xref: /dports/math/blaze/blaze-3.8/blaze/math/expressions/TSVecDMatMultExpr.h

//=================================================================================================
/*!
//  \file blaze/math/expressions/TSVecDMatMultExpr.h
//  \brief Header file for the transpose sparse vector/dense matrix multiplication expression
//
//  Copyright (C) 2012-2020 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_TSVECDMATMULTEXPR_H_
#define _BLAZE_MATH_EXPRESSIONS_TSVECDMATMULTEXPR_H_


//*************************************************************************************************
// Includes
//*************************************************************************************************

#include <blaze/math/Aliases.h>
#include <blaze/math/constraints/DenseMatrix.h>
#include <blaze/math/constraints/DenseVector.h>
#include <blaze/math/constraints/MatMatMultExpr.h>
#include <blaze/math/constraints/RequiresEvaluation.h>
#include <blaze/math/constraints/RowMajorMatrix.h>
#include <blaze/math/constraints/RowVector.h>
#include <blaze/math/constraints/SparseVector.h>
#include <blaze/math/constraints/TVecMatMultExpr.h>
#include <blaze/math/constraints/Zero.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/MatMatMultExpr.h>
#include <blaze/math/expressions/TVecMatMultExpr.h>
#include <blaze/math/shims/PrevMultiple.h>
#include <blaze/math/shims/Reset.h>
#include <blaze/math/shims/Serial.h>
#include <blaze/math/SIMD.h>
#include <blaze/math/traits/MultTrait.h>
#include <blaze/math/typetraits/HasSIMDAdd.h>
#include <blaze/math/typetraits/HasSIMDMult.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/IsPadded.h>
#include <blaze/math/typetraits/IsResizable.h>
#include <blaze/math/typetraits/IsSIMDCombinable.h>
#include <blaze/math/typetraits/IsStrictlyLower.h>
#include <blaze/math/typetraits/IsStrictlyUpper.h>
#include <blaze/math/typetraits/IsUpper.h>
#include <blaze/math/typetraits/IsZero.h>
#include <blaze/math/typetraits/RequiresEvaluation.h>
#include <blaze/math/views/Check.h>
#include <blaze/system/MacroDisable.h>
#include <blaze/system/Optimizations.h>
#include <blaze/system/Thresholds.h>
#include <blaze/util/Assert.h>
#include <blaze/util/EnableIf.h>
#include <blaze/util/FunctionTrace.h>
#include <blaze/util/MaybeUnused.h>
#include <blaze/util/mpl/If.h>
#include <blaze/util/Types.h>


namespace blaze {

//=================================================================================================
//
//  CLASS TSVECDMATMULTEXPR
//
//=================================================================================================

//*************************************************************************************************
/*!\brief Expression object for transpose sparse vector-dense matrix multiplications.
// \ingroup dense_vector_expression
//
// The TSVecDMatMultExpr class represents the compile time expression for multiplications
// between transpose sparse vectors and row-major dense matrices.
*/
template< typename VT    // Type of the left-hand side sparse vector
        , typename MT >  // Type of the right-hand side dense matrix
class TSVecDMatMultExpr
   : public TVecMatMultExpr< DenseVector< TSVecDMatMultExpr<VT,MT>, true > >
   , private Computation
{
 private:
   //**Type definitions****************************************************************************
   using VRT = ResultType_t<VT>;     //!< Result type of the left-hand side sparse vector expression.
   using MRT = ResultType_t<MT>;     //!< Result type of the right-hand side dense matrix expression.
   using VET = ElementType_t<VRT>;   //!< Element type of the left-hand side sparse vector expression.
   using MET = ElementType_t<MRT>;   //!< Element type of the right-hand side dense matrix expression.
   using VCT = CompositeType_t<VT>;  //!< Composite type of the left-hand side sparse vector expression.
   using MCT = CompositeType_t<MT>;  //!< Composite type of the right-hand side dense matrix expression.
   //**********************************************************************************************

   //**********************************************************************************************
   //! Compilation switch for the composite type of the left-hand side sparse vector expression.
   static constexpr bool evaluateVector = ( IsComputation_v<VT> || RequiresEvaluation_v<VT> );
   //**********************************************************************************************

   //**********************************************************************************************
   //! Compilation switch for the composite type of the right-hand side dense matrix expression.
   static constexpr bool evaluateMatrix = RequiresEvaluation_v<MT>;
   //**********************************************************************************************

   //**********************************************************************************************
   /*! \cond BLAZE_INTERNAL */
   //! Helper variable template for the explicit application of the SFINAE principle.
   /*! This variable template is a helper for the selection of the parallel evaluation strategy.
       In case either the vector or the matrix operand requires an intermediate evaluation, the
       variable will be set to 1, otherwise it will be 0. */
   template< typename T1 >
   static constexpr bool UseSMPAssign_v = ( evaluateVector || evaluateMatrix );
   /*! \endcond */
   //**********************************************************************************************

   //**********************************************************************************************
   /*! \cond BLAZE_INTERNAL */
   //! Helper variable template for the explicit application of the SFINAE principle.
   /*! In case the matrix type and the two involved vector types are suited for a vectorized
       computation of the vector/matrix multiplication, the variable will be set to 1, otherwise
       it will be 0. */
   template< typename T1, typename T2, typename T3 >
   static constexpr bool UseVectorizedKernel_v =
      ( useOptimizedKernels &&
        !IsDiagonal_v<T3> &&
        T1::simdEnabled && T3::simdEnabled &&
        IsSIMDCombinable_v< ElementType_t<T1>
                          , ElementType_t<T2>
                          , ElementType_t<T3> > &&
        HasSIMDAdd_v< ElementType_t<T2>, ElementType_t<T3> > &&
        HasSIMDMult_v< ElementType_t<T2>, ElementType_t<T3> > );
   /*! \endcond */
   //**********************************************************************************************

   //**********************************************************************************************
   /*! \cond BLAZE_INTERNAL */
   //! Helper variable template for the explicit application of the SFINAE principle.
   /*! In case a vectorized computation of the vector/matrix multiplication is not possible, but
       a loop-unrolled computation is feasible, the variable will be set to 1, otherwise it will
       be 0. */
   template< typename T1, typename T2, typename T3 >
   static constexpr bool UseOptimizedKernel_v =
      ( useOptimizedKernels &&
        !UseVectorizedKernel_v<T1,T2,T3> &&
        !IsDiagonal_v<T3> &&
        !IsResizable_v< ElementType_t<T1> > &&
        !IsResizable_v<VET> );
   /*! \endcond */
   //**********************************************************************************************

   //**********************************************************************************************
   /*! \cond BLAZE_INTERNAL */
   //! Helper variable template for the explicit application of the SFINAE principle.
   /*! In case neither a vectorized nor optimized computation is possible, the variable will be
       set to 1, otherwise it will be 0. */
   template< typename T1, typename T2, typename T3 >
   static constexpr bool UseDefaultKernel_v =
      ( !UseVectorizedKernel_v<T1,T2,T3> && !UseOptimizedKernel_v<T1,T2,T3> );
   /*! \endcond */
   //**********************************************************************************************

 public:
   //**Type definitions****************************************************************************
   //! Type of this TSVecDMatMultExpr instance.
   using This = TSVecDMatMultExpr<VT,MT>;

   //! Base type of this TSVecDMatMultExpr instance.
   using BaseType = TVecMatMultExpr< DenseVector<This,true> >;

   using ResultType    = MultTrait_t<VRT,MRT>;         //!< Result type for expression template evaluations.
   using TransposeType = TransposeType_t<ResultType>;  //!< Transpose type for expression template evaluations.
   using ElementType   = ElementType_t<ResultType>;    //!< Resulting element type.
   using SIMDType      = SIMDTrait_t<ElementType>;     //!< Resulting SIMD 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 vector expression.
   using LeftOperand = If_t< IsExpression_v<VT>, const VT, const VT& >;

   //! Composite type of the right-hand side sparse matrix expression.
   using RightOperand = If_t< IsExpression_v<MT>, const MT, const MT& >;

   //! Type for the assignment of the left-hand side sparse vector operand.
   using LT = If_t< evaluateVector, const VRT, VCT >;

   //! Type for the assignment of the right-hand side dense matrix operand.
   using RT = If_t< evaluateMatrix, const MRT, MCT >;
   //**********************************************************************************************

   //**Compilation flags***************************************************************************
   //! Compilation switch for the expression template evaluation strategy.
   static constexpr bool simdEnabled =
      ( !IsDiagonal_v<MT> &&
        MT::simdEnabled &&
        HasSIMDAdd_v<VET,MET> &&
        HasSIMDMult_v<VET,MET> );

   //! Compilation switch for the expression template assignment strategy.
   static constexpr bool smpAssignable =
      ( !evaluateVector && VT::smpAssignable && !evaluateMatrix && MT::smpAssignable );
   //**********************************************************************************************

   //**SIMD properties*****************************************************************************
   //! The number of elements packed within a single SIMD element.
   static constexpr size_t SIMDSIZE = SIMDTrait<ElementType>::size;
   //**********************************************************************************************

   //**Constructor*********************************************************************************
   /*!\brief Constructor for the TSVecDMatMultExpr class.
   //
   // \param vec The left-hand side sparse vector operand of the multiplication expression.
   // \param mat The right-hand side dense matrix operand of the multiplication expression.
   */
   inline TSVecDMatMultExpr( const VT& vec, const MT& mat ) noexcept
      : vec_( vec )  // Left-hand side sparse vector of the multiplication expression
      , mat_( mat )  // Right-hand side dense matrix of the multiplication expression
   {
      BLAZE_INTERNAL_ASSERT( vec_.size() == mat_.rows(), "Invalid vector and matrix 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_.columns(), "Invalid vector access index" );

      if( IsDiagonal_v<MT> )
      {
         return vec_[index] * mat_(index,index);
      }
      else if( IsLower_v<MT> )
      {
         const size_t begin( IsStrictlyLower_v<MT> ? index+1UL : index );
         const size_t n    ( mat_.rows() - begin );
         return subvector( vec_, begin, n, unchecked ) *
                subvector( column( mat_, index, unchecked ), begin, n, unchecked );
      }
      else if( IsUpper_v<MT> )
      {
         const size_t n( IsStrictlyUpper_v<MT> ? index : index+1UL );
         return subvector( vec_, 0UL, n, unchecked ) *
                subvector( column( mat_, index, unchecked ), 0UL, n, unchecked );
      }
      else
      {
         return vec_ * column( mat_, index, unchecked );
      }
   }
   //**********************************************************************************************

   //**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_.columns() ) {
         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_.columns();
   }
   //**********************************************************************************************

   //**Left operand access*************************************************************************
   /*!\brief Returns the left-hand side sparse vector operand.
   //
   // \return The left-hand side sparse vector operand.
   */
   inline LeftOperand leftOperand() const noexcept {
      return vec_;
   }
   //**********************************************************************************************

   //**Right operand access************************************************************************
   /*!\brief Returns the right-hand side dense matrix operand.
   //
   // \return The right-hand side dense matrix operand.
   */
   inline RightOperand rightOperand() const noexcept {
      return mat_;
   }
   //**********************************************************************************************

   //**********************************************************************************************
   /*!\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 vec_.isAliased( alias ) || mat_.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 vec_.isAliased( alias ) || mat_.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 mat_.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_TSVECDMATMULT_THRESHOLD );
   }
   //**********************************************************************************************

 private:
   //**Member variables****************************************************************************
   LeftOperand  vec_;  //!< Left-hand side sparse vector of the multiplication expression.
   RightOperand mat_;  //!< Right-hand side dense matrix of the multiplication expression.
   //**********************************************************************************************

   //**Assignment to dense vectors*****************************************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief Assignment of a transpose sparse vector-dense matrix multiplication to a dense vector
   //        (\f$ \vec{y}^T=\vec{x}^T*A \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 vector-
   // dense matrix multiplication expression to a dense vector.
   */
   template< typename VT2 >  // Type of the target dense vector
   friend inline void assign( DenseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_INTERNAL_ASSERT( (*lhs).size() == rhs.size(), "Invalid vector sizes" );

      // Evaluation of the left-hand side sparse vector operand
      LT x( serial( rhs.vec_ ) );
      if( x.nonZeros() == 0UL ) {
         reset( *lhs );
         return;
      }

      // Evaluation of the right-hand side dense matrix operand
      RT A( serial( rhs.mat_ ) );

      // Checking the evaluated operands
      BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
      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( A.columns() == (*lhs).size()     , "Invalid vector size"       );

      // Performing the sparse vector-dense matrix multiplication
      TSVecDMatMultExpr::selectAssignKernel( *lhs, x, A );
   }
   /*! \endcond */
   //**********************************************************************************************

   //**Default assignment to dense vectors*********************************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief Default assignment of a transpose sparse vector-dense matrix multiplication
   //        (\f$ \vec{y}^T=\vec{x}^T*A \f$).
   // \ingroup dense_vector
   //
   // \param y The target left-hand side dense vector.
   // \param x The left-hand side sparse vector operand.
   // \param A The right-hand side dense matrix operand.
   // \return void
   //
   // This function implements the default assignment kernel for the transpose sparse vector-
   // dense matrix multiplication.
   */
   template< typename VT1    // Type of the left-hand side target vector
           , typename VT2    // Type of the left-hand side vector operand
           , typename MT1 >  // Type of the right-hand side matrix operand
   static inline auto selectAssignKernel( VT1& y, const VT2& x, const MT1& A )
      -> EnableIf_t< UseDefaultKernel_v<VT1,VT2,MT1> >
   {
      BLAZE_INTERNAL_ASSERT( x.nonZeros() != 0UL, "Invalid number of non-zero elements" );

      const size_t N( A.columns() );

      auto element( x.begin() );
      const auto end( x.end() );

      size_t last( 0UL );

      if( IsUpper_v<MT1> ) {
         const size_t jend( IsStrictlyUpper_v<MT1> ? element->index()+1UL : element->index() );
         for( size_t j=0UL; j<jend; ++j )
            reset( y[j] );
      }

      for( ; element!=end; ++element )
      {
         const size_t index( element->index() );

         if( IsDiagonal_v<MT1> )
         {
            for( size_t j=last; j<index; ++j )
               reset( y[j] );

            y[index] = element->value() * A(index,index);
            last = index + 1UL;
         }
         else
         {
            const size_t jbegin( ( IsUpper_v<MT1> )
                                 ?( IsStrictlyUpper_v<MT1> ? index+1UL : index )
                                 :( 0UL ) );
            const size_t jend( ( IsLower_v<MT1> )
                               ?( IsStrictlyLower_v<MT1> ? index : index+1UL )
                               :( N ) );
            BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

            for( size_t j=jbegin; j<last; ++j ) {
               y[j] += element->value() * A(index,j);
            }
            for( size_t j=last; j<jend; ++j ) {
               y[j] = element->value() * A(index,j);
            }

            last = jend;
         }
      }

      if( IsLower_v<MT1> ) {
         for( size_t j=last; j<N; ++j )
            reset( y[j] );
      }
   }
   /*! \endcond */
   //**********************************************************************************************

   //**Optimized assignment to dense vectors*******************************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief Optimized assignment of a transpose sparse vector-dense matrix multiplication
   //        (\f$ \vec{y}^T=\vec{x}^T*A \f$).
   // \ingroup dense_vector
   //
   // \param y The target left-hand side dense vector.
   // \param x The left-hand side sparse vector operand.
   // \param A The right-hand side dense matrix operand.
   // \return void
   //
   // This function implements the optimized assignment kernel for the transpose sparse vector-
   // dense matrix multiplication.
   */
   template< typename VT1    // Type of the left-hand side target vector
           , typename VT2    // Type of the left-hand side vector operand
           , typename MT1 >  // Type of the right-hand side matrix operand
   static inline auto selectAssignKernel( VT1& y, const VT2& x, const MT1& A )
      -> EnableIf_t< UseOptimizedKernel_v<VT1,VT2,MT1> >
   {
      BLAZE_INTERNAL_ASSERT( x.nonZeros() != 0UL, "Invalid number of non-zero elements" );

      const size_t N( A.columns() );

      auto element( x.begin() );
      const auto end( x.end() );

      const size_t ipos( prevMultiple( x.nonZeros(), 4UL ) );
      BLAZE_INTERNAL_ASSERT( ipos <= x.nonZeros(), "Invalid end calculation" );

      if( ipos > 3UL )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );
         ++element;
         const size_t i2( element->index() );
         const VET    v2( element->value() );
         ++element;
         const size_t i3( element->index() );
         const VET    v3( element->value() );
         ++element;
         const size_t i4( element->index() );
         const VET    v4( element->value() );
         ++element;

         BLAZE_INTERNAL_ASSERT( i1 < i2 && i2 < i3 && i3 < i4, "Invalid sparse vector index detected" );

         for( size_t j=0UL; j<N; ++j ) {
            y[j] = v1 * A(i1,j) + v2 * A(i2,j) + v3 * A(i3,j) + v4 * A(i4,j);
         }
      }
      else
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );
         ++element;

         for( size_t j=0UL; j<N; ++j ) {
            y[j] = v1 * A(i1,j);
         }
      }

      for( size_t i=(ipos>3UL)?(4UL):(1UL); (i+4UL)<=ipos; i+=4UL )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );
         ++element;
         const size_t i2( element->index() );
         const VET    v2( element->value() );
         ++element;
         const size_t i3( element->index() );
         const VET    v3( element->value() );
         ++element;
         const size_t i4( element->index() );
         const VET    v4( element->value() );
         ++element;

         BLAZE_INTERNAL_ASSERT( i1 < i2 && i2 < i3 && i3 < i4, "Invalid sparse vector index detected" );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( IsStrictlyUpper_v<MT1> ? i1+1UL : i1 )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i4 : i4+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         for( size_t j=jbegin; j<jend; ++j ) {
            y[j] += v1 * A(i1,j) + v2 * A(i2,j) + v3 * A(i3,j) + v4 * A(i4,j);
         }
      }
      for( ; element!=end; ++element )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( IsStrictlyUpper_v<MT1> ? i1+1UL : i1 )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i1 : i1+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         for( size_t j=jbegin; j<jend; ++j ) {
            y[j] += v1 * A(i1,j);
         }
      }
   }
   /*! \endcond */
   //**********************************************************************************************

   //**Vectorized assignment to dense vectors******************************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief Vectorized assignment of a transpose sparse vector-dense matrix multiplication
   //        (\f$ \vec{y}^T=\vec{x}^T*A \f$).
   // \ingroup dense_vector
   //
   // \param y The target left-hand side dense vector.
   // \param x The left-hand side sparse vector operand.
   // \param A The right-hand side dense matrix operand.
   // \return void
   //
   // This function implements the vectorized assignment kernel for the transpose sparse vector-
   // dense matrix multiplication.
   */
   template< typename VT1    // Type of the left-hand side target vector
           , typename VT2    // Type of the left-hand side vector operand
           , typename MT1 >  // Type of the right-hand side matrix operand
   static inline auto selectAssignKernel( VT1& y, const VT2& x, const MT1& A )
      -> EnableIf_t< UseVectorizedKernel_v<VT1,VT2,MT1> >
   {
      BLAZE_INTERNAL_ASSERT( x.nonZeros() != 0UL, "Invalid number of non-zero elements" );

      constexpr bool remainder( !IsPadded_v<VT1> || !IsPadded_v<MT1> );

      const size_t N( A.columns() );

      auto element( x.begin() );
      const auto end( x.end() );

      const size_t ipos( prevMultiple( x.nonZeros(), 4UL ) );
      BLAZE_INTERNAL_ASSERT( ipos <= x.nonZeros(), "Invalid end calculation" );

      if( ipos > 3UL )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );
         ++element;
         const size_t i2( element->index() );
         const VET    v2( element->value() );
         ++element;
         const size_t i3( element->index() );
         const VET    v3( element->value() );
         ++element;
         const size_t i4( element->index() );
         const VET    v4( element->value() );
         ++element;

         BLAZE_INTERNAL_ASSERT( i1 < i2 && i2 < i3 && i3 < i4, "Invalid sparse vector index detected" );

         const SIMDType xmm1( set( v1 ) );
         const SIMDType xmm2( set( v2 ) );
         const SIMDType xmm3( set( v3 ) );
         const SIMDType xmm4( set( v4 ) );

         const size_t jpos( remainder ? prevMultiple( N, SIMDSIZE ) : N );
         BLAZE_INTERNAL_ASSERT( jpos <= N, "Invalid end calculation" );

         size_t j( 0UL );

         for( ; j<jpos; j+=SIMDSIZE ) {
            y.store( j, xmm1 * A.load(i1,j) + xmm2 * A.load(i2,j) + xmm3 * A.load(i3,j) + xmm4 * A.load(i4,j) );
         }
         for( ; remainder && j<N; ++j ) {
            y[j] = v1 * A(i1,j) + v2 * A(i2,j) + v3 * A(i3,j) + v4 * A(i4,j);
         }
      }
      else
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );
         ++element;

         const SIMDType xmm1( set( v1 ) );

         const size_t jpos( remainder ? prevMultiple( N, SIMDSIZE ) : N );
         BLAZE_INTERNAL_ASSERT( jpos <= N, "Invalid end calculation" );

         size_t j( 0UL );

         for( ; j<jpos; j+=SIMDSIZE ) {
            y.store( j, xmm1 * A.load(i1,j) );
         }
         for( ; remainder && j<N; ++j ) {
            y[j] = v1 * A(i1,j);
         }
      }

      for( size_t i=(ipos>3UL)?(4UL):(1UL); (i+4UL)<=ipos; i+=4UL )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );
         ++element;
         const size_t i2( element->index() );
         const VET    v2( element->value() );
         ++element;
         const size_t i3( element->index() );
         const VET    v3( element->value() );
         ++element;
         const size_t i4( element->index() );
         const VET    v4( element->value() );
         ++element;

         BLAZE_INTERNAL_ASSERT( i1 < i2 && i2 < i3 && i3 < i4, "Invalid sparse vector index detected" );

         const SIMDType xmm1( set( v1 ) );
         const SIMDType xmm2( set( v2 ) );
         const SIMDType xmm3( set( v3 ) );
         const SIMDType xmm4( set( v4 ) );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( prevMultiple( ( IsStrictlyUpper_v<MT1> ? i1+1UL : i1 ), SIMDSIZE ) )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i4 : i4+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         const size_t jpos( remainder ? prevMultiple( jend, SIMDSIZE ) : jend );
         BLAZE_INTERNAL_ASSERT( jpos <= jend, "Invalid end calculation" );

         size_t j( jbegin );

         for( ; j<jpos; j+=SIMDSIZE ) {
            y.store( j, y.load(j) + xmm1 * A.load(i1,j) + xmm2 * A.load(i2,j) + xmm3 * A.load(i3,j) + xmm4 * A.load(i4,j) );
         }
         for( ; remainder && j<jend; ++j ) {
            y[j] += v1 * A(i1,j) + v2 * A(i2,j) + v3 * A(i3,j) + v4 * A(i4,j);
         }
      }
      for( ; element!=end; ++element )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );

         const SIMDType xmm1( set( v1 ) );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( prevMultiple( ( IsStrictlyUpper_v<MT1> ? i1+1UL : i1 ), SIMDSIZE ) )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i1 : i1+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         const size_t jpos( remainder ? prevMultiple( jend, SIMDSIZE ) : jend );
         BLAZE_INTERNAL_ASSERT( jpos <= jend, "Invalid end calculation" );

         size_t j( jbegin );

         for( ; j<jpos; j+=SIMDSIZE ) {
            y.store( j, y.load(j) + xmm1 * A.load(i1,j) );
         }
         for( ; remainder && j<jend; ++j ) {
            y[j] += v1 * A(i1,j);
         }
      }
   }
   /*! \endcond */
   //**********************************************************************************************

   //**Assignment to sparse vectors****************************************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief Assignment of a transpose sparse vector-dense matrix multiplication to a sparse
   //        vector (\f$ \vec{y}^T=\vec{x}^T*A \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 vector-
   // dense matrix multiplication expression to a sparse vector.
   */
   template< typename VT2 >  // Type of the target sparse vector
   friend inline void assign( SparseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
      BLAZE_CONSTRAINT_MUST_BE_ROW_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********************************************************
   /*!\brief Addition assignment of a transpose sparse vector-dense matrix multiplication to a
   //        dense vector (\f$ \vec{y}^T+=\vec{x}^T*A \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 transpose sparse
   // vector-dense matrix multiplication expression to a dense vector.
   */
   template< typename VT2 >  // Type of the target dense vector
   friend inline void addAssign( DenseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_INTERNAL_ASSERT( (*lhs).size() == rhs.size(), "Invalid vector sizes" );

      // Evaluation of the left-hand side sparse vector operand
      LT x( serial( rhs.vec_ ) );
      if( x.nonZeros() == 0UL ) return;

      // Evaluation of the right-hand side dense matrix operand
      RT A( serial( rhs.mat_ ) );

      // Checking the evaluated operands
      BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
      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( A.columns() == (*lhs).size()     , "Invalid vector size"       );

      // Performing the sparse vector-dense matrix multiplication
      TSVecDMatMultExpr::selectAddAssignKernel( *lhs, x, A );
   }
   //**********************************************************************************************

   //**Default addition assignment to dense vectors************************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief Default addition assignment of a transpose sparse vector-dense matrix multiplication
   //        (\f$ \vec{y}^T+=\vec{x}^T*A \f$).
   // \ingroup dense_vector
   //
   // \param y The target left-hand side dense vector.
   // \param x The left-hand side sparse vector operand.
   // \param A The right-hand side dense matrix operand.
   // \return void
   //
   // This function implements the default addition assignment kernel for the transpose sparse
   // vector-dense matrix multiplication.
   */
   template< typename VT1    // Type of the left-hand side target vector
           , typename VT2    // Type of the left-hand side vector operand
           , typename MT1 >  // Type of the right-hand side matrix operand
   static inline auto selectAddAssignKernel( VT1& y, const VT2& x, const MT1& A )
      -> EnableIf_t< UseDefaultKernel_v<VT1,VT2,MT1> >
   {
      BLAZE_INTERNAL_ASSERT( x.nonZeros() != 0UL, "Invalid number of non-zero elements" );

      const size_t N( A.columns() );

      auto element( x.begin() );
      const auto end( x.end() );

      for( ; element!=end; ++element )
      {
         const size_t index( element->index() );

         if( IsDiagonal_v<MT1> )
         {
            y[index] += A(index,index) * element->value();
         }
         else
         {
            const size_t jbegin( ( IsUpper_v<MT1> )
                                 ?( IsStrictlyUpper_v<MT1> ? index+1UL : index )
                                 :( 0UL ) );
            const size_t jend( ( IsLower_v<MT1> )
                               ?( IsStrictlyLower_v<MT1> ? index : index+1UL )
                               :( N ) );
            BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

            for( size_t j=jbegin; j<jend; ++j ) {
               y[j] += element->value() * A(index,j);
            }
         }
      }
   }
   /*! \endcond */
   //**********************************************************************************************

   //**Optimized addition assignment to dense vectors**********************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief Optimized addition assignment of a transpose sparse vector-dense matrix multiplication
   //        (\f$ \vec{y}^T+=\vec{x}^T*A \f$).
   // \ingroup dense_vector
   //
   // \param y The target left-hand side dense vector.
   // \param x The left-hand side sparse vector operand.
   // \param A The right-hand side dense matrix operand.
   // \return void
   //
   // This function implements the optimized addition assignment kernel for the transpose sparse
   // vector-dense matrix multiplication.
   */
   template< typename VT1    // Type of the left-hand side target vector
           , typename VT2    // Type of the left-hand side vector operand
           , typename MT1 >  // Type of the right-hand side matrix operand
   static inline auto selectAddAssignKernel( VT1& y, const VT2& x, const MT1& A )
      -> EnableIf_t< UseOptimizedKernel_v<VT1,VT2,MT1> >
   {
      BLAZE_INTERNAL_ASSERT( x.nonZeros() != 0UL, "Invalid number of non-zero elements" );

      const size_t N( A.columns() );

      auto element( x.begin() );
      const auto end( x.end() );

      const size_t ipos( prevMultiple( x.nonZeros(), 4UL ) );
      BLAZE_INTERNAL_ASSERT( ipos <= x.nonZeros(), "Invalid end calculation" );

      for( size_t i=0UL; (i+4UL)<=ipos; i+=4UL )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );
         ++element;
         const size_t i2( element->index() );
         const VET    v2( element->value() );
         ++element;
         const size_t i3( element->index() );
         const VET    v3( element->value() );
         ++element;
         const size_t i4( element->index() );
         const VET    v4( element->value() );
         ++element;

         BLAZE_INTERNAL_ASSERT( i1 < i2 && i2 < i3 && i3 < i4, "Invalid sparse vector index detected" );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( IsStrictlyUpper_v<MT1> ? i+1UL : i1 )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i4 : i4+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         for( size_t j=jbegin; j<jend; ++j ) {
            y[j] += v1 * A(i1,j) + v2 * A(i2,j) + v3 * A(i3,j) + v4 * A(i4,j);
         }
      }
      for( ; element!=end; ++element )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( IsStrictlyUpper_v<MT1> ? i1+1UL : i1 )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i1 : i1+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         for( size_t j=jbegin; j<jend; ++j ) {
            y[j] += v1 * A(i1,j);
         }
      }
   }
   /*! \endcond */
   //**********************************************************************************************

   //**Vectorized addition assignment to dense vectors*********************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief Vectorized addition assignment of a transpose sparse vector-dense matrix multiplication
   //        (\f$ \vec{y}^T+=\vec{x}^T*A \f$).
   // \ingroup dense_vector
   //
   // \param y The target left-hand side dense vector.
   // \param x The left-hand side sparse vector operand.
   // \param A The right-hand side dense matrix operand.
   // \return void
   //
   // This function implements the vectorized addition assignment kernel for the transpose sparse
   // vector-dense matrix multiplication.
   */
   template< typename VT1    // Type of the left-hand side target vector
           , typename VT2    // Type of the left-hand side vector operand
           , typename MT1 >  // Type of the right-hand side matrix operand
   static inline auto selectAddAssignKernel( VT1& y, const VT2& x, const MT1& A )
      -> EnableIf_t< UseVectorizedKernel_v<VT1,VT2,MT1> >
   {
      BLAZE_INTERNAL_ASSERT( x.nonZeros() != 0UL, "Invalid number of non-zero elements" );

      constexpr bool remainder( !IsPadded_v<VT1> || !IsPadded_v<MT1> );

      const size_t N( A.columns() );

      auto element( x.begin() );
      const auto end( x.end() );

      const size_t ipos( prevMultiple( x.nonZeros(), 4UL ) );
      BLAZE_INTERNAL_ASSERT( ipos <= x.nonZeros(), "Invalid end calculation" );

      for( size_t i=0UL; (i+4UL)<=ipos; i+=4UL )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );
         ++element;
         const size_t i2( element->index() );
         const VET    v2( element->value() );
         ++element;
         const size_t i3( element->index() );
         const VET    v3( element->value() );
         ++element;
         const size_t i4( element->index() );
         const VET    v4( element->value() );
         ++element;

         BLAZE_INTERNAL_ASSERT( i1 < i2 && i2 < i3 && i3 < i4, "Invalid sparse vector index detected" );

         const SIMDType xmm1( set( v1 ) );
         const SIMDType xmm2( set( v2 ) );
         const SIMDType xmm3( set( v3 ) );
         const SIMDType xmm4( set( v4 ) );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( prevMultiple( ( IsStrictlyUpper_v<MT1> ? i1+1UL : i1 ), SIMDSIZE ) )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i4 : i4+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         const size_t jpos( remainder ? prevMultiple( jend, SIMDSIZE ) : jend );
         BLAZE_INTERNAL_ASSERT( jpos <= jend, "Invalid end calculation" );

         size_t j( jbegin );

         for( ; j<jpos; j+=SIMDSIZE ) {
            y.store( j, y.load(j) + xmm1 * A.load(i1,j) + xmm2 * A.load(i2,j) + xmm3 * A.load(i3,j) + xmm4 * A.load(i4,j) );
         }
         for( ; remainder && j<jend; ++j ) {
            y[j] += v1 * A(i1,j) + v2 * A(i2,j) + v3 * A(i3,j) + v4 * A(i4,j);
         }
      }
      for( ; element!=end; ++element )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );

         const SIMDType xmm1( set( v1 ) );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( prevMultiple( ( IsStrictlyUpper_v<MT1> ? i1+1UL : i1 ), SIMDSIZE ) )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i1 : i1+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         const size_t jpos( remainder ? prevMultiple( jend, SIMDSIZE ) : jend );
         BLAZE_INTERNAL_ASSERT( jpos <= jend, "Invalid end calculation" );

         size_t j( jbegin );

         for( ; j<jpos; j+=SIMDSIZE ) {
            y.store( j, y.load(j) + xmm1 * A.load(i1,j) );
         }
         for( ; remainder && j<jend; ++j ) {
            y[j] += v1 * A(i1,j);
         }
      }
   }
   /*! \endcond */
   //**********************************************************************************************

   //**Addition assignment to sparse vectors*******************************************************
   // No special implementation for the addition assignment to sparse vectors.
   //**********************************************************************************************

   //**Subtraction assignment to dense vectors*****************************************************
   /*!\brief Subtraction assignment of a transpose sparse vector-dense matrix multiplication to a
   //        dense vector (\f$ \vec{y}^T-=\vec{x}^T*A \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 transpose
   // sparse vector-dense matrix multiplication expression to a dense vector.
   */
   template< typename VT2 >  // Type of the target dense vector
   friend inline void subAssign( DenseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_INTERNAL_ASSERT( (*lhs).size() == rhs.size(), "Invalid vector sizes" );

      // Evaluation of the left-hand side sparse vector operand
      LT x( serial( rhs.vec_ ) );
      if( x.nonZeros() == 0UL ) return;

      // Evaluation of the right-hand side dense matrix operand
      RT A( serial( rhs.mat_ ) );

      // Checking the evaluated operands
      BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
      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( A.columns() == (*lhs).size()     , "Invalid vector size"       );

      // Performing the sparse vector-dense matrix multiplication
      TSVecDMatMultExpr::selectSubAssignKernel( *lhs, x, A );
   }
   //**********************************************************************************************

   //**Default subtraction assignment to dense vectors*********************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief Default subtraction assignment of a transpose sparse vector-dense matrix
   //        multiplication (\f$ \vec{y}^T-=\vec{x}^T*A \f$).
   // \ingroup dense_vector
   //
   // \param y The target left-hand side dense vector.
   // \param x The left-hand side sparse vector operand.
   // \param A The right-hand side dense matrix operand.
   // \return void
   //
   // This function implements the default subtraction assignment kernel for the transpose
   // sparse vector-dense matrix multiplication.
   */
   template< typename VT1    // Type of the left-hand side target vector
           , typename VT2    // Type of the left-hand side vector operand
           , typename MT1 >  // Type of the right-hand side matrix operand
   static inline auto selectSubAssignKernel( VT1& y, const VT2& x, const MT1& A )
      -> EnableIf_t< UseDefaultKernel_v<VT1,VT2,MT1> >
   {
      BLAZE_INTERNAL_ASSERT( x.nonZeros() != 0UL, "Invalid number of non-zero elements" );

      const size_t N( A.columns() );

      auto element( x.begin() );
      const auto end( x.end() );

      for( ; element!=end; ++element )
      {
         const size_t index( element->index() );

         if( IsDiagonal_v<MT1> )
         {
            y[index] -= A(index,index) * element->value();
         }
         else
         {
            const size_t jbegin( ( IsUpper_v<MT1> )
                                 ?( IsStrictlyUpper_v<MT1> ? index+1UL : index )
                                 :( 0UL ) );
            const size_t jend( ( IsLower_v<MT1> )
                               ?( IsStrictlyLower_v<MT1> ? index : index+1UL )
                               :( N ) );
            BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

            for( size_t j=jbegin; j<jend; ++j ) {
               y[j] -= element->value() * A(index,j);
            }
         }
      }
   }
   /*! \endcond */
   //**********************************************************************************************

   //**Optimized subtraction assignment to dense vectors*******************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief Optimized subtraction assignment of a transpose sparse vector-dense matrix
   //        multiplication (\f$ \vec{y}^T-=\vec{x}^T*A \f$).
   // \ingroup dense_vector
   //
   // \param y The target left-hand side dense vector.
   // \param x The left-hand side sparse vector operand.
   // \param A The right-hand side dense matrix operand.
   // \return void
   //
   // This function implements the optimized subtraction assignment kernel for the transpose
   // sparse vector-dense matrix multiplication.
   */
   template< typename VT1    // Type of the left-hand side target vector
           , typename VT2    // Type of the left-hand side vector operand
           , typename MT1 >  // Type of the right-hand side matrix operand
   static inline auto selectSubAssignKernel( VT1& y, const VT2& x, const MT1& A )
      -> EnableIf_t< UseOptimizedKernel_v<VT1,VT2,MT1> >
   {
      BLAZE_INTERNAL_ASSERT( x.nonZeros() != 0UL, "Invalid number of non-zero elements" );

      const size_t N( A.columns() );

      auto element( x.begin() );
      const auto end( x.end() );

      const size_t ipos( prevMultiple( x.nonZeros(), 4UL ) );
      BLAZE_INTERNAL_ASSERT( ipos <= x.nonZeros(), "Invalid end calculation" );

      for( size_t i=0UL; (i+4UL)<=ipos; i+=4UL )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );
         ++element;
         const size_t i2( element->index() );
         const VET    v2( element->value() );
         ++element;
         const size_t i3( element->index() );
         const VET    v3( element->value() );
         ++element;
         const size_t i4( element->index() );
         const VET    v4( element->value() );
         ++element;

         BLAZE_INTERNAL_ASSERT( i1 < i2 && i2 < i3 && i3 < i4, "Invalid sparse vector index detected" );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( IsStrictlyUpper_v<MT1> ? i+1UL : i1 )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i4 : i4+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         for( size_t j=jbegin; j<jend; ++j ) {
            y[j] -= v1 * A(i1,j) + v2 * A(i2,j) + v3 * A(i3,j) + v4 * A(i4,j);
         }
      }
      for( ; element!=end; ++element )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( IsStrictlyUpper_v<MT1> ? i1+1UL : i1 )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i1 : i1+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         for( size_t j=jbegin; j<jend; ++j ) {
            y[j] -= v1 * A(i1,j);
         }
      }
   }
   /*! \endcond */
   //**********************************************************************************************

   //**Vectorized subtraction assignment to dense vectors******************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief Vectorized subtraction assignment of a transpose sparse vector-dense matrix
   //        multiplication (\f$ \vec{y}^T-=\vec{x}^T*A \f$).
   // \ingroup dense_vector
   //
   // \param y The target left-hand side dense vector.
   // \param x The left-hand side sparse vector operand.
   // \param A The right-hand side dense matrix operand.
   // \return void
   //
   // This function implements the vectorized subtraction assignment kernel for the transpose
   // sparse vector-dense matrix multiplication.
   */
   template< typename VT1    // Type of the left-hand side target vector
           , typename VT2    // Type of the left-hand side vector operand
           , typename MT1 >  // Type of the right-hand side matrix operand
   static inline auto selectSubAssignKernel( VT1& y, const VT2& x, const MT1& A )
      -> EnableIf_t< UseVectorizedKernel_v<VT1,VT2,MT1> >
   {
      BLAZE_INTERNAL_ASSERT( x.nonZeros() != 0UL, "Invalid number of non-zero elements" );

      constexpr bool remainder( !IsPadded_v<VT1> || !IsPadded_v<MT1> );

      const size_t N( A.columns() );

      auto element( x.begin() );
      const auto end( x.end() );

      const size_t ipos( prevMultiple( x.nonZeros(), 4UL ) );
      BLAZE_INTERNAL_ASSERT( ipos <= x.nonZeros(), "Invalid end calculation" );

      for( size_t i=0UL; (i+4UL)<=ipos; i+=4UL )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );
         ++element;
         const size_t i2( element->index() );
         const VET    v2( element->value() );
         ++element;
         const size_t i3( element->index() );
         const VET    v3( element->value() );
         ++element;
         const size_t i4( element->index() );
         const VET    v4( element->value() );
         ++element;

         BLAZE_INTERNAL_ASSERT( i1 < i2 && i2 < i3 && i3 < i4, "Invalid sparse vector index detected" );

         const SIMDType xmm1( set( v1 ) );
         const SIMDType xmm2( set( v2 ) );
         const SIMDType xmm3( set( v3 ) );
         const SIMDType xmm4( set( v4 ) );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( prevMultiple( ( IsStrictlyUpper_v<MT1> ? i1+1UL : i1 ), SIMDSIZE ) )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i4 : i4+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         const size_t jpos( remainder ? prevMultiple( jend, SIMDSIZE ) : jend );
         BLAZE_INTERNAL_ASSERT( jpos <= jend, "Invalid end calculation" );

         size_t j( jbegin );

         for( ; j<jpos; j+=SIMDSIZE ) {
            y.store( j, y.load(j) - xmm1 * A.load(i1,j) - xmm2 * A.load(i2,j) - xmm3 * A.load(i3,j) - xmm4 * A.load(i4,j) );
         }
         for( ; remainder && j<jend; ++j ) {
            y[j] -= v1 * A(i1,j) + v2 * A(i2,j) + v3 * A(i3,j) + v4 * A(i4,j);
         }
      }
      for( ; element!=end; ++element )
      {
         const size_t i1( element->index() );
         const VET    v1( element->value() );

         const SIMDType xmm1( set( v1 ) );

         const size_t jbegin( ( IsUpper_v<MT1> )
                              ?( prevMultiple( ( IsStrictlyUpper_v<MT1> ? i1+1UL : i1 ), SIMDSIZE ) )
                              :( 0UL ) );
         const size_t jend( ( IsLower_v<MT1> )
                            ?( IsStrictlyLower_v<MT1> ? i1 : i1+1UL )
                            :( N ) );
         BLAZE_INTERNAL_ASSERT( jbegin <= jend, "Invalid loop indices detected" );

         const size_t jpos( remainder ? prevMultiple( jend, SIMDSIZE ) : jend );
         BLAZE_INTERNAL_ASSERT( jpos <= jend, "Invalid end calculation" );

         size_t j( jbegin );

         for( ; j<jpos; j+=SIMDSIZE ) {
            y.store( j, y.load(j) - xmm1 * A.load(i1,j) );
         }
         for( ; remainder && j<jend; ++j ) {
            y[j] -= v1 * A(i1,j);
         }
      }
   }
   /*! \endcond */
   //**********************************************************************************************

   //**Subtraction assignment to sparse vectors****************************************************
   // No special implementation for the subtraction assignment to sparse vectors.
   //**********************************************************************************************

   //**Multiplication assignment to dense vectors**************************************************
   /*!\brief Multiplication assignment of a transpose sparse vector-dense matrix multiplication
   //        to a dense vector (\f$ \vec{y}^T*=\vec{x}^T*A \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 transpose
   // sparse vector-dense matrix multiplication expression to a dense vector.
   */
   template< typename VT2 >  // Type of the target dense vector
   friend inline void multAssign( DenseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
      BLAZE_CONSTRAINT_MUST_BE_ROW_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 );
   }
   //**********************************************************************************************

   //**Multiplication assignment to sparse vectors*************************************************
   // No special implementation for the multiplication assignment to sparse vectors.
   //**********************************************************************************************

   //**Division assignment to dense vectors********************************************************
   /*!\brief Division assignment of a transpose sparse vector-dense matrix multiplication to a
   //        dense vector (\f$ \vec{y}^T/=\vec{x}^T*A \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 transpose sparse
   // vector-dense matrix multiplication expression to a dense vector.
   */
   template< typename VT2 >  // Type of the target dense vector
   friend inline void divAssign( DenseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
      BLAZE_CONSTRAINT_MUST_BE_ROW_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 );
   }
   //**********************************************************************************************

   //**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 vector-dense matrix multiplication to a dense
   //        vector (\f$ \vec{y}^T=\vec{x}^T*A \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
   // vector-dense matrix 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 auto smpAssign( DenseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
      -> EnableIf_t< UseSMPAssign_v<VT2> >
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_INTERNAL_ASSERT( (*lhs).size() == rhs.size(), "Invalid vector sizes" );

      // Evaluation of the left-hand side sparse vector operand
      LT x( rhs.vec_ );
      if( x.nonZeros() == 0UL ) {
         reset( *lhs );
         return;
      }

      // Evaluation of the right-hand side dense matrix operand
      RT A( rhs.mat_ );

      // Checking the evaluated operands
      BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
      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( A.columns() == (*lhs).size()     , "Invalid vector size"       );

      // Performing the sparse vector-dense matrix multiplication
      smpAssign( *lhs, x * A );
   }
   /*! \endcond */
   //**********************************************************************************************

   //**SMP assignment to sparse vectors************************************************************
   /*! \cond BLAZE_INTERNAL */
   /*!\brief SMP assignment of a transpose sparse vector-dense matrix multiplication to a sparse
   //        vector (\f$ \vec{y}^T=\vec{x}^T*A \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
   // vector-dense matrix 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 auto smpAssign( SparseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
      -> EnableIf_t< UseSMPAssign_v<VT2> >
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
      BLAZE_CONSTRAINT_MUST_BE_ROW_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****************************************************
   /*!\brief SMP addition assignment of a transpose sparse vector-dense matrix multiplication to
   //        a dense vector (\f$ \vec{y}^T+=\vec{x}^T*A \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 transpose
   // sparse vector-dense matrix 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 auto smpAddAssign( DenseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
      -> EnableIf_t< UseSMPAssign_v<VT2> >
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_INTERNAL_ASSERT( (*lhs).size() == rhs.size(), "Invalid vector sizes" );

      // Evaluation of the left-hand side sparse vector operand
      LT x( rhs.vec_ );
      if( x.nonZeros() == 0UL ) return;

      // Evaluation of the right-hand side dense matrix operand
      RT A( rhs.mat_ );

      // Checking the evaluated operands
      BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
      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( A.columns() == (*lhs).size()     , "Invalid vector size"       );

      // Performing the sparse vector-dense matrix multiplication
      smpAddAssign( *lhs, x * A );
   }
   //**********************************************************************************************

   //**SMP addition assignment to sparse vectors***************************************************
   // No special implementation for the SMP addition assignment to sparse vectors.
   //**********************************************************************************************

   //**SMP subtraction assignment to dense vectors*************************************************
   /*!\brief SMP subtraction assignment of a transpose sparse vector-dense matrix multiplication
   //        to a dense vector (\f$ \vec{y}^T-=\vec{x}^T*A \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 transpose
   // sparse vector-dense matrix 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 auto smpSubAssign( DenseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
      -> EnableIf_t< UseSMPAssign_v<VT2> >
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_INTERNAL_ASSERT( (*lhs).size() == rhs.size(), "Invalid vector sizes" );

      // Evaluation of the left-hand side sparse vector operand
      LT x( rhs.vec_ );
      if( x.nonZeros() == 0UL ) return;

      // Evaluation of the right-hand side dense matrix operand
      RT A( rhs.mat_ );

      // Checking the evaluated operands
      BLAZE_INTERNAL_ASSERT( x.size()    == rhs.vec_.size()   , "Invalid vector size"       );
      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( A.columns() == (*lhs).size()     , "Invalid vector size"       );

      // Performing the sparse vector-dense matrix multiplication
      smpSubAssign( *lhs, x * A );
   }
   //**********************************************************************************************

   //**SMP subtraction assignment to sparse vectors************************************************
   // No special implementation for the SMP subtraction assignment to sparse vectors.
   //**********************************************************************************************

   //**SMP multiplication assignment to dense vectors**********************************************
   /*!\brief SMP multiplication assignment of a transpose sparse vector-dense matrix multiplication
   //        to a dense vector (\f$ \vec{y}^T*=\vec{x}^T*A \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
   // transpose sparse vector-dense matrix 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 auto smpMultAssign( DenseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
      -> EnableIf_t< UseSMPAssign_v<VT2> >
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
      BLAZE_CONSTRAINT_MUST_BE_ROW_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 );
   }
   //**********************************************************************************************

   //**SMP multiplication assignment to sparse vectors*********************************************
   // No special implementation for the SMP multiplication assignment to sparse vectors.
   //**********************************************************************************************

   //**SMP division assignment to dense vectors****************************************************
   /*!\brief SMP division assignment of a transpose sparse vector-dense matrix multiplication to
   //        a dense vector (\f$ \vec{y}^T/=\vec{x}^T*A \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 transpose
   // sparse vector-dense matrix 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 auto smpDivAssign( DenseVector<VT2,true>& lhs, const TSVecDMatMultExpr& rhs )
      -> EnableIf_t< UseSMPAssign_v<VT2> >
   {
      BLAZE_FUNCTION_TRACE;

      BLAZE_CONSTRAINT_MUST_BE_DENSE_VECTOR_TYPE( ResultType );
      BLAZE_CONSTRAINT_MUST_BE_ROW_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 );
   }
   //**********************************************************************************************

   //**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_VECTOR_TYPE( VT );
   BLAZE_CONSTRAINT_MUST_BE_ROW_VECTOR_TYPE( VT );
   BLAZE_CONSTRAINT_MUST_NOT_BE_ZERO_TYPE( VT );
   BLAZE_CONSTRAINT_MUST_BE_DENSE_MATRIX_TYPE( MT );
   BLAZE_CONSTRAINT_MUST_BE_ROW_MAJOR_MATRIX_TYPE( MT );
   BLAZE_CONSTRAINT_MUST_FORM_VALID_TVECMATMULTEXPR( VT, MT );
   /*! \endcond */
   //**********************************************************************************************
};
//*************************************************************************************************




//=================================================================================================
//
//  GLOBAL BINARY ARITHMETIC OPERATORS
//
//=================================================================================================

//*************************************************************************************************
/*! \cond BLAZE_INTERNAL */
/*!\brief Backend implementation of the multiplication of a transpose sparse vector
//        and a row-major dense matrix (\f$ \vec{a}=B*\vec{c} \f$).
// \ingroup dense_vector
//
// \param vec The left-hand side transpose sparse vector for the multiplication.
// \param mat The right-hand side row-major dense matrix for the multiplication.
// \return The resulting transpose vector.
//
// This function implements the performance optimized treatment of the multiplication of a
// transpose sparse vector and a row-major dense matrix.
*/
template< typename VT  // Type of the left-hand side sparse vector
        , typename MT  // Type of the right-hand side dense matrix
        , DisableIf_t< IsZero_v<VT> >* = nullptr >
inline const TSVecDMatMultExpr<VT,MT>
   tsvecdmatmult( const SparseVector<VT,true>& vec, const DenseMatrix<MT,false>& mat )
{
   BLAZE_FUNCTION_TRACE;

   BLAZE_INTERNAL_ASSERT( (*vec).size() == (*mat).rows(), "Invalid vector and matrix sizes" );

   return TSVecDMatMultExpr<VT,MT>( *vec, *mat );
}
/*! \endcond */
//*************************************************************************************************


//*************************************************************************************************
/*! \cond BLAZE_INTERNAL */
/*!\brief Backend implementation of the multiplication of a transpose zero vector
//        and a row-major dense matrix (\f$ \vec{a}=B*\vec{c} \f$).
// \ingroup dense_vector
//
// \param vec The left-hand side transpose zero vector for the multiplication.
// \param mat The right-hand side row-major dense matrix for the multiplication.
// \return The resulting zero vector.
//
// This function implements the performance optimized treatment of the multiplication of a
// transpose zero vector and a row-major dense matrix. It returns a zero vector.
*/
template< typename VT  // Type of the left-hand side sparse vector
        , typename MT  // Type of the right-hand side dense matrix
        , EnableIf_t< IsZero_v<VT> >* = nullptr >
inline decltype(auto)
   tsvecdmatmult( const SparseVector<VT,true>& vec, const DenseMatrix<MT,false>& mat )
{
   BLAZE_FUNCTION_TRACE;

   MAYBE_UNUSED( vec );

   BLAZE_INTERNAL_ASSERT( (*vec).size() == (*mat).rows(), "Invalid vector and matrix sizes" );

   using ReturnType = const MultTrait_t< ResultType_t<VT>, ResultType_t<MT> >;

   BLAZE_CONSTRAINT_MUST_BE_ROW_VECTOR_TYPE( ReturnType );
   BLAZE_CONSTRAINT_MUST_BE_ZERO_TYPE( ReturnType );

   return ReturnType( (*mat).columns() );
}
/*! \endcond */
//*************************************************************************************************


//*************************************************************************************************
/*!\brief Multiplication operator for the multiplication of a transpose sparse vector and a
//        row-major dense matrix (\f$ \vec{y}^T=\vec{x}^T*A \f$).
// \ingroup dense_matrix
//
// \param vec The left-hand side transpose sparse vector for the multiplication.
// \param mat The right-hand side row-major dense matrix for the multiplication.
// \return The resulting transpose vector.
// \exception std::invalid_argument Vector and matrix sizes do not match.
//
// This operator represents the multiplication between a transpose sparse vector and a row-major
// dense matrix:

   \code
   using blaze::rowVector;
   using blaze::rowMajor;

   blaze::CompressedVector<double,rowVector> x, y;
   blaze::DynamicMatrix<double,rowMajor> A;
   // ... Resizing and initialization
   y = x * A;
   \endcode

// The operator returns an expression representing a transpose sparse vector of the higher-order
// element type of the two involved element types \a VT::ElementType and \a MT::ElementType.
// Both the dense matrix type \a VT and the dense vector type \a MT as well as the two element
// types \a VT::ElementType and \a MT::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 rows of
// the matrix \a mat, a \a std::invalid_argument is thrown.
*/
template< typename VT    // Type of the left-hand side sparse vector
        , typename MT >  // Type of the right-hand side dense matrix
inline decltype(auto)
   operator*( const SparseVector<VT,true>& vec, const DenseMatrix<MT,false>& mat )
{
   BLAZE_FUNCTION_TRACE;

   BLAZE_CONSTRAINT_MUST_NOT_BE_MATMATMULTEXPR_TYPE( MT );

   if( (*vec).size() != (*mat).rows() ) {
      BLAZE_THROW_INVALID_ARGUMENT( "Vector and matrix sizes do not match" );
   }

   return tsvecdmatmult( *vec, *mat );
}
//*************************************************************************************************




//=================================================================================================
//
//  ISALIGNED SPECIALIZATIONS
//
//=================================================================================================

//*************************************************************************************************
/*! \cond BLAZE_INTERNAL */
template< typename VT, typename MT >
struct IsAligned< TSVecDMatMultExpr<VT,MT> >
   : public IsAligned<MT>
{};
/*! \endcond */
//*************************************************************************************************

} // namespace blaze

#endif