text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
  //
  // This Source Code Form is subject to the terms of the Mozilla
  // Public License v. 2.0. If a copy of the MPL was not distributed
  // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
  
  #ifndef EIGEN_GENERAL_MATRIX_VECTOR_H
  #define EIGEN_GENERAL_MATRIX_VECTOR_H
  
  namespace Eigen {
  
  namespace internal {
  
  /* Optimized col-major matrix * vector product:
   * This algorithm processes 4 columns at onces that allows to both reduce
   * the number of load/stores of the result by a factor 4 and to reduce
   * the instruction dependency. Moreover, we know that all bands have the
   * same alignment pattern.
   *
   * Mixing type logic: C += alpha * A * B
   *  |  A  |  B  |alpha| comments
   *  |real |cplx |cplx | no vectorization
   *  |real |cplx |real | alpha is converted to a cplx when calling the run function, no vectorization
   *  |cplx |real |cplx | invalid, the caller has to do tmp: = A * B; C += alpha*tmp
   *  |cplx |real |real | optimal case, vectorization possible via real-cplx mul
   *
   * Accesses to the matrix coefficients follow the following logic:
   *
   * - if all columns have the same alignment then
   *   - if the columns have the same alignment as the result vector, then easy! (-> AllAligned case)
   *   - otherwise perform unaligned loads only (-> NoneAligned case)
   * - otherwise
   *   - if even columns have the same alignment then
   *     // odd columns are guaranteed to have the same alignment too
   *     - if even or odd columns have the same alignment as the result, then
   *       // for a register size of 2 scalars, this is guarantee to be the case (e.g., SSE with double)
   *       - perform half aligned and half unaligned loads (-> EvenAligned case)
   *     - otherwise perform unaligned loads only (-> NoneAligned case)
   *   - otherwise, if the register size is 4 scalars (e.g., SSE with float) then
   *     - one over 4 consecutive columns is guaranteed to be aligned with the result vector,
   *       perform simple aligned loads for this column and aligned loads plus re-alignment for the other. (-> FirstAligned case)
   *       // this re-alignment is done by the palign function implemented for SSE in Eigen/src/Core/arch/SSE/PacketMath.h
   *   - otherwise,
   *     // if we get here, this means the register size is greater than 4 (e.g., AVX with floats),
   *     // we currently fall back to the NoneAligned case
   *
   * The same reasoning apply for the transposed case.
   *
   * The last case (PacketSize>4) could probably be improved by generalizing the FirstAligned case, but since we do not support AVX yet...
   * One might also wonder why in the EvenAligned case we perform unaligned loads instead of using the aligned-loads plus re-alignment
   * strategy as in the FirstAligned case. The reason is that we observed that unaligned loads on a 8 byte boundary are not too slow
   * compared to unaligned loads on a 4 byte boundary.
   *
   */
  template<typename Index, typename LhsScalar, typename LhsMapper, bool ConjugateLhs, typename RhsScalar, typename RhsMapper, bool ConjugateRhs, int Version>
  struct general_matrix_vector_product<Index,LhsScalar,LhsMapper,ColMajor,ConjugateLhs,RhsScalar,RhsMapper,ConjugateRhs,Version>
  {
    typedef typename ScalarBinaryOpTraits<LhsScalar, RhsScalar>::ReturnType ResScalar;
  
  enum {
    Vectorizable = packet_traits<LhsScalar>::Vectorizable && packet_traits<RhsScalar>::Vectorizable
                && int(packet_traits<LhsScalar>::size)==int(packet_traits<RhsScalar>::size),
    LhsPacketSize = Vectorizable ? packet_traits<LhsScalar>::size : 1,
    RhsPacketSize = Vectorizable ? packet_traits<RhsScalar>::size : 1,
    ResPacketSize = Vectorizable ? packet_traits<ResScalar>::size : 1
  };
  
  typedef typename packet_traits<LhsScalar>::type  _LhsPacket;
  typedef typename packet_traits<RhsScalar>::type  _RhsPacket;
  typedef typename packet_traits<ResScalar>::type  _ResPacket;
  
  typedef typename conditional<Vectorizable,_LhsPacket,LhsScalar>::type LhsPacket;
  typedef typename conditional<Vectorizable,_RhsPacket,RhsScalar>::type RhsPacket;
  typedef typename conditional<Vectorizable,_ResPacket,ResScalar>::type ResPacket;
  
  EIGEN_DONT_INLINE static void run(
    Index rows, Index cols,
    const LhsMapper& lhs,
    const RhsMapper& rhs,
          ResScalar* res, Index resIncr,
    RhsScalar alpha);
  };
  
  template<typename Index, typename LhsScalar, typename LhsMapper, bool ConjugateLhs, typename RhsScalar, typename RhsMapper, bool ConjugateRhs, int Version>
  EIGEN_DONT_INLINE void general_matrix_vector_product<Index,LhsScalar,LhsMapper,ColMajor,ConjugateLhs,RhsScalar,RhsMapper,ConjugateRhs,Version>::run(
    Index rows, Index cols,
    const LhsMapper& lhs,
    const RhsMapper& rhs,
          ResScalar* res, Index resIncr,
    RhsScalar alpha)
  {
    EIGEN_UNUSED_VARIABLE(resIncr);
    eigen_internal_assert(resIncr==1);
    #ifdef _EIGEN_ACCUMULATE_PACKETS
    #error _EIGEN_ACCUMULATE_PACKETS has already been defined
    #endif
    #define _EIGEN_ACCUMULATE_PACKETS(Alignment0,Alignment13,Alignment2) \
      pstore(&res[j], \
        padd(pload<ResPacket>(&res[j]), \
          padd( \
        padd(pcj.pmul(lhs0.template load<LhsPacket, Alignment0>(j),    ptmp0), \
        pcj.pmul(lhs1.template load<LhsPacket, Alignment13>(j),   ptmp1)),   \
        padd(pcj.pmul(lhs2.template load<LhsPacket, Alignment2>(j),    ptmp2), \
        pcj.pmul(lhs3.template load<LhsPacket, Alignment13>(j),   ptmp3)) )))
  
    typedef typename LhsMapper::VectorMapper LhsScalars;
  
    conj_helper<LhsScalar,RhsScalar,ConjugateLhs,ConjugateRhs> cj;
    conj_helper<LhsPacket,RhsPacket,ConjugateLhs,ConjugateRhs> pcj;
    if(ConjugateRhs)
      alpha = numext::conj(alpha);
  
    enum { AllAligned = 0, EvenAligned, FirstAligned, NoneAligned };
    const Index columnsAtOnce = 4;
    const Index peels = 2;
    const Index LhsPacketAlignedMask = LhsPacketSize-1;
    const Index ResPacketAlignedMask = ResPacketSize-1;
  //  const Index PeelAlignedMask = ResPacketSize*peels-1;
    const Index size = rows;
  
    const Index lhsStride = lhs.stride();
  
    // How many coeffs of the result do we have to skip to be aligned.
    // Here we assume data are at least aligned on the base scalar type.
    Index alignedStart = internal::first_default_aligned(res,size);
    Index alignedSize = ResPacketSize>1 ? alignedStart + ((size-alignedStart) & ~ResPacketAlignedMask) : 0;
    const Index peeledSize = alignedSize - RhsPacketSize*peels - RhsPacketSize + 1;
  
    const Index alignmentStep = LhsPacketSize>1 ? (LhsPacketSize - lhsStride % LhsPacketSize) & LhsPacketAlignedMask : 0;
    Index alignmentPattern = alignmentStep==0 ? AllAligned
                         : alignmentStep==(LhsPacketSize/2) ? EvenAligned
                         : FirstAligned;
  
    // we cannot assume the first element is aligned because of sub-matrices
    const Index lhsAlignmentOffset = lhs.firstAligned(size);
  
    // find how many columns do we have to skip to be aligned with the result (if possible)
    Index skipColumns = 0;
    // if the data cannot be aligned (TODO add some compile time tests when possible, e.g. for floats)
    if( (lhsAlignmentOffset < 0) || (lhsAlignmentOffset == size) || (UIntPtr(res)%sizeof(ResScalar)) )
    {
      alignedSize = 0;
      alignedStart = 0;
      alignmentPattern = NoneAligned;
    }
    else if(LhsPacketSize > 4)
    {
      // TODO: extend the code to support aligned loads whenever possible when LhsPacketSize > 4.
      // Currently, it seems to be better to perform unaligned loads anyway
      alignmentPattern = NoneAligned;
    }
    else if (LhsPacketSize>1)
    {
    //    eigen_internal_assert(size_t(firstLhs+lhsAlignmentOffset)%sizeof(LhsPacket)==0 || size<LhsPacketSize);
  
      while (skipColumns<LhsPacketSize &&
            alignedStart != ((lhsAlignmentOffset + alignmentStep*skipColumns)%LhsPacketSize))
        ++skipColumns;
      if (skipColumns==LhsPacketSize)
      {
        // nothing can be aligned, no need to skip any column
        alignmentPattern = NoneAligned;
        skipColumns = 0;
      }
      else
      {
        skipColumns = (std::min)(skipColumns,cols);
        // note that the skiped columns are processed later.
      }
  
      /*    eigen_internal_assert(  (alignmentPattern==NoneAligned)
                        || (skipColumns + columnsAtOnce >= cols)
                        || LhsPacketSize > size
                        || (size_t(firstLhs+alignedStart+lhsStride*skipColumns)%sizeof(LhsPacket))==0);*/
    }
    else if(Vectorizable)
    {
      alignedStart = 0;
      alignedSize = size;
      alignmentPattern = AllAligned;
    }
  
    const Index offset1 = (bool(FirstAligned) && bool(alignmentStep)==1)?3:1;
    const Index offset3 = (bool(FirstAligned) && bool(alignmentStep)==1)?1:3;
  
    Index columnBound = ((cols-skipColumns)/columnsAtOnce)*columnsAtOnce + skipColumns;
    for (Index i=skipColumns; i<columnBound; i+=columnsAtOnce)
    {
      RhsPacket ptmp0 = pset1<RhsPacket>(alpha*rhs(i, 0)),
                ptmp1 = pset1<RhsPacket>(alpha*rhs(i+offset1, 0)),
                ptmp2 = pset1<RhsPacket>(alpha*rhs(i+2, 0)),
                ptmp3 = pset1<RhsPacket>(alpha*rhs(i+offset3, 0));
  
      // this helps a lot generating better binary code
      const LhsScalars lhs0 = lhs.getVectorMapper(0, i+0),   lhs1 = lhs.getVectorMapper(0, i+offset1),
                       lhs2 = lhs.getVectorMapper(0, i+2),   lhs3 = lhs.getVectorMapper(0, i+offset3);
  
      if (Vectorizable)
      {
        /* explicit vectorization */
        // process initial unaligned coeffs
        for (Index j=0; j<alignedStart; ++j)
        {
          res[j] = cj.pmadd(lhs0(j), pfirst(ptmp0), res[j]);
          res[j] = cj.pmadd(lhs1(j), pfirst(ptmp1), res[j]);
          res[j] = cj.pmadd(lhs2(j), pfirst(ptmp2), res[j]);
          res[j] = cj.pmadd(lhs3(j), pfirst(ptmp3), res[j]);
        }
  
        if (alignedSize>alignedStart)
        {
          switch(alignmentPattern)
          {
            case AllAligned:
              for (Index j = alignedStart; j<alignedSize; j+=ResPacketSize)
                _EIGEN_ACCUMULATE_PACKETS(Aligned,Aligned,Aligned);
              break;
            case EvenAligned:
              for (Index j = alignedStart; j<alignedSize; j+=ResPacketSize)
                _EIGEN_ACCUMULATE_PACKETS(Aligned,Unaligned,Aligned);
              break;
            case FirstAligned:
            {
              Index j = alignedStart;
              if(peels>1)
              {
                LhsPacket A00, A01, A02, A03, A10, A11, A12, A13;
                ResPacket T0, T1;
  
                A01 = lhs1.template load<LhsPacket, Aligned>(alignedStart-1);
                A02 = lhs2.template load<LhsPacket, Aligned>(alignedStart-2);
                A03 = lhs3.template load<LhsPacket, Aligned>(alignedStart-3);
  
                for (; j<peeledSize; j+=peels*ResPacketSize)
                {
                  A11 = lhs1.template load<LhsPacket, Aligned>(j-1+LhsPacketSize);  palign<1>(A01,A11);
                  A12 = lhs2.template load<LhsPacket, Aligned>(j-2+LhsPacketSize);  palign<2>(A02,A12);
                  A13 = lhs3.template load<LhsPacket, Aligned>(j-3+LhsPacketSize);  palign<3>(A03,A13);
  
                  A00 = lhs0.template load<LhsPacket, Aligned>(j);
                  A10 = lhs0.template load<LhsPacket, Aligned>(j+LhsPacketSize);
                  T0  = pcj.pmadd(A00, ptmp0, pload<ResPacket>(&res[j]));
                  T1  = pcj.pmadd(A10, ptmp0, pload<ResPacket>(&res[j+ResPacketSize]));
  
                  T0  = pcj.pmadd(A01, ptmp1, T0);
                  A01 = lhs1.template load<LhsPacket, Aligned>(j-1+2*LhsPacketSize);  palign<1>(A11,A01);
                  T0  = pcj.pmadd(A02, ptmp2, T0);
                  A02 = lhs2.template load<LhsPacket, Aligned>(j-2+2*LhsPacketSize);  palign<2>(A12,A02);
                  T0  = pcj.pmadd(A03, ptmp3, T0);
                  pstore(&res[j],T0);
                  A03 = lhs3.template load<LhsPacket, Aligned>(j-3+2*LhsPacketSize);  palign<3>(A13,A03);
                  T1  = pcj.pmadd(A11, ptmp1, T1);
                  T1  = pcj.pmadd(A12, ptmp2, T1);
                  T1  = pcj.pmadd(A13, ptmp3, T1);
                  pstore(&res[j+ResPacketSize],T1);
                }
              }
              for (; j<alignedSize; j+=ResPacketSize)
                _EIGEN_ACCUMULATE_PACKETS(Aligned,Unaligned,Unaligned);
              break;
            }
            default:
              for (Index j = alignedStart; j<alignedSize; j+=ResPacketSize)
                _EIGEN_ACCUMULATE_PACKETS(Unaligned,Unaligned,Unaligned);
              break;
          }
        }
      } // end explicit vectorization
  
      /* process remaining coeffs (or all if there is no explicit vectorization) */
      for (Index j=alignedSize; j<size; ++j)
      {
        res[j] = cj.pmadd(lhs0(j), pfirst(ptmp0), res[j]);
        res[j] = cj.pmadd(lhs1(j), pfirst(ptmp1), res[j]);
        res[j] = cj.pmadd(lhs2(j), pfirst(ptmp2), res[j]);
        res[j] = cj.pmadd(lhs3(j), pfirst(ptmp3), res[j]);
      }
    }
  
    // process remaining first and last columns (at most columnsAtOnce-1)
    Index end = cols;
    Index start = columnBound;
    do
    {
      for (Index k=start; k<end; ++k)
      {
        RhsPacket ptmp0 = pset1<RhsPacket>(alpha*rhs(k, 0));
        const LhsScalars lhs0 = lhs.getVectorMapper(0, k);
  
        if (Vectorizable)
        {
          /* explicit vectorization */
          // process first unaligned result's coeffs
          for (Index j=0; j<alignedStart; ++j)
            res[j] += cj.pmul(lhs0(j), pfirst(ptmp0));
          // process aligned result's coeffs
          if (lhs0.template aligned<LhsPacket>(alignedStart))
            for (Index i = alignedStart;i<alignedSize;i+=ResPacketSize)
              pstore(&res[i], pcj.pmadd(lhs0.template load<LhsPacket, Aligned>(i), ptmp0, pload<ResPacket>(&res[i])));
          else
            for (Index i = alignedStart;i<alignedSize;i+=ResPacketSize)
              pstore(&res[i], pcj.pmadd(lhs0.template load<LhsPacket, Unaligned>(i), ptmp0, pload<ResPacket>(&res[i])));
        }
  
        // process remaining scalars (or all if no explicit vectorization)
        for (Index i=alignedSize; i<size; ++i)
          res[i] += cj.pmul(lhs0(i), pfirst(ptmp0));
      }
      if (skipColumns)
      {
        start = 0;
        end = skipColumns;
        skipColumns = 0;
      }
      else
        break;
    } while(Vectorizable);
    #undef _EIGEN_ACCUMULATE_PACKETS
  }
  
  /* Optimized row-major matrix * vector product:
   * This algorithm processes 4 rows at onces that allows to both reduce
   * the number of load/stores of the result by a factor 4 and to reduce
   * the instruction dependency. Moreover, we know that all bands have the
   * same alignment pattern.
   *
   * Mixing type logic:
   *  - alpha is always a complex (or converted to a complex)
   *  - no vectorization
   */
  template<typename Index, typename LhsScalar, typename LhsMapper, bool ConjugateLhs, typename RhsScalar, typename RhsMapper, bool ConjugateRhs, int Version>
  struct general_matrix_vector_product<Index,LhsScalar,LhsMapper,RowMajor,ConjugateLhs,RhsScalar,RhsMapper,ConjugateRhs,Version>
  {
  typedef typename ScalarBinaryOpTraits<LhsScalar, RhsScalar>::ReturnType ResScalar;
  
  enum {
    Vectorizable = packet_traits<LhsScalar>::Vectorizable && packet_traits<RhsScalar>::Vectorizable
                && int(packet_traits<LhsScalar>::size)==int(packet_traits<RhsScalar>::size),
    LhsPacketSize = Vectorizable ? packet_traits<LhsScalar>::size : 1,
    RhsPacketSize = Vectorizable ? packet_traits<RhsScalar>::size : 1,
    ResPacketSize = Vectorizable ? packet_traits<ResScalar>::size : 1
  };
  
  typedef typename packet_traits<LhsScalar>::type  _LhsPacket;
  typedef typename packet_traits<RhsScalar>::type  _RhsPacket;
  typedef typename packet_traits<ResScalar>::type  _ResPacket;
  
  typedef typename conditional<Vectorizable,_LhsPacket,LhsScalar>::type LhsPacket;
  typedef typename conditional<Vectorizable,_RhsPacket,RhsScalar>::type RhsPacket;
  typedef typename conditional<Vectorizable,_ResPacket,ResScalar>::type ResPacket;
  
  EIGEN_DONT_INLINE static void run(
    Index rows, Index cols,
    const LhsMapper& lhs,
    const RhsMapper& rhs,
          ResScalar* res, Index resIncr,
    ResScalar alpha);
  };
  
  template<typename Index, typename LhsScalar, typename LhsMapper, bool ConjugateLhs, typename RhsScalar, typename RhsMapper, bool ConjugateRhs, int Version>
  EIGEN_DONT_INLINE void general_matrix_vector_product<Index,LhsScalar,LhsMapper,RowMajor,ConjugateLhs,RhsScalar,RhsMapper,ConjugateRhs,Version>::run(
    Index rows, Index cols,
    const LhsMapper& lhs,
    const RhsMapper& rhs,
    ResScalar* res, Index resIncr,
    ResScalar alpha)
  {
    eigen_internal_assert(rhs.stride()==1);
  
    #ifdef _EIGEN_ACCUMULATE_PACKETS
    #error _EIGEN_ACCUMULATE_PACKETS has already been defined
    #endif
  
    #define _EIGEN_ACCUMULATE_PACKETS(Alignment0,Alignment13,Alignment2) {\
      RhsPacket b = rhs.getVectorMapper(j, 0).template load<RhsPacket, Aligned>(0);  \
      ptmp0 = pcj.pmadd(lhs0.template load<LhsPacket, Alignment0>(j), b, ptmp0); \
      ptmp1 = pcj.pmadd(lhs1.template load<LhsPacket, Alignment13>(j), b, ptmp1); \
      ptmp2 = pcj.pmadd(lhs2.template load<LhsPacket, Alignment2>(j), b, ptmp2); \
      ptmp3 = pcj.pmadd(lhs3.template load<LhsPacket, Alignment13>(j), b, ptmp3); }
  
    conj_helper<LhsScalar,RhsScalar,ConjugateLhs,ConjugateRhs> cj;
    conj_helper<LhsPacket,RhsPacket,ConjugateLhs,ConjugateRhs> pcj;
  
    typedef typename LhsMapper::VectorMapper LhsScalars;
  
    enum { AllAligned=0, EvenAligned=1, FirstAligned=2, NoneAligned=3 };
    const Index rowsAtOnce = 4;
    const Index peels = 2;
    const Index RhsPacketAlignedMask = RhsPacketSize-1;
    const Index LhsPacketAlignedMask = LhsPacketSize-1;
    const Index depth = cols;
    const Index lhsStride = lhs.stride();
  
    // How many coeffs of the result do we have to skip to be aligned.
    // Here we assume data are at least aligned on the base scalar type
    // if that's not the case then vectorization is discarded, see below.
    Index alignedStart = rhs.firstAligned(depth);
    Index alignedSize = RhsPacketSize>1 ? alignedStart + ((depth-alignedStart) & ~RhsPacketAlignedMask) : 0;
    const Index peeledSize = alignedSize - RhsPacketSize*peels - RhsPacketSize + 1;
  
    const Index alignmentStep = LhsPacketSize>1 ? (LhsPacketSize - lhsStride % LhsPacketSize) & LhsPacketAlignedMask : 0;
    Index alignmentPattern = alignmentStep==0 ? AllAligned
                             : alignmentStep==(LhsPacketSize/2) ? EvenAligned
                             : FirstAligned;
  
    // we cannot assume the first element is aligned because of sub-matrices
    const Index lhsAlignmentOffset = lhs.firstAligned(depth);
    const Index rhsAlignmentOffset = rhs.firstAligned(rows);
  
    // find how many rows do we have to skip to be aligned with rhs (if possible)
    Index skipRows = 0;
    // if the data cannot be aligned (TODO add some compile time tests when possible, e.g. for floats)
    if( (sizeof(LhsScalar)!=sizeof(RhsScalar)) ||
        (lhsAlignmentOffset < 0) || (lhsAlignmentOffset == depth) ||
        (rhsAlignmentOffset < 0) || (rhsAlignmentOffset == rows) )
    {
      alignedSize = 0;
      alignedStart = 0;
      alignmentPattern = NoneAligned;
    }
    else if(LhsPacketSize > 4)
    {
      // TODO: extend the code to support aligned loads whenever possible when LhsPacketSize > 4.
      alignmentPattern = NoneAligned;
    }
    else if (LhsPacketSize>1)
    {
    //    eigen_internal_assert(size_t(firstLhs+lhsAlignmentOffset)%sizeof(LhsPacket)==0  || depth<LhsPacketSize);
  
      while (skipRows<LhsPacketSize &&
             alignedStart != ((lhsAlignmentOffset + alignmentStep*skipRows)%LhsPacketSize))
        ++skipRows;
      if (skipRows==LhsPacketSize)
      {
        // nothing can be aligned, no need to skip any column
        alignmentPattern = NoneAligned;
        skipRows = 0;
      }
      else
      {
        skipRows = (std::min)(skipRows,Index(rows));
        // note that the skiped columns are processed later.
      }
      /*    eigen_internal_assert(  alignmentPattern==NoneAligned
                        || LhsPacketSize==1
                        || (skipRows + rowsAtOnce >= rows)
                        || LhsPacketSize > depth
                        || (size_t(firstLhs+alignedStart+lhsStride*skipRows)%sizeof(LhsPacket))==0);*/
    }
    else if(Vectorizable)
    {
      alignedStart = 0;
      alignedSize = depth;
      alignmentPattern = AllAligned;
    }
  
    const Index offset1 = (bool(FirstAligned) && bool(alignmentStep)==1)?3:1;
    const Index offset3 = (bool(FirstAligned) && bool(alignmentStep)==1)?1:3;
  
    Index rowBound = ((rows-skipRows)/rowsAtOnce)*rowsAtOnce + skipRows;
    for (Index i=skipRows; i<rowBound; i+=rowsAtOnce)
    {
      // FIXME: what is the purpose of this EIGEN_ALIGN_DEFAULT ??
      EIGEN_ALIGN_MAX ResScalar tmp0 = ResScalar(0);
      ResScalar tmp1 = ResScalar(0), tmp2 = ResScalar(0), tmp3 = ResScalar(0);
  
      // this helps the compiler generating good binary code
      const LhsScalars lhs0 = lhs.getVectorMapper(i+0, 0),    lhs1 = lhs.getVectorMapper(i+offset1, 0),
                       lhs2 = lhs.getVectorMapper(i+2, 0),    lhs3 = lhs.getVectorMapper(i+offset3, 0);
  
      if (Vectorizable)
      {
        /* explicit vectorization */
        ResPacket ptmp0 = pset1<ResPacket>(ResScalar(0)), ptmp1 = pset1<ResPacket>(ResScalar(0)),
                  ptmp2 = pset1<ResPacket>(ResScalar(0)), ptmp3 = pset1<ResPacket>(ResScalar(0));
  
        // process initial unaligned coeffs
        // FIXME this loop get vectorized by the compiler !
        for (Index j=0; j<alignedStart; ++j)
        {
          RhsScalar b = rhs(j, 0);
          tmp0 += cj.pmul(lhs0(j),b); tmp1 += cj.pmul(lhs1(j),b);
          tmp2 += cj.pmul(lhs2(j),b); tmp3 += cj.pmul(lhs3(j),b);
        }
  
        if (alignedSize>alignedStart)
        {
          switch(alignmentPattern)
          {
            case AllAligned:
              for (Index j = alignedStart; j<alignedSize; j+=RhsPacketSize)
                _EIGEN_ACCUMULATE_PACKETS(Aligned,Aligned,Aligned);
              break;
            case EvenAligned:
              for (Index j = alignedStart; j<alignedSize; j+=RhsPacketSize)
                _EIGEN_ACCUMULATE_PACKETS(Aligned,Unaligned,Aligned);
              break;
            case FirstAligned:
            {
              Index j = alignedStart;
              if (peels>1)
              {
                /* Here we proccess 4 rows with with two peeled iterations to hide
                 * the overhead of unaligned loads. Moreover unaligned loads are handled
                 * using special shift/move operations between the two aligned packets
                 * overlaping the desired unaligned packet. This is *much* more efficient
                 * than basic unaligned loads.
                 */
                LhsPacket A01, A02, A03, A11, A12, A13;
                A01 = lhs1.template load<LhsPacket, Aligned>(alignedStart-1);
                A02 = lhs2.template load<LhsPacket, Aligned>(alignedStart-2);
                A03 = lhs3.template load<LhsPacket, Aligned>(alignedStart-3);
  
                for (; j<peeledSize; j+=peels*RhsPacketSize)
                {
                  RhsPacket b = rhs.getVectorMapper(j, 0).template load<RhsPacket, Aligned>(0);
                  A11 = lhs1.template load<LhsPacket, Aligned>(j-1+LhsPacketSize);  palign<1>(A01,A11);
                  A12 = lhs2.template load<LhsPacket, Aligned>(j-2+LhsPacketSize);  palign<2>(A02,A12);
                  A13 = lhs3.template load<LhsPacket, Aligned>(j-3+LhsPacketSize);  palign<3>(A03,A13);
  
                  ptmp0 = pcj.pmadd(lhs0.template load<LhsPacket, Aligned>(j), b, ptmp0);
                  ptmp1 = pcj.pmadd(A01, b, ptmp1);
                  A01 = lhs1.template load<LhsPacket, Aligned>(j-1+2*LhsPacketSize);  palign<1>(A11,A01);
                  ptmp2 = pcj.pmadd(A02, b, ptmp2);
                  A02 = lhs2.template load<LhsPacket, Aligned>(j-2+2*LhsPacketSize);  palign<2>(A12,A02);
                  ptmp3 = pcj.pmadd(A03, b, ptmp3);
                  A03 = lhs3.template load<LhsPacket, Aligned>(j-3+2*LhsPacketSize);  palign<3>(A13,A03);
  
                  b = rhs.getVectorMapper(j+RhsPacketSize, 0).template load<RhsPacket, Aligned>(0);
                  ptmp0 = pcj.pmadd(lhs0.template load<LhsPacket, Aligned>(j+LhsPacketSize), b, ptmp0);
                  ptmp1 = pcj.pmadd(A11, b, ptmp1);
                  ptmp2 = pcj.pmadd(A12, b, ptmp2);
                  ptmp3 = pcj.pmadd(A13, b, ptmp3);
                }
              }
              for (; j<alignedSize; j+=RhsPacketSize)
                _EIGEN_ACCUMULATE_PACKETS(Aligned,Unaligned,Unaligned);
              break;
            }
            default:
              for (Index j = alignedStart; j<alignedSize; j+=RhsPacketSize)
                _EIGEN_ACCUMULATE_PACKETS(Unaligned,Unaligned,Unaligned);
              break;
          }
          tmp0 += predux(ptmp0);
          tmp1 += predux(ptmp1);
          tmp2 += predux(ptmp2);
          tmp3 += predux(ptmp3);
        }
      } // end explicit vectorization
  
      // process remaining coeffs (or all if no explicit vectorization)
      // FIXME this loop get vectorized by the compiler !
      for (Index j=alignedSize; j<depth; ++j)
      {
        RhsScalar b = rhs(j, 0);
        tmp0 += cj.pmul(lhs0(j),b); tmp1 += cj.pmul(lhs1(j),b);
        tmp2 += cj.pmul(lhs2(j),b); tmp3 += cj.pmul(lhs3(j),b);
      }
      res[i*resIncr]            += alpha*tmp0;
      res[(i+offset1)*resIncr]  += alpha*tmp1;
      res[(i+2)*resIncr]        += alpha*tmp2;
      res[(i+offset3)*resIncr]  += alpha*tmp3;
    }
  
    // process remaining first and last rows (at most columnsAtOnce-1)
    Index end = rows;
    Index start = rowBound;
    do
    {
      for (Index i=start; i<end; ++i)
      {
        EIGEN_ALIGN_MAX ResScalar tmp0 = ResScalar(0);
        ResPacket ptmp0 = pset1<ResPacket>(tmp0);
        const LhsScalars lhs0 = lhs.getVectorMapper(i, 0);
        // process first unaligned result's coeffs
        // FIXME this loop get vectorized by the compiler !
        for (Index j=0; j<alignedStart; ++j)
          tmp0 += cj.pmul(lhs0(j), rhs(j, 0));
  
        if (alignedSize>alignedStart)
        {
          // process aligned rhs coeffs
          if (lhs0.template aligned<LhsPacket>(alignedStart))
            for (Index j = alignedStart;j<alignedSize;j+=RhsPacketSize)
              ptmp0 = pcj.pmadd(lhs0.template load<LhsPacket, Aligned>(j), rhs.getVectorMapper(j, 0).template load<RhsPacket, Aligned>(0), ptmp0);
          else
            for (Index j = alignedStart;j<alignedSize;j+=RhsPacketSize)
              ptmp0 = pcj.pmadd(lhs0.template load<LhsPacket, Unaligned>(j), rhs.getVectorMapper(j, 0).template load<RhsPacket, Aligned>(0), ptmp0);
          tmp0 += predux(ptmp0);
        }
  
        // process remaining scalars
        // FIXME this loop get vectorized by the compiler !
        for (Index j=alignedSize; j<depth; ++j)
          tmp0 += cj.pmul(lhs0(j), rhs(j, 0));
        res[i*resIncr] += alpha*tmp0;
      }
      if (skipRows)
      {
        start = 0;
        end = skipRows;
        skipRows = 0;
      }
      else
        break;
    } while(Vectorizable);
  
    #undef _EIGEN_ACCUMULATE_PACKETS
  }
  
  } // end namespace internal
  
  } // end namespace Eigen
  
  #endif // EIGEN_GENERAL_MATRIX_VECTOR_H
  