TriangularSolverMatrix.h

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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Modifications Copyright (C) 2022 Intel Corporation
//
// 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_TRIANGULAR_SOLVER_MATRIX_H
#define EIGEN_TRIANGULAR_SOLVER_MATRIX_H
 
#include "../InternalHeaderCheck.h"
 
namespace Eigen {
 
namespace internal {
 
template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder,int OtherInnerStride, bool Specialized>
struct trsmKernelL {
  // Generic Implementation of triangular solve for triangular matrix on left and multiple rhs.
  // Handles non-packed matrices.
  static void kernel(
    Index size, Index otherSize,
    const Scalar* _tri, Index triStride,
    Scalar* _other, Index otherIncr, Index otherStride);
};
 
template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder,int OtherInnerStride, bool Specialized>
struct trsmKernelR {
  // Generic Implementation of triangular solve for triangular matrix on right and multiple lhs.
  // Handles non-packed matrices.
  static void kernel(
    Index size, Index otherSize,
    const Scalar* _tri, Index triStride,
    Scalar* _other, Index otherIncr, Index otherStride);
};
 
template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder,int OtherInnerStride, bool Specialized>
EIGEN_STRONG_INLINE void trsmKernelL<Scalar, Index, Mode, Conjugate, TriStorageOrder, OtherInnerStride, Specialized>::kernel(
    Index size, Index otherSize,
    const Scalar* _tri, Index triStride,
    Scalar* _other, Index otherIncr, Index otherStride)
  {
    typedef const_blas_data_mapper<Scalar, Index, TriStorageOrder> TriMapper;
    typedef blas_data_mapper<Scalar, Index, ColMajor, Unaligned, OtherInnerStride> OtherMapper;
    TriMapper tri(_tri, triStride);
    OtherMapper other(_other, otherStride, otherIncr);
 
    enum { IsLower = (Mode&Lower) == Lower };
    conj_if<Conjugate> conj;
 
    // tr solve
    for (Index k=0; k<size; ++k)
    {
      // TODO write a small kernel handling this (can be shared with trsv)
      Index i  = IsLower ? k : -k-1;
      Index rs = size - k - 1; // remaining size
      Index s  = TriStorageOrder==RowMajor ? (IsLower ? 0 : i+1)
        :  IsLower ? i+1 : i-rs;
 
      Scalar a = (Mode & UnitDiag) ? Scalar(1) : Scalar(1)/conj(tri(i,i));
      for (Index j=0; j<otherSize; ++j)
      {
        if (TriStorageOrder==RowMajor)
        {
          Scalar b(0);
          const Scalar* l = &tri(i,s);
          typename OtherMapper::LinearMapper r = other.getLinearMapper(s,j);
          for (Index i3=0; i3<k; ++i3)
            b += conj(l[i3]) * r(i3);
 
          other(i,j) = (other(i,j) - b)*a;
        }
        else
        {
          Scalar& otherij = other(i,j);
          otherij *= a;
          Scalar b = otherij;
          typename OtherMapper::LinearMapper r = other.getLinearMapper(s,j);
          typename TriMapper::LinearMapper l = tri.getLinearMapper(s,i);
          for (Index i3=0;i3<rs;++i3)
            r(i3) -= b * conj(l(i3));
        }
      }
    }
  }
 
 
template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride, bool Specialized>
EIGEN_STRONG_INLINE void trsmKernelR<Scalar, Index, Mode, Conjugate, TriStorageOrder, OtherInnerStride, Specialized>::kernel(
    Index size, Index otherSize,
    const Scalar* _tri, Index triStride,
    Scalar* _other, Index otherIncr, Index otherStride)
{
  typedef typename NumTraits<Scalar>::Real RealScalar;
  typedef blas_data_mapper<Scalar, Index, ColMajor, Unaligned, OtherInnerStride> LhsMapper;
  typedef const_blas_data_mapper<Scalar, Index, TriStorageOrder> RhsMapper;
  LhsMapper lhs(_other, otherStride, otherIncr);
  RhsMapper rhs(_tri, triStride);
 
  enum {
    RhsStorageOrder   = TriStorageOrder,
    IsLower = (Mode&Lower) == Lower
  };
  conj_if<Conjugate> conj;
 
  for (Index k=0; k<size; ++k)
  {
    Index j = IsLower ? size-k-1 : k;
 
    typename LhsMapper::LinearMapper r = lhs.getLinearMapper(0,j);
    for (Index k3=0; k3<k; ++k3)
    {
      Scalar b = conj(rhs(IsLower ? j+1+k3 : k3,j));
      typename LhsMapper::LinearMapper a = lhs.getLinearMapper(0,IsLower ? j+1+k3 : k3);
      for (Index i=0; i<otherSize; ++i)
                    r(i) -= a(i) * b;
    }
    if((Mode & UnitDiag)==0)
    {
      Scalar inv_rjj = RealScalar(1)/conj(rhs(j,j));
      for (Index i=0; i<otherSize; ++i)
        r(i) *= inv_rjj;
    }
  }
}
 
 
// if the rhs is row major, let's transpose the product
template <typename Scalar, typename Index, int Side, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride>
struct triangular_solve_matrix<Scalar,Index,Side,Mode,Conjugate,TriStorageOrder,RowMajor,OtherInnerStride>
{
  static void run(
    Index size, Index cols,
    const Scalar*  tri, Index triStride,
    Scalar* _other, Index otherIncr, Index otherStride,
    level3_blocking<Scalar,Scalar>& blocking)
  {
    triangular_solve_matrix<
      Scalar, Index, Side==OnTheLeft?OnTheRight:OnTheLeft,
      (Mode&UnitDiag) | ((Mode&Upper) ? Lower : Upper),
      NumTraits<Scalar>::IsComplex && Conjugate,
      TriStorageOrder==RowMajor ? ColMajor : RowMajor, ColMajor, OtherInnerStride>
      ::run(size, cols, tri, triStride, _other, otherIncr, otherStride, blocking);
  }
};
 
/* Optimized triangular solver with multiple right hand side and the triangular matrix on the left
 */
template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder,int OtherInnerStride>
struct triangular_solve_matrix<Scalar,Index,OnTheLeft,Mode,Conjugate,TriStorageOrder,ColMajor,OtherInnerStride>
{
  static EIGEN_DONT_INLINE void run(
    Index size, Index otherSize,
    const Scalar* _tri, Index triStride,
    Scalar* _other, Index otherIncr, Index otherStride,
    level3_blocking<Scalar,Scalar>& blocking);
};
 
template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride>
EIGEN_DONT_INLINE void triangular_solve_matrix<Scalar,Index,OnTheLeft,Mode,Conjugate,TriStorageOrder,ColMajor,OtherInnerStride>::run(
    Index size, Index otherSize,
    const Scalar* _tri, Index triStride,
    Scalar* _other, Index otherIncr, Index otherStride,
    level3_blocking<Scalar,Scalar>& blocking)
  {
    Index cols = otherSize;
 
    std::ptrdiff_t l1, l2, l3;
    manage_caching_sizes(GetAction, &l1, &l2, &l3);
 
#if defined(EIGEN_VECTORIZE_AVX512) && EIGEN_USE_AVX512_TRSM_L_KERNELS && EIGEN_ENABLE_AVX512_NOCOPY_TRSM_L_CUTOFFS
    EIGEN_IF_CONSTEXPR( (OtherInnerStride == 1 &&
                       (std::is_same<Scalar,float>::value ||
                        std::is_same<Scalar,double>::value)) ) {
      // Very rough cutoffs to determine when to call trsm w/o packing
      // For small problem sizes trsmKernel compiled with clang is generally faster.
      // TODO: Investigate better heuristics for cutoffs.
      double L2Cap = 0.5; // 50% of L2 size
      if (size < avx512_trsm_cutoff<Scalar>(l2, cols, L2Cap)) {
        trsmKernelL<Scalar, Index, Mode, Conjugate, TriStorageOrder, 1, /*Specialized=*/true>::kernel(
          size, cols, _tri, triStride, _other, 1, otherStride);
        return;
      }
    }
#endif
 
    typedef const_blas_data_mapper<Scalar, Index, TriStorageOrder> TriMapper;
    typedef blas_data_mapper<Scalar, Index, ColMajor, Unaligned, OtherInnerStride> OtherMapper;
    TriMapper tri(_tri, triStride);
    OtherMapper other(_other, otherStride, otherIncr);
 
    typedef gebp_traits<Scalar,Scalar> Traits;
 
    enum {
      SmallPanelWidth   = plain_enum_max(Traits::mr, Traits::nr),
      IsLower = (Mode&Lower) == Lower
    };
 
    Index kc = blocking.kc();                   // cache block size along the K direction
    Index mc = (std::min)(size,blocking.mc());  // cache block size along the M direction
 
    std::size_t sizeA = kc*mc;
    std::size_t sizeB = kc*cols;
 
    ei_declare_aligned_stack_constructed_variable(Scalar, blockA, sizeA, blocking.blockA());
    ei_declare_aligned_stack_constructed_variable(Scalar, blockB, sizeB, blocking.blockB());
 
    gebp_kernel<Scalar, Scalar, Index, OtherMapper, Traits::mr, Traits::nr, Conjugate, false> gebp_kernel;
    gemm_pack_lhs<Scalar, Index, TriMapper, Traits::mr, Traits::LhsProgress, typename Traits::LhsPacket4Packing, TriStorageOrder> pack_lhs;
    gemm_pack_rhs<Scalar, Index, OtherMapper, Traits::nr, ColMajor, false, true> pack_rhs;
 
    // the goal here is to subdivise the Rhs panels such that we keep some cache
    // coherence when accessing the rhs elements
    Index subcols = cols>0 ? l2/(4 * sizeof(Scalar) * std::max<Index>(otherStride,size)) : 0;
    subcols = std::max<Index>((subcols/Traits::nr)*Traits::nr, Traits::nr);
 
    for(Index k2=IsLower ? 0 : size;
        IsLower ? k2<size : k2>0;
        IsLower ? k2+=kc : k2-=kc)
    {
      const Index actual_kc = (std::min)(IsLower ? size-k2 : k2, kc);
 
      // We have selected and packed a big horizontal panel R1 of rhs. Let B be the packed copy of this panel,
      // and R2 the remaining part of rhs. The corresponding vertical panel of lhs is split into
      // A11 (the triangular part) and A21 the remaining rectangular part.
      // Then the high level algorithm is:
      //  - B = R1                    => general block copy (done during the next step)
      //  - R1 = A11^-1 B             => tricky part
      //  - update B from the new R1  => actually this has to be performed continuously during the above step
      //  - R2 -= A21 * B             => GEPP
 
      // The tricky part: compute R1 = A11^-1 B while updating B from R1
      // The idea is to split A11 into multiple small vertical panels.
      // Each panel can be split into a small triangular part T1k which is processed without optimization,
      // and the remaining small part T2k which is processed using gebp with appropriate block strides
      for(Index j2=0; j2<cols; j2+=subcols)
      {
        Index actual_cols = (std::min)(cols-j2,subcols);
        // for each small vertical panels [T1k^T, T2k^T]^T of lhs
        for (Index k1=0; k1<actual_kc; k1+=SmallPanelWidth)
        {
          Index actualPanelWidth = std::min<Index>(actual_kc-k1, SmallPanelWidth);
          // tr solve
          {
            Index i  = IsLower ? k2+k1 : k2-k1;
#if defined(EIGEN_VECTORIZE_AVX512) && EIGEN_USE_AVX512_TRSM_L_KERNELS
            EIGEN_IF_CONSTEXPR( (OtherInnerStride == 1 &&
                                 (std::is_same<Scalar,float>::value ||
                                  std::is_same<Scalar,double>::value)) ) {
              i  = IsLower ? k2 + k1: k2 - k1 - actualPanelWidth;
            }
#endif
            trsmKernelL<Scalar, Index, Mode, Conjugate, TriStorageOrder, OtherInnerStride, /*Specialized=*/true>::kernel(
              actualPanelWidth, actual_cols,
              _tri + i + (i)*triStride, triStride,
              _other + i*OtherInnerStride + j2*otherStride, otherIncr, otherStride);
          }
 
          Index lengthTarget = actual_kc-k1-actualPanelWidth;
          Index startBlock   = IsLower ? k2+k1 : k2-k1-actualPanelWidth;
          Index blockBOffset = IsLower ? k1 : lengthTarget;
 
          // update the respective rows of B from other
          pack_rhs(blockB+actual_kc*j2, other.getSubMapper(startBlock,j2), actualPanelWidth, actual_cols, actual_kc, blockBOffset);
 
          // GEBP
          if (lengthTarget>0)
          {
            Index startTarget  = IsLower ? k2+k1+actualPanelWidth : k2-actual_kc;
 
            pack_lhs(blockA, tri.getSubMapper(startTarget,startBlock), actualPanelWidth, lengthTarget);
 
            gebp_kernel(other.getSubMapper(startTarget,j2), blockA, blockB+actual_kc*j2, lengthTarget, actualPanelWidth, actual_cols, Scalar(-1),
                        actualPanelWidth, actual_kc, 0, blockBOffset);
          }
        }
      }
 
      // R2 -= A21 * B => GEPP
      {
        Index start = IsLower ? k2+kc : 0;
        Index end   = IsLower ? size : k2-kc;
        for(Index i2=start; i2<end; i2+=mc)
        {
          const Index actual_mc = (std::min)(mc,end-i2);
          if (actual_mc>0)
          {
            pack_lhs(blockA, tri.getSubMapper(i2, IsLower ? k2 : k2-kc), actual_kc, actual_mc);
 
            gebp_kernel(other.getSubMapper(i2, 0), blockA, blockB, actual_mc, actual_kc, cols, Scalar(-1), -1, -1, 0, 0);
          }
        }
      }
    }
  }
 
/* Optimized triangular solver with multiple left hand sides and the triangular matrix on the right
 */
template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride>
struct triangular_solve_matrix<Scalar,Index,OnTheRight,Mode,Conjugate,TriStorageOrder,ColMajor,OtherInnerStride>
{
  static EIGEN_DONT_INLINE void run(
    Index size, Index otherSize,
    const Scalar* _tri, Index triStride,
    Scalar* _other, Index otherIncr, Index otherStride,
    level3_blocking<Scalar,Scalar>& blocking);
};
 
template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride>
EIGEN_DONT_INLINE void triangular_solve_matrix<Scalar,Index,OnTheRight,Mode,Conjugate,TriStorageOrder,ColMajor,OtherInnerStride>::run(
    Index size, Index otherSize,
    const Scalar* _tri, Index triStride,
    Scalar* _other, Index otherIncr, Index otherStride,
    level3_blocking<Scalar,Scalar>& blocking)
  {
    Index rows = otherSize;
 
#if defined(EIGEN_VECTORIZE_AVX512) && EIGEN_USE_AVX512_TRSM_R_KERNELS && EIGEN_ENABLE_AVX512_NOCOPY_TRSM_R_CUTOFFS
    EIGEN_IF_CONSTEXPR( (OtherInnerStride == 1 &&
                 (std::is_same<Scalar,float>::value ||
                  std::is_same<Scalar,double>::value)) ) {
      // TODO: Investigate better heuristics for cutoffs.
      std::ptrdiff_t l1, l2, l3;
      manage_caching_sizes(GetAction, &l1, &l2, &l3);
      double L2Cap = 0.5; // 50% of L2 size
      if (size < avx512_trsm_cutoff<Scalar>(l2, rows, L2Cap)) {
        trsmKernelR<Scalar, Index, Mode, Conjugate, TriStorageOrder, OtherInnerStride, /*Specialized=*/true>::
          kernel(size, rows, _tri, triStride, _other, 1, otherStride);
        return;
      }
    }
#endif
 
    typedef blas_data_mapper<Scalar, Index, ColMajor, Unaligned, OtherInnerStride> LhsMapper;
    typedef const_blas_data_mapper<Scalar, Index, TriStorageOrder> RhsMapper;
    LhsMapper lhs(_other, otherStride, otherIncr);
    RhsMapper rhs(_tri, triStride);
 
    typedef gebp_traits<Scalar,Scalar> Traits;
    enum {
      RhsStorageOrder   = TriStorageOrder,
      SmallPanelWidth   = plain_enum_max(Traits::mr, Traits::nr),
      IsLower = (Mode&Lower) == Lower
    };
 
    Index kc = blocking.kc();                   // cache block size along the K direction
    Index mc = (std::min)(rows,blocking.mc());  // cache block size along the M direction
 
    std::size_t sizeA = kc*mc;
    std::size_t sizeB = kc*size;
 
    ei_declare_aligned_stack_constructed_variable(Scalar, blockA, sizeA, blocking.blockA());
    ei_declare_aligned_stack_constructed_variable(Scalar, blockB, sizeB, blocking.blockB());
 
    gebp_kernel<Scalar, Scalar, Index, LhsMapper, Traits::mr, Traits::nr, false, Conjugate> gebp_kernel;
    gemm_pack_rhs<Scalar, Index, RhsMapper, Traits::nr, RhsStorageOrder> pack_rhs;
    gemm_pack_rhs<Scalar, Index, RhsMapper, Traits::nr, RhsStorageOrder,false,true> pack_rhs_panel;
    gemm_pack_lhs<Scalar, Index, LhsMapper, Traits::mr, Traits::LhsProgress, typename Traits::LhsPacket4Packing, ColMajor, false, true> pack_lhs_panel;
 
    for(Index k2=IsLower ? size : 0;
        IsLower ? k2>0 : k2<size;
        IsLower ? k2-=kc : k2+=kc)
    {
      const Index actual_kc = (std::min)(IsLower ? k2 : size-k2, kc);
      Index actual_k2 = IsLower ? k2-actual_kc : k2 ;
 
      Index startPanel = IsLower ? 0 : k2+actual_kc;
      Index rs = IsLower ? actual_k2 : size - actual_k2 - actual_kc;
      Scalar* geb = blockB+actual_kc*actual_kc;
 
      if (rs>0) pack_rhs(geb, rhs.getSubMapper(actual_k2,startPanel), actual_kc, rs);
 
      // triangular packing (we only pack the panels off the diagonal,
      // neglecting the blocks overlapping the diagonal
      {
        for (Index j2=0; j2<actual_kc; j2+=SmallPanelWidth)
        {
          Index actualPanelWidth = std::min<Index>(actual_kc-j2, SmallPanelWidth);
          Index actual_j2 = actual_k2 + j2;
          Index panelOffset = IsLower ? j2+actualPanelWidth : 0;
          Index panelLength = IsLower ? actual_kc-j2-actualPanelWidth : j2;
 
          if (panelLength>0)
          pack_rhs_panel(blockB+j2*actual_kc,
                         rhs.getSubMapper(actual_k2+panelOffset, actual_j2),
                         panelLength, actualPanelWidth,
                         actual_kc, panelOffset);
        }
      }
 
      for(Index i2=0; i2<rows; i2+=mc)
      {
        const Index actual_mc = (std::min)(mc,rows-i2);
 
        // triangular solver kernel
        {
          // for each small block of the diagonal (=> vertical panels of rhs)
          for (Index j2 = IsLower
                      ? (actual_kc - ((actual_kc%SmallPanelWidth) ? Index(actual_kc%SmallPanelWidth)
                                                                  : Index(SmallPanelWidth)))
                      : 0;
               IsLower ? j2>=0 : j2<actual_kc;
               IsLower ? j2-=SmallPanelWidth : j2+=SmallPanelWidth)
          {
            Index actualPanelWidth = std::min<Index>(actual_kc-j2, SmallPanelWidth);
            Index absolute_j2 = actual_k2 + j2;
            Index panelOffset = IsLower ? j2+actualPanelWidth : 0;
            Index panelLength = IsLower ? actual_kc - j2 - actualPanelWidth : j2;
 
            // GEBP
            if(panelLength>0)
            {
              gebp_kernel(lhs.getSubMapper(i2,absolute_j2),
                          blockA, blockB+j2*actual_kc,
                          actual_mc, panelLength, actualPanelWidth,
                          Scalar(-1),
                          actual_kc, actual_kc, // strides
                          panelOffset, panelOffset); // offsets
            }
 
            {
              // unblocked triangular solve
              trsmKernelR<Scalar, Index, Mode, Conjugate, TriStorageOrder, OtherInnerStride, /*Specialized=*/true>::
                kernel(actualPanelWidth, actual_mc,
                            _tri + absolute_j2 + absolute_j2*triStride, triStride,
                            _other + i2*OtherInnerStride + absolute_j2*otherStride, otherIncr, otherStride);
            }
            // pack the just computed part of lhs to A
            pack_lhs_panel(blockA, lhs.getSubMapper(i2,absolute_j2),
                           actualPanelWidth, actual_mc,
                           actual_kc, j2);
          }
        }
 
        if (rs>0)
          gebp_kernel(lhs.getSubMapper(i2, startPanel), blockA, geb,
                      actual_mc, actual_kc, rs, Scalar(-1),
                      -1, -1, 0, 0);
      }
    }
  }
} // end namespace internal
 
} // end namespace Eigen
 
#endif // EIGEN_TRIANGULAR_SOLVER_MATRIX_H