), 0 );
        size_t nonzeros( 0UL );
  
        const VectorIterator vend ( x.end() );
        VectorIterator       velem( x.begin() );
  
        for( ; velem!=vend; ++velem )
        {
           const MatrixIterator mend ( A.end  ( velem->index() ) );
           MatrixIterator       melem( A.begin( velem->index() ) );
  
           for( ; melem!=mend; ++melem ) {
              if( !indices[melem->index()] ) {
                 indices[melem->index()] = 1;
                 ++nonzeros;
                 tmp[melem->index()] = melem->value() * velem->value();
              }
              else {
                 tmp[melem->index()] += melem->value() * velem->value();
              }
           }
        }
  
        (~lhs).reserve( nonzeros );
  
        for( size_t i=0UL; i<(~lhs).size(); ++i ) {
           if( indices[i] ) {
              (~lhs).append( i, tmp[i] );
           }
        }
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Addition assignment to dense vectors********************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Addition assignment of a transpose sparse matrix-sparse vector multiplication to a
     //        dense vector.
     // \ingroup sparse_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 transpose
     // sparse matrix-sparse vector multiplication expression to a dense vector.
     */
     template< typename VT1 >  // Type of the target dense vector
     friend inline void addAssign( DenseVector<VT1,false>& lhs, const TSMatSVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        // Evaluation of the right-hand side sparse vector operand
        RT x( serial( rhs.vec_ ) );
        if( x.nonZeros() == 0UL ) return;
  
        // Evaluation of the left-hand side sparse matrix operand
        LT A( serial( rhs.mat_ ) );
  
        // Checking the evaluated operators
        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"       );
  
        // Performing the transpose sparse matrix-sparse vector multiplication
        TSMatSVecMultExpr::selectAddAssignKernel( ~lhs, A, x );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Default addition assignment to dense vectors************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Default addition assignment of a transpose sparse matrix-sparse vector multiplication
     //        (\f$ \vec{y}+=A*\vec{x} \f$).
     // \ingroup sparse_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 sparse vector operand.
     // \return void
     //
     // This function implements the default addition assignment kernel for the sparse matrix-sparse
     // 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 )
     {
        typedef typename RemoveReference<MT1>::Type::ConstIterator  MatrixIterator;
        typedef typename RemoveReference<VT2>::Type::ConstIterator  VectorIterator;
  
        const VectorIterator vend ( x.end() );
        VectorIterator       velem( x.begin() );
  
        for( ; velem!=vend; ++velem )
        {
           const MatrixIterator mend ( A.end  ( velem->index() ) );
           MatrixIterator       melem( A.begin( velem->index() ) );
  
           for( ; melem!=mend; ++melem ) {
              y[melem->index()] += melem->value() * velem->value();
           }
        }
     }
     /*! \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 transpose sparse matrix-sparse vector multiplication to
     //        a dense vector.
     // \ingroup sparse_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 transpose
     // sparse matrix-sparse vector multiplication expression to a dense vector.
     */
     template< typename VT1 >  // Type of the target dense vector
     friend inline void subAssign( DenseVector<VT1,false>& lhs, const TSMatSVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        // Evaluation of the right-hand side sparse vector operand
        RT x( serial( rhs.vec_ ) );
        if( x.nonZeros() == 0UL ) return;
  
        // Evaluation of the left-hand side sparse matrix operand
        LT A( serial( rhs.mat_ ) );
  
        // Checking the evaluated operators
        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"       );
  
        // Performing the transpose sparse matrix-sparse vector multiplication
        TSMatSVecMultExpr::selectSubAssignKernel( ~lhs, A, x );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**Default subtraction assignment to dense vectors*********************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief Default subtraction assignment of a transpose sparse matrix-sparse vector
     //        multiplication (\f$ \vec{y}-=A*\vec{x} \f$).
     // \ingroup sparse_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 sparse vector operand.
     // \return void
     //
     // This function implements the default subtraction assignment kernel for the sparse matrix-
     // sparse 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 )
     {
        typedef typename RemoveReference<MT1>::Type::ConstIterator  MatrixIterator;
        typedef typename RemoveReference<VT2>::Type::ConstIterator  VectorIterator;
  
        const VectorIterator vend ( x.end() );
        VectorIterator       velem( x.begin() );
  
        for( ; velem!=vend; ++velem )
        {
           const MatrixIterator mend ( A.end  ( velem->index() ) );
           MatrixIterator       melem( A.begin( velem->index() ) );
  
           for( ; melem!=mend; ++melem ) {
              y[melem->index()] -= melem->value() * velem->value();
           }
        }
     }
     /*! \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 transpose sparse matrix-sparse vector multiplication
     //        to a dense vector.
     // \ingroup sparse_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 transpose
     // sparse matrix-sparse vector multiplication expression to a dense vector.
     */
     template< typename VT1 >  // Type of the target dense vector
     friend inline void multAssign( DenseVector<VT1,false>& lhs, const TSMatSVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_CONSTRAINT_MUST_BE_SPARSE_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_BE_COLUMN_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_BE_REFERENCE_TYPE( typename ResultType::CompositeType );
  
        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.
     //**********************************************************************************************
  
     //**SMP assignment to dense vectors*************************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief SMP assignment of a transpose sparse matrix-sparse vector multiplication to a dense
     //        vector.
     // \ingroup sparse_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-sparse 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 VT1 >  // Type of the target dense vector
     friend inline typename EnableIf< UseSMPAssign<VT1> >::Type
        smpAssign( DenseVector<VT1,false>& lhs, const TSMatSVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        // Resetting the left-hand side target dense vector
        reset( ~lhs );
  
        // Evaluation of the right-hand side sparse vector operand
        RT x( rhs.vec_ );
        if( x.nonZeros() == 0UL ) return;
  
        // Evaluation of the left-hand side sparse matrix operand
        LT A( rhs.mat_ );
  
        // Checking the evaluated operators
        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"       );
  
        // Performing the transpose sparse matrix-sparse vector multiplication
        smpAssign( ~lhs, A * x );
     }
     /*! \endcond */
     //**********************************************************************************************
  
     //**SMP assignment to sparse vectors************************************************************
     // No special implementation for the SMP assignment to sparse vectors.
     //**********************************************************************************************
  
     //**SMP addition assignment to dense vectors****************************************************
     /*! \cond BLAZE_INTERNAL */
     /*!\brief SMP addition assignment of a transpose sparse matrix-sparse vector multiplication
     //        to a dense vector.
     // \ingroup sparse_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 transpose
     // sparse matrix-sparse 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 VT1 >  // Type of the target dense vector
     friend inline typename EnableIf< UseSMPAssign<VT1> >::Type
        smpAddAssign( DenseVector<VT1,false>& lhs, const TSMatSVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        // Evaluation of the right-hand side sparse vector operand
        RT x( rhs.vec_ );
        if( x.nonZeros() == 0UL ) return;
  
        // Evaluation of the left-hand side sparse matrix operand
        LT A( rhs.mat_ );
  
        // Checking the evaluated operators
        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"       );
  
        // Performing the transpose sparse matrix-sparse vector multiplication
        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 transpose sparse matrix-sparse vector multiplication
     //        to a dense vector.
     // \ingroup sparse_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 transpose
     // sparse matrix-sparse 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 VT1 >  // Type of the target dense vector
     friend inline typename EnableIf< UseSMPAssign<VT1> >::Type
        smpSubAssign( DenseVector<VT1,false>& lhs, const TSMatSVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_INTERNAL_ASSERT( (~lhs).size() == rhs.size(), "Invalid vector sizes" );
  
        // Evaluation of the right-hand side sparse vector operand
        RT x( rhs.vec_ );
        if( x.nonZeros() == 0UL ) return;
  
        // Evaluation of the left-hand side sparse matrix operand
        LT A( rhs.mat_ );
  
        // Checking the evaluated operators
        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"       );
  
        // Performing the transpose sparse matrix-sparse vector multiplication
        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 transpose sparse matrix-sparse vector multiplication
     //        to a dense vector.
     // \ingroup sparse_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
     // transpose sparse matrix-sparse 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 VT1 >  // Type of the target dense vector
     friend inline typename EnableIf< UseSMPAssign<VT1> >::Type
        smpMultAssign( DenseVector<VT1,false>& lhs, const TSMatSVecMultExpr& rhs )
     {
        BLAZE_FUNCTION_TRACE;
  
        BLAZE_CONSTRAINT_MUST_BE_SPARSE_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_BE_COLUMN_VECTOR_TYPE( ResultType );
        BLAZE_CONSTRAINT_MUST_BE_REFERENCE_TYPE( typename ResultType::CompositeType );
  
        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.
     //**********************************************************************************************
  
     //**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_BE_SPARSE_VECTOR_TYPE( VT );
     BLAZE_CONSTRAINT_MUST_BE_COLUMN_VECTOR_TYPE( VT );
     BLAZE_CONSTRAINT_MUST_FORM_VALID_MATVECMULTEXPR( MT, VT );
     /*! \endcond */
     //**********************************************************************************************
  };
  //*************************************************************************************************
  
  
  
  
  //=================================================================================================
  //
  //  GLOBAL BINARY ARITHMETIC OPERATORS
  //
  //=================================================================================================
  
  //*************************************************************************************************
  /*!\brief Multiplication operator for the multiplication of a transpose sparse matrix and a
  //        sparse vector (\f$ \vec{y}=A*\vec{x} \f$).
  // \ingroup sparse_vector
  //
  // \param mat The left-hand side sparse matrix for the multiplication.
  // \param vec The right-hand side sparse 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 transpose sparse matrix and a sparse
  // vector:
  
     \code
     using blaze::columnMajor;
     using blaze::columnVector;
  
     blaze::CompressedMatrix<double,columnMajor> A;
     blaze::CompressedVector<double,columnVector> x, y;
     // ... Resizing and initialization
     y = A * x;
     \endcode
  
  // The operator returns an expression representing a sparse vector of the higher-order element
  // type of the two involved element types \a T1::ElementType and \a T2::ElementType. Both the
  // sparse matrix type \a T1 and the sparse vector type \a T2 as well as the two element types
  // \a T1::ElementType and \a T2::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 T1    // Type of the left-hand side sparse matrix
          , typename T2 >  // Type of the right-hand side sparse vector
  inline const typename DisableIf< IsMatMatMultExpr<T1>, TSMatSVecMultExpr<T1,T2> >::Type
     operator*( const SparseMatrix<T1,true>& mat, const SparseVector<T2,false>& vec )
  {
     BLAZE_FUNCTION_TRACE;
  
     if( (~mat).columns() != (~vec).size() ) {
        BLAZE_THROW_INVALID_ARGUMENT( "Matrix and vector sizes do not match" );
     }
  
     return TSMatSVecMultExpr<T1,T2>( ~mat, ~vec );
  }
  //*************************************************************************************************
  
  
  
  
  //=================================================================================================
  //
  //  SIZE SPECIALIZATIONS
  //
  //=================================================================================================
  
  //*************************************************************************************************
  /*! \cond BLAZE_INTERNAL */
  template< typename MT, typename VT >
  struct Size< TSMatSVecMultExpr<MT,VT> > : public Rows<MT>
  {};
  /*! \endcond */
  //*************************************************************************************************
  
  
  
  
  //=================================================================================================
  //
  //  EXPRESSION TRAIT SPECIALIZATIONS
  //
  //=================================================================================================
  
  //*************************************************************************************************
  /*! \cond BLAZE_INTERNAL */
  template< typename MT, typename VT, bool AF >
  struct SubvectorExprTrait< TSMatSVecMultExpr<MT,VT>, AF >
  {
   public:
     //**********************************************************************************************
     typedef typename MultExprTrait< typename SubmatrixExprTrait<const MT,AF>::Type
                                   , typename SubvectorExprTrait<const VT,AF>::Type >::Type  Type;
     //**********************************************************************************************
  };
  /*! \endcond */
  //*************************************************************************************************
  
  } // namespace blaze
  
  #endif
  