SparseDenseProduct.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2015 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/.
// SPDX-License-Identifier: MPL-2.0
 
#ifndef EIGEN_SPARSEDENSEPRODUCT_H
#define EIGEN_SPARSEDENSEPRODUCT_H
 
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
 
namespace Eigen {
 
namespace internal {
 
template <>
struct product_promote_storage_type<Sparse, Dense, OuterProduct> {
  using ret = Sparse;
};
template <>
struct product_promote_storage_type<Dense, Sparse, OuterProduct> {
  using ret = Sparse;
};
 
// Type trait to detect if a sparse type supports direct compressed storage access
// (i.e., has valuePtr(), innerIndexPtr(), outerIndexPtr(), isCompressed()).
// All types deriving from SparseCompressedBase provide these methods.
template <typename T>
struct has_compressed_storage : std::is_base_of<SparseCompressedBase<T>, T> {};
 
template <typename SparseLhsType, typename DenseRhsType, typename DenseResType, typename AlphaType,
          int LhsStorageOrder = ((SparseLhsType::Flags & RowMajorBit) == RowMajorBit) ? RowMajor : ColMajor,
          bool ColPerCol = ((DenseRhsType::Flags & RowMajorBit) == 0) || DenseRhsType::ColsAtCompileTime == 1>
struct sparse_time_dense_product_impl;
 
// RowMajor, single column (ColPerCol=true): CSR SpMV
template <typename SparseLhsType, typename DenseRhsType, typename DenseResType>
struct sparse_time_dense_product_impl<SparseLhsType, DenseRhsType, DenseResType, typename DenseResType::Scalar,
                                      RowMajor, true> {
  using Lhs = internal::remove_all_t<SparseLhsType>;
  using Res = internal::remove_all_t<DenseResType>;
  using LhsInnerIterator = typename evaluator<Lhs>::InnerIterator;
  using LhsEval = evaluator<Lhs>;
  using ResScalar = typename Res::Scalar;
 
  static void run(const SparseLhsType& lhs, const DenseRhsType& rhs, DenseResType& res,
                  const typename Res::Scalar& alpha) {
    LhsEval lhsEval(lhs);
    Index n = lhs.outerSize();
 
    for (Index c = 0; c < rhs.cols(); ++c) {
      runCol(lhsEval, lhs, rhs, res, alpha, n, c, bool_constant<has_compressed_storage<Lhs>::value>());
    }
  }
 
  // Direct pointer path: works for both compressed and non-compressed storage.
  static void runCol(const LhsEval& /*lhsEval*/, const SparseLhsType& lhs, const DenseRhsType& rhs, DenseResType& res,
                     const ResScalar& alpha, Index n, Index c, std::true_type /* has_compressed_storage */) {
    runColImpl(lhs, rhs, res, alpha, n, c, bool_constant<bool(DenseRhsType::Flags & DirectAccessBit)>());
  }
 
  template <typename RhsT>
  static void runColImpl(const SparseLhsType& lhs, const RhsT& rhs, DenseResType& res, const ResScalar& alpha, Index n,
                         Index c, std::true_type) {
    const Lhs& mat = lhs;
    const auto* vals = mat.valuePtr();
    const auto* inds = mat.innerIndexPtr();
    // Sparse vectors don't store outer indices.
    const auto* outer = mat.outerIndexPtr();
    const auto* innerNnz = mat.innerNonZeroPtr();
    // The fast rhs pointer path requires unit inner stride (common case: VectorXd, contiguous matrix column).
    if (rhs.innerStride() == 1) {
      const auto* x = rhs.data() + c * rhs.outerStride();
#ifdef EIGEN_HAS_OPENMP
      Index threads = Eigen::nbThreads();
      if (threads > 1 && mat.nonZeros() > 20000) {
#pragma omp parallel for schedule(dynamic, (n + threads * 4 - 1) / (threads * 4)) num_threads(threads)
        for (Index i = 0; i < n; ++i) {
          Index k = outer ? outer[i] : 0;
          const Index end = innerNnz ? (outer ? outer[i] : 0) + innerNnz[i] : (outer ? outer[i + 1] : mat.nonZeros());
          ResScalar sum0(0), sum1(0);
          for (; k < end; ++k) {
            sum0 += vals[k] * x[inds[k]];
            ++k;
            if (k < end) {
              sum1 += vals[k] * x[inds[k]];
            }
          }
          res.coeffRef(i, c) += alpha * (sum0 + sum1);
        }
      } else
#endif
      {
        for (Index i = 0; i < n; ++i) {
          Index k = outer ? outer[i] : 0;
          const Index end = innerNnz ? (outer ? outer[i] : 0) + innerNnz[i] : (outer ? outer[i + 1] : mat.nonZeros());
          // Two independent accumulators to break the dependency chain
          ResScalar sum0(0), sum1(0);
          for (; k < end; ++k) {
            sum0 += vals[k] * x[inds[k]];
            ++k;
            if (k < end) {
              sum1 += vals[k] * x[inds[k]];
            }
          }
          res.coeffRef(i, c) += alpha * (sum0 + sum1);
        }
      }
    } else {
      runColImpl(lhs, rhs, res, alpha, n, c, std::false_type());
    }
  }
 
  // Use fall-back path without direct access to rhs.
  template <typename RhsT>
  static void runColImpl(const SparseLhsType& lhs, const RhsT& rhs, DenseResType& res, const ResScalar& alpha, Index n,
                         Index c, std::false_type) {
    const Lhs& mat = lhs;
    const auto* vals = mat.valuePtr();
    const auto* inds = mat.innerIndexPtr();
    const auto* outer = mat.outerIndexPtr();
    const auto* innerNnz = mat.innerNonZeroPtr();
    // Non-unit rhs stride (or no direct access): use direct pointers for sparse side, coeff() for rhs
    for (Index i = 0; i < n; ++i) {
      Index k = outer ? outer[i] : 0;
      const Index end = innerNnz ? (outer ? outer[i] : 0) + innerNnz[i] : (outer ? outer[i + 1] : mat.nonZeros());
      ResScalar sum0(0), sum1(0);
      for (; k < end; ++k) {
        sum0 += vals[k] * rhs.coeff(inds[k], c);
        ++k;
        if (k < end) {
          sum1 += vals[k] * rhs.coeff(inds[k], c);
        }
      }
      res.coeffRef(i, c) += alpha * (sum0 + sum1);
    }
  }
 
  // Iterator fallback path
  static void runCol(const LhsEval& lhsEval, const SparseLhsType& /*lhs*/, const DenseRhsType& rhs, DenseResType& res,
                     const ResScalar& alpha, Index n, Index c, std::false_type /* has_compressed_storage */) {
#ifdef EIGEN_HAS_OPENMP
    Index threads = Eigen::nbThreads();
    if (threads > 1 && lhsEval.nonZerosEstimate() > 20000) {
#pragma omp parallel for schedule(dynamic, (n + threads * 4 - 1) / (threads * 4)) num_threads(threads)
      for (Index i = 0; i < n; ++i) processRow(lhsEval, rhs, res, alpha, i, c);
    } else
#endif
    {
      for (Index i = 0; i < n; ++i) processRow(lhsEval, rhs, res, alpha, i, c);
    }
  }
 
  static void processRow(const LhsEval& lhsEval, const DenseRhsType& rhs, DenseResType& res, const ResScalar& alpha,
                         Index i, Index col) {
    ResScalar tmp_a(0);
    ResScalar tmp_b(0);
    for (LhsInnerIterator it(lhsEval, i); it; ++it) {
      tmp_a += it.value() * rhs.coeff(it.index(), col);
      ++it;
      if (it) {
        tmp_b += it.value() * rhs.coeff(it.index(), col);
      }
    }
    res.coeffRef(i, col) += alpha * (tmp_a + tmp_b);
  }
};
 
// ColMajor, single column (ColPerCol=true): CSC SpMV
template <typename SparseLhsType, typename DenseRhsType, typename DenseResType, typename AlphaType>
struct sparse_time_dense_product_impl<SparseLhsType, DenseRhsType, DenseResType, AlphaType, ColMajor, true> {
  using Lhs = internal::remove_all_t<SparseLhsType>;
  using Rhs = internal::remove_all_t<DenseRhsType>;
  using Res = internal::remove_all_t<DenseResType>;
  using LhsEval = evaluator<Lhs>;
  using LhsInnerIterator = typename LhsEval::InnerIterator;
 
  static void run(const SparseLhsType& lhs, const DenseRhsType& rhs, DenseResType& res, const AlphaType& alpha) {
    runImpl(lhs, rhs, res, alpha, bool_constant<has_compressed_storage<Lhs>::value>());
  }
 
  // Direct pointer path: works for both compressed and non-compressed storage.
  static void runImpl(const SparseLhsType& lhs, const DenseRhsType& rhs, DenseResType& res, const AlphaType& alpha,
                      std::true_type /* has_compressed_storage */) {
    using LhsScalar = typename Lhs::Scalar;
    using StorageIndex = typename Lhs::StorageIndex;
    const Lhs& mat = lhs;
    const LhsScalar* vals = mat.valuePtr();
    const StorageIndex* inds = mat.innerIndexPtr();
    // Sparse vectors don't store outer indices.
    const auto* outer = mat.outerIndexPtr();
    const auto* innerNnz = mat.innerNonZeroPtr();
    // The fast result pointer path requires contiguous ColMajor result layout.
    // Transpose<ColMajor> reports innerStride()==1 but is actually RowMajor, so check both.
    EIGEN_IF_CONSTEXPR (!(Res::Flags & RowMajorBit)) {
      if (res.innerStride() == 1) {
        const Index n = lhs.outerSize();
        // The threaded scatter+reduce path relies on a thread_local scratch buffer for
        // host-thread safety (see below), so it is only available where thread_local is
        // usable; under EIGEN_AVOID_THREAD_LOCAL fall through to the serial scatter.
#if defined(EIGEN_HAS_OPENMP) && !defined(EIGEN_AVOID_THREAD_LOCAL)
        using ResScalar = typename Res::Scalar;
        const Index m = res.rows();
        const Index threads = Eigen::nbThreads();
        // Per-thread scratch + reduction: the natural per-column partition would
        // race on the output (writes to y[inds[k]] are scattered across rows),
        // so each thread accumulates into its own m-sized output buffer and the
        // results are summed at the end. Activated above the same 20000-nnz
        // threshold as the RowMajor kernel, plus a second gate on per-thread
        // scratch size: `threads * m` scalars are touched by the reduction,
        // and on tall / very-sparse matrices that can dwarf the SpMV cost --
        // require avg nnz per row >= threads so the reduction can't dominate.
        // `outer` is null for SparseVector lhs; the nnz-balanced partition needs
        // the outer-index array, so fall back to the serial scatter below.
        if (outer && threads > 1 && mat.nonZeros() > 20000 && mat.nonZeros() >= Index(threads) * m) {
          // Per-calling-thread persistent scratch (per template instantiation).
          // Grows monotonically; reused across calls. The buffer is left at
          // all-zeros after each call by folding the zero-out into the reduction
          // step, which avoids a separate init pass on every SpMV. `thread_local`
          // is required: two unrelated host threads concurrently calling
          // y = A*x would otherwise race on this static buffer (and a reallocating
          // grow on one would dangle the other's scratch_ptr).
          thread_local static std::vector<ResScalar> scratch_buf;
          const std::size_t need = static_cast<std::size_t>(threads) * static_cast<std::size_t>(m);
          if (scratch_buf.size() < need) scratch_buf.assign(need, ResScalar(0));
          ResScalar* scratch_ptr = scratch_buf.data();
          // nnz-balanced column partition: each thread t owns the contiguous
          // column range [part[t], part[t+1]). Deterministic mapping of j to
          // thread (required for bit-reproducible reduction below) AND
          // nnz-balanced load (dynamic scheduling would balance but break
          // determinism; static round-robin would be deterministic but
          // imbalanced on skewed matrices).
          std::vector<Index> part(static_cast<std::size_t>(threads) + 1);
          part[threads] = n;
          part[0] = 0;
          // Targets are monotonically increasing in t, so each lower_bound starts
          // from the previous result; total work is O(T + log n) rather than
          // T * log n.
          const StorageIndex* const part_last = outer + n + 1;
          const StorageIndex* part_lo = outer;
          for (Index t = 1; t < threads; ++t) {
            const Index target = (t * mat.nonZeros()) / threads;
            part_lo = std::lower_bound(part_lo, part_last, StorageIndex(target));
            part[t] = part_lo - outer;
          }
          for (Index c = 0; c < rhs.cols(); ++c) {
            typename Res::Scalar* y = res.data() + c * res.outerStride();
#pragma omp parallel for schedule(static, 1) num_threads(threads)
            for (Index t = 0; t < threads; ++t) {
              ResScalar* yt = scratch_ptr + t * m;
              const Index j_lo = part[t], j_hi = part[t + 1];
              for (Index j = j_lo; j < j_hi; ++j) {
                typename ScalarBinaryOpTraits<AlphaType, typename Rhs::Scalar>::ReturnType rhs_j(alpha *
                                                                                                 rhs.coeff(j, c));
                const Index start = outer ? outer[j] : 0;
                const Index end = innerNnz ? start + innerNnz[j] : (outer ? outer[j + 1] : mat.nonZeros());
                Index k = start;
                for (; k + 3 < end; k += 4) {
                  yt[inds[k]] += vals[k] * rhs_j;
                  yt[inds[k + 1]] += vals[k + 1] * rhs_j;
                  yt[inds[k + 2]] += vals[k + 2] * rhs_j;
                  yt[inds[k + 3]] += vals[k + 3] * rhs_j;
                }
                for (; k < end; ++k) yt[inds[k]] += vals[k] * rhs_j;
              }
            }
            // Reduce per-thread buffers into y AND zero them, so the next call
            // doesn't have to re-init. Process rows in cache-resident blocks: the
            // natural [t*m+i] scratch layout makes the cross-thread read for a
            // single row a stride-m gather (one cache line per thread, ~threads*m
            // bytes apart, unvectorizable). Blocking lets each thread's stripe be
            // swept as a unit-stride, vectorizable stream into a small per-block
            // accumulator. Each thread owns a contiguous range of row blocks
            // (static schedule) so the zero stores stay independent, and the
            // accumulator is summed in the exact t = 0..threads-1 order, keeping
            // the result bit-identical to the scalar reduction it replaces.
            constexpr Index kReduceBlock = 512;
#pragma omp parallel for schedule(static) num_threads(threads)
            for (Index i0 = 0; i0 < m; i0 += kReduceBlock) {
              const Index i1 = numext::mini(i0 + kReduceBlock, m);
              const Index len = i1 - i0;
              EIGEN_ALIGN_MAX ResScalar acc[kReduceBlock];
              for (Index ii = 0; ii < len; ++ii) acc[ii] = ResScalar(0);
              for (Index t = 0; t < threads; ++t) {
                ResScalar* row = scratch_ptr + t * m + i0;
                for (Index ii = 0; ii < len; ++ii) {
                  acc[ii] += row[ii];
                  row[ii] = ResScalar(0);
                }
              }
              for (Index ii = 0; ii < len; ++ii) y[i0 + ii] += acc[ii];
            }
          }
        } else
#endif
        {
          for (Index c = 0; c < rhs.cols(); ++c) {
            typename Res::Scalar* y = res.data() + c * res.outerStride();
            for (Index j = 0; j < n; ++j) {
              typename ScalarBinaryOpTraits<AlphaType, typename Rhs::Scalar>::ReturnType rhs_j(alpha * rhs.coeff(j, c));
              const Index start = outer ? outer[j] : 0;
              const Index end = innerNnz ? start + innerNnz[j] : (outer ? outer[j + 1] : mat.nonZeros());
              Index k = start;
              // 4-way unrolled scatter-add (no SIMD: writes are scattered)
              for (; k + 3 < end; k += 4) {
                y[inds[k]] += vals[k] * rhs_j;
                y[inds[k + 1]] += vals[k + 1] * rhs_j;
                y[inds[k + 2]] += vals[k + 2] * rhs_j;
                y[inds[k + 3]] += vals[k + 3] * rhs_j;
              }
              for (; k < end; ++k) y[inds[k]] += vals[k] * rhs_j;
            }
          }
        }
        return;
      }
    }
    // Non-unit result stride: use coeffRef() for result access
    for (Index c = 0; c < rhs.cols(); ++c) {
      for (Index j = 0; j < lhs.outerSize(); ++j) {
        typename ScalarBinaryOpTraits<AlphaType, typename Rhs::Scalar>::ReturnType rhs_j(alpha * rhs.coeff(j, c));
        const Index start = outer ? outer[j] : 0;
        const Index end = innerNnz ? start + innerNnz[j] : (outer ? outer[j + 1] : mat.nonZeros());
        for (Index k = start; k < end; ++k) res.coeffRef(inds[k], c) += vals[k] * rhs_j;
      }
    }
  }
 
  // Iterator-based fallback
  static void runImpl(const SparseLhsType& lhs, const DenseRhsType& rhs, DenseResType& res, const AlphaType& alpha,
                      std::false_type /* has_compressed_storage */) {
    LhsEval lhsEval(lhs);
    for (Index c = 0; c < rhs.cols(); ++c) {
      for (Index j = 0; j < lhs.outerSize(); ++j) {
        typename ScalarBinaryOpTraits<AlphaType, typename Rhs::Scalar>::ReturnType rhs_j(alpha * rhs.coeff(j, c));
        for (LhsInnerIterator it(lhsEval, j); it; ++it) res.coeffRef(it.index(), c) += it.value() * rhs_j;
      }
    }
  }
};
 
// RowMajor, multiple columns (ColPerCol=false): sparse * dense_matrix
template <typename SparseLhsType, typename DenseRhsType, typename DenseResType>
struct sparse_time_dense_product_impl<SparseLhsType, DenseRhsType, DenseResType, typename DenseResType::Scalar,
                                      RowMajor, false> {
  using Lhs = internal::remove_all_t<SparseLhsType>;
  using Res = internal::remove_all_t<DenseResType>;
  using LhsEval = evaluator<Lhs>;
  using LhsInnerIterator = typename LhsEval::InnerIterator;
 
  static constexpr bool IsCompressedLhs = has_compressed_storage<Lhs>::value;
 
  static void run(const SparseLhsType& lhs, const DenseRhsType& rhs, DenseResType& res,
                  const typename Res::Scalar& alpha) {
    Index n = lhs.rows();
    LhsEval lhsEval(lhs);
 
#ifdef EIGEN_HAS_OPENMP
    Index threads = Eigen::nbThreads();
    // This 20000 threshold has been found experimentally on 2D and 3D Poisson problems.
    // It basically represents the minimal amount of work to be done to be worth it.
    if (threads > 1 && lhsEval.nonZerosEstimate() * rhs.cols() > 20000) {
#pragma omp parallel for schedule(dynamic, (n + threads * 4 - 1) / (threads * 4)) num_threads(threads)
      for (Index i = 0; i < n; ++i) processRow(lhsEval, lhs, rhs, res, alpha, i, bool_constant<IsCompressedLhs>());
    } else
#endif
    {
      for (Index i = 0; i < n; ++i) processRow(lhsEval, lhs, rhs, res, alpha, i, bool_constant<IsCompressedLhs>());
    }
  }
 
  // Direct pointer path: works for both compressed and non-compressed storage.
  static void processRow(const LhsEval& /*lhsEval*/, const SparseLhsType& lhs, const DenseRhsType& rhs, Res& res,
                         const typename Res::Scalar& alpha, Index i, std::true_type /* has_compressed_storage */) {
    using LhsScalar = typename Lhs::Scalar;
    using StorageIndex = typename Lhs::StorageIndex;
    const Lhs& mat = lhs;
    const LhsScalar* vals = mat.valuePtr();
    const StorageIndex* inds = mat.innerIndexPtr();
    // Sparse vectors don't store outer indices.
    const Index start = mat.outerIndexPtr() ? mat.outerIndexPtr()[i] : 0;
    const auto* innerNnz = mat.innerNonZeroPtr();
    const Index end =
        innerNnz ? start + innerNnz[i] : (mat.outerIndexPtr() ? mat.outerIndexPtr()[i + 1] : mat.nonZeros());
    typename Res::RowXpr res_i(res.row(i));
    for (Index k = start; k < end; ++k) res_i += (alpha * vals[k]) * rhs.row(inds[k]);
  }
 
  static void processRow(const LhsEval& lhsEval, const SparseLhsType& /*lhs*/, const DenseRhsType& rhs, Res& res,
                         const typename Res::Scalar& alpha, Index i, std::false_type /* has_compressed_storage */) {
    typename Res::RowXpr res_i(res.row(i));
    for (LhsInnerIterator it(lhsEval, i); it; ++it) res_i += (alpha * it.value()) * rhs.row(it.index());
  }
};
 
// ColMajor, multiple columns (ColPerCol=false): sparse * dense_matrix
template <typename SparseLhsType, typename DenseRhsType, typename DenseResType>
struct sparse_time_dense_product_impl<SparseLhsType, DenseRhsType, DenseResType, typename DenseResType::Scalar,
                                      ColMajor, false> {
  using Lhs = internal::remove_all_t<SparseLhsType>;
  using Rhs = internal::remove_all_t<DenseRhsType>;
  using Res = internal::remove_all_t<DenseResType>;
  using LhsInnerIterator = typename evaluator<Lhs>::InnerIterator;
 
  static void run(const SparseLhsType& lhs, const DenseRhsType& rhs, DenseResType& res,
                  const typename Res::Scalar& alpha) {
    runImpl(lhs, rhs, res, alpha, bool_constant<has_compressed_storage<Lhs>::value>());
  }
 
  // Direct pointer path: works for both compressed and non-compressed storage.
  static void runImpl(const SparseLhsType& lhs, const DenseRhsType& rhs, DenseResType& res,
                      const typename Res::Scalar& alpha, std::true_type /* has_compressed_storage */) {
    using LhsScalar = typename Lhs::Scalar;
    using StorageIndex = typename Lhs::StorageIndex;
    const Lhs& mat = lhs;
    const LhsScalar* vals = mat.valuePtr();
    const StorageIndex* inds = mat.innerIndexPtr();
    // Sparse vectors don't store outer indices.
    const auto* outer = mat.outerIndexPtr();
    const auto* innerNnz = mat.innerNonZeroPtr();
    for (Index j = 0; j < lhs.outerSize(); ++j) {
      typename Rhs::ConstRowXpr rhs_j(rhs.row(j));
      const Index start = outer ? outer[j] : 0;
      const Index end = innerNnz ? start + innerNnz[j] : (outer ? outer[j + 1] : mat.nonZeros());
      for (Index k = start; k < end; ++k) res.row(inds[k]) += (alpha * vals[k]) * rhs_j;
    }
  }
 
  static void runImpl(const SparseLhsType& lhs, const DenseRhsType& rhs, DenseResType& res,
                      const typename Res::Scalar& alpha, std::false_type /* has_compressed_storage */) {
    evaluator<Lhs> lhsEval(lhs);
    for (Index j = 0; j < lhs.outerSize(); ++j) {
      typename Rhs::ConstRowXpr rhs_j(rhs.row(j));
      for (LhsInnerIterator it(lhsEval, j); it; ++it) res.row(it.index()) += (alpha * it.value()) * rhs_j;
    }
  }
};
 
template <typename SparseLhsType, typename DenseRhsType, typename DenseResType, typename AlphaType>
inline void sparse_time_dense_product(const SparseLhsType& lhs, const DenseRhsType& rhs, DenseResType& res,
                                      const AlphaType& alpha) {
  sparse_time_dense_product_impl<SparseLhsType, DenseRhsType, DenseResType, AlphaType>::run(lhs, rhs, res, alpha);
}
 
}  // end namespace internal
 
namespace internal {
 
template <typename Lhs, typename Rhs, int ProductType>
struct generic_product_impl<Lhs, Rhs, SparseShape, DenseShape, ProductType>
    : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, SparseShape, DenseShape, ProductType> > {
  using Scalar = typename Product<Lhs, Rhs>::Scalar;
 
  template <typename Dest>
  static void scaleAndAddTo(Dest& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha) {
    using LhsNested = typename nested_eval<Lhs, ((Rhs::Flags & RowMajorBit) == 0) ? 1 : Rhs::ColsAtCompileTime>::type;
    using RhsNested = typename nested_eval<Rhs, ((Lhs::Flags & RowMajorBit) == 0) ? 1 : Dynamic>::type;
    LhsNested lhsNested(lhs);
    RhsNested rhsNested(rhs);
    internal::sparse_time_dense_product(lhsNested, rhsNested, dst, alpha);
  }
};
 
template <typename Lhs, typename Rhs, int ProductType>
struct generic_product_impl<Lhs, Rhs, SparseTriangularShape, DenseShape, ProductType>
    : generic_product_impl<Lhs, Rhs, SparseShape, DenseShape, ProductType> {};
 
template <typename Lhs, typename Rhs, int ProductType>
struct generic_product_impl<Lhs, Rhs, DenseShape, SparseShape, ProductType>
    : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, DenseShape, SparseShape, ProductType> > {
  using Scalar = typename Product<Lhs, Rhs>::Scalar;
 
  template <typename Dst>
  static void scaleAndAddTo(Dst& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha) {
    using LhsNested = typename nested_eval<Lhs, ((Rhs::Flags & RowMajorBit) == 0) ? Dynamic : 1>::type;
    using RhsNested =
        typename nested_eval<Rhs, ((Lhs::Flags & RowMajorBit) == RowMajorBit) ? 1 : Lhs::RowsAtCompileTime>::type;
    LhsNested lhsNested(lhs);
    RhsNested rhsNested(rhs);
 
    // transpose everything
    Transpose<Dst> dstT(dst);
    internal::sparse_time_dense_product(rhsNested.transpose(), lhsNested.transpose(), dstT, alpha);
  }
};
 
template <typename Lhs, typename Rhs, int ProductType>
struct generic_product_impl<Lhs, Rhs, DenseShape, SparseTriangularShape, ProductType>
    : generic_product_impl<Lhs, Rhs, DenseShape, SparseShape, ProductType> {};
 
template <typename LhsT, typename RhsT, bool NeedToTranspose>
struct sparse_dense_outer_product_evaluator {
 protected:
  using Lhs1 = std::conditional_t<NeedToTranspose, RhsT, LhsT>;
  using ActualRhs = std::conditional_t<NeedToTranspose, LhsT, RhsT>;
  using ProdXprType = Product<LhsT, RhsT, DefaultProduct>;
 
  // if the actual left-hand side is a dense vector,
  // then build a sparse-view so that we can seamlessly iterate over it.
  using ActualLhs = std::conditional_t<std::is_same<typename internal::traits<Lhs1>::StorageKind, Sparse>::value, Lhs1,
                                       SparseView<Lhs1>>;
  using LhsArg = std::conditional_t<std::is_same<typename internal::traits<Lhs1>::StorageKind, Sparse>::value,
                                    const Lhs1&, SparseView<Lhs1>>;
 
  using LhsEval = evaluator<ActualLhs>;
  using RhsEval = evaluator<ActualRhs>;
  using LhsIterator = typename evaluator<ActualLhs>::InnerIterator;
  using Scalar = typename ProdXprType::Scalar;
 
 public:
  enum { Flags = NeedToTranspose ? RowMajorBit : 0, CoeffReadCost = HugeCost };
 
  class InnerIterator : public LhsIterator {
   public:
    InnerIterator(const sparse_dense_outer_product_evaluator& xprEval, Index outer)
        : LhsIterator(xprEval.m_lhsXprImpl, 0),
          m_outer(outer),
          m_empty(false),
          m_factor(get(xprEval.m_rhsXprImpl, outer, typename internal::traits<ActualRhs>::StorageKind())) {}
 
    EIGEN_STRONG_INLINE Index outer() const { return m_outer; }
    EIGEN_STRONG_INLINE Index row() const { return NeedToTranspose ? m_outer : LhsIterator::index(); }
    EIGEN_STRONG_INLINE Index col() const { return NeedToTranspose ? LhsIterator::index() : m_outer; }
 
    EIGEN_STRONG_INLINE Scalar value() const { return LhsIterator::value() * m_factor; }
    EIGEN_STRONG_INLINE operator bool() const { return LhsIterator::operator bool() && (!m_empty); }
 
   protected:
    Scalar get(const RhsEval& rhs, Index outer, Dense = Dense()) const { return rhs.coeff(outer); }
 
    Scalar get(const RhsEval& rhs, Index outer, Sparse = Sparse()) {
      typename RhsEval::InnerIterator it(rhs, outer);
      if (it && it.index() == 0 && it.value() != Scalar(0)) return it.value();
      m_empty = true;
      return Scalar(0);
    }
 
    Index m_outer;
    bool m_empty;
    Scalar m_factor;
  };
 
  sparse_dense_outer_product_evaluator(const Lhs1& lhs, const ActualRhs& rhs)
      : m_lhs(lhs), m_lhsXprImpl(m_lhs), m_rhsXprImpl(rhs) {
    EIGEN_INTERNAL_CHECK_COST_VALUE(CoeffReadCost);
  }
 
  // transpose case
  sparse_dense_outer_product_evaluator(const ActualRhs& rhs, const Lhs1& lhs)
      : m_lhs(lhs), m_lhsXprImpl(m_lhs), m_rhsXprImpl(rhs) {
    EIGEN_INTERNAL_CHECK_COST_VALUE(CoeffReadCost);
  }
 
 protected:
  const LhsArg m_lhs;
  evaluator<ActualLhs> m_lhsXprImpl;
  evaluator<ActualRhs> m_rhsXprImpl;
};
 
// sparse * dense outer product
template <typename Lhs, typename Rhs>
struct product_evaluator<Product<Lhs, Rhs, DefaultProduct>, OuterProduct, SparseShape, DenseShape>
    : sparse_dense_outer_product_evaluator<Lhs, Rhs, Lhs::IsRowMajor> {
  using Base = sparse_dense_outer_product_evaluator<Lhs, Rhs, Lhs::IsRowMajor>;
 
  using XprType = Product<Lhs, Rhs>;
  using PlainObject = typename XprType::PlainObject;
 
  explicit product_evaluator(const XprType& xpr) : Base(xpr.lhs(), xpr.rhs()) {}
};
 
template <typename Lhs, typename Rhs>
struct product_evaluator<Product<Lhs, Rhs, DefaultProduct>, OuterProduct, DenseShape, SparseShape>
    : sparse_dense_outer_product_evaluator<Lhs, Rhs, Rhs::IsRowMajor> {
  using Base = sparse_dense_outer_product_evaluator<Lhs, Rhs, Rhs::IsRowMajor>;
 
  using XprType = Product<Lhs, Rhs>;
  using PlainObject = typename XprType::PlainObject;
 
  explicit product_evaluator(const XprType& xpr) : Base(xpr.lhs(), xpr.rhs()) {}
};
 
}  // end namespace internal
 
}  // end namespace Eigen
 
#endif  // EIGEN_SPARSEDENSEPRODUCT_H