test/sparse_basic.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2011 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2008 Daniel Gomez Ferro <dgomezferro@gmail.com>
// Copyright (C) 2013 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@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_SPARSE_TEST_INCLUDED_FROM_SPARSE_EXTRA
static long g_realloc_count = 0;
#define EIGEN_SPARSE_COMPRESSED_STORAGE_REALLOCATE_PLUGIN g_realloc_count++;

static long g_dense_op_sparse_count = 0;
#define EIGEN_SPARSE_ASSIGNMENT_FROM_DENSE_OP_SPARSE_PLUGIN g_dense_op_sparse_count++;
#define EIGEN_SPARSE_ASSIGNMENT_FROM_SPARSE_ADD_DENSE_PLUGIN g_dense_op_sparse_count += 10;
#define EIGEN_SPARSE_ASSIGNMENT_FROM_SPARSE_SUB_DENSE_PLUGIN g_dense_op_sparse_count += 20;
#endif

#include "sparse.h"

template <typename SparseMatrixType>
void sparse_basic(const SparseMatrixType& ref) {
  typedef typename SparseMatrixType::StorageIndex StorageIndex;
  typedef Matrix<StorageIndex, 2, 1> Vector2;

  const Index rows = ref.rows();
  const Index cols = ref.cols();
  const Index inner = ref.innerSize();
  const Index outer = ref.outerSize();

  typedef typename SparseMatrixType::Scalar Scalar;
  typedef typename SparseMatrixType::RealScalar RealScalar;
  enum { Flags = SparseMatrixType::Flags };

  double density = (std::max)(8. / (rows * cols), 0.01);
  typedef Matrix<Scalar, Dynamic, Dynamic> DenseMatrix;
  typedef Matrix<Scalar, Dynamic, 1> DenseVector;
  typedef Matrix<Scalar, Dynamic, Dynamic, SparseMatrixType::IsRowMajor ? RowMajor : ColMajor> CompatibleDenseMatrix;
  Scalar eps = Scalar(1e-6);

  Scalar s1 = internal::random<Scalar>();
  {
    SparseMatrixType m(rows, cols);
    DenseMatrix refMat = DenseMatrix::Zero(rows, cols);
    DenseVector vec1 = DenseVector::Random(rows);

    std::vector<Vector2> zeroCoords;
    std::vector<Vector2> nonzeroCoords;
    initSparse<Scalar>(density, refMat, m, 0, &zeroCoords, &nonzeroCoords);

    // test coeff and coeffRef
    for (std::size_t i = 0; i < zeroCoords.size(); ++i) {
      VERIFY_IS_MUCH_SMALLER_THAN(m.coeff(zeroCoords[i].x(), zeroCoords[i].y()), eps);
      if (internal::is_same<SparseMatrixType, SparseMatrix<Scalar, Flags>>::value)
        VERIFY_RAISES_ASSERT(m.coeffRef(zeroCoords[i].x(), zeroCoords[i].y()) = 5);
    }
    VERIFY_IS_APPROX(m, refMat);

    if (!nonzeroCoords.empty()) {
      m.coeffRef(nonzeroCoords[0].x(), nonzeroCoords[0].y()) = Scalar(5);
      refMat.coeffRef(nonzeroCoords[0].x(), nonzeroCoords[0].y()) = Scalar(5);
    }

    VERIFY_IS_APPROX(m, refMat);

    // test assertion
    VERIFY_RAISES_ASSERT(m.coeffRef(-1, 1) = 0);
    VERIFY_RAISES_ASSERT(m.coeffRef(0, m.cols()) = 0);
  }

  // test insert (inner random)
  {
    DenseMatrix m1(rows, cols);
    m1.setZero();
    SparseMatrixType m2(rows, cols);
    bool call_reserve = internal::random<int>() % 2;
    Index nnz = internal::random<int>(1, int(rows) / 2);
    if (call_reserve) {
      if (internal::random<int>() % 2)
        m2.reserve(VectorXi::Constant(m2.outerSize(), int(nnz)));
      else
        m2.reserve(m2.outerSize() * nnz);
    }
    g_realloc_count = 0;
    for (Index j = 0; j < cols; ++j) {
      for (Index k = 0; k < nnz; ++k) {
        Index i = internal::random<Index>(0, rows - 1);
        if (m1.coeff(i, j) == Scalar(0)) {
          Scalar v = internal::random<Scalar>();
          if (v == Scalar(0)) v = Scalar(1);
          m1(i, j) = v;
          m2.insert(i, j) = v;
        }
      }
    }

    if (call_reserve && !SparseMatrixType::IsRowMajor) {
      VERIFY(g_realloc_count == 0);
    }

    VERIFY_IS_APPROX(m2, m1);
  }

  // test insert (fully random)
  {
    DenseMatrix m1(rows, cols);
    m1.setZero();
    SparseMatrixType m2(rows, cols);
    if (internal::random<int>() % 2) m2.reserve(VectorXi::Constant(m2.outerSize(), 2));
    for (int k = 0; k < rows * cols; ++k) {
      Index i = internal::random<Index>(0, rows - 1);
      Index j = internal::random<Index>(0, cols - 1);
      if ((m1.coeff(i, j) == Scalar(0)) && (internal::random<int>() % 2)) {
        Scalar v = internal::random<Scalar>();
        if (v == Scalar(0)) v = Scalar(1);
        m1(i, j) = v;
        m2.insert(i, j) = v;
      } else {
        Scalar v = internal::random<Scalar>();
        if (v == Scalar(0)) v = Scalar(1);
        m1(i, j) = v;
        m2.coeffRef(i, j) = v;
      }
    }
    VERIFY_IS_APPROX(m2, m1);
  }

  // test insert (un-compressed)
  for (int mode = 0; mode < 4; ++mode) {
    DenseMatrix m1(rows, cols);
    m1.setZero();
    SparseMatrixType m2(rows, cols);
    VectorXi r(VectorXi::Constant(m2.outerSize(),
                                  ((mode % 2) == 0) ? int(m2.innerSize()) : std::max<int>(1, int(m2.innerSize()) / 8)));
    m2.reserve(r);
    for (Index k = 0; k < rows * cols; ++k) {
      Index i = internal::random<Index>(0, rows - 1);
      Index j = internal::random<Index>(0, cols - 1);
      if (m1.coeff(i, j) == Scalar(0)) {
        Scalar v = internal::random<Scalar>();
        if (v == Scalar(0)) v = Scalar(1);
        m1(i, j) = v;
        m2.insert(i, j) = v;
      }
      if (mode == 3) m2.reserve(r);
    }
    if (internal::random<int>() % 2) m2.makeCompressed();
    VERIFY_IS_APPROX(m2, m1);
  }

  // test removeOuterVectors / insertEmptyOuterVectors
  {
    for (int mode = 0; mode < 4; mode++) {
      CompatibleDenseMatrix m1(rows, cols);
      m1.setZero();
      SparseMatrixType m2(rows, cols);
      Vector<Index, Dynamic> reserveSizes(outer);
      for (Index j = 0; j < outer; j++) reserveSizes(j) = internal::random<Index>(1, inner - 1);
      m2.reserve(reserveSizes);
      for (Index j = 0; j < outer; j++) {
        Index i = internal::random<Index>(0, inner - 1);
        Scalar val = internal::random<Scalar>();
        m1.coeffRefByOuterInner(j, i) = val;
        m2.insertByOuterInner(j, i) = val;
      }
      if (mode % 2 == 0) m2.makeCompressed();

      if (mode < 2) {
        Index num = internal::random<Index>(0, outer - 1);
        Index start = internal::random<Index>(0, outer - num);

        Index newRows = SparseMatrixType::IsRowMajor ? rows - num : rows;
        Index newCols = SparseMatrixType::IsRowMajor ? cols : cols - num;

        CompatibleDenseMatrix m3(newRows, newCols);
        m3.setConstant(Scalar(NumTraits<RealScalar>::quiet_NaN()));

        if (SparseMatrixType::IsRowMajor) {
          m3.topRows(start) = m1.topRows(start);
          m3.bottomRows(newRows - start) = m1.bottomRows(newRows - start);
        } else {
          m3.leftCols(start) = m1.leftCols(start);
          m3.rightCols(newCols - start) = m1.rightCols(newCols - start);
        }

        SparseMatrixType m4 = m2;
        m4.removeOuterVectors(start, num);

        VERIFY_IS_CWISE_EQUAL(m3, m4.toDense());
      } else {
        Index num = internal::random<Index>(0, outer - 1);
        Index start = internal::random<Index>(0, outer - 1);

        Index newRows = SparseMatrixType::IsRowMajor ? rows + num : rows;
        Index newCols = SparseMatrixType::IsRowMajor ? cols : cols + num;

        CompatibleDenseMatrix m3(newRows, newCols);
        m3.setConstant(Scalar(NumTraits<RealScalar>::quiet_NaN()));

        if (SparseMatrixType::IsRowMajor) {
          m3.topRows(start) = m1.topRows(start);
          m3.middleRows(start, num).setZero();
          m3.bottomRows(rows - start) = m1.bottomRows(rows - start);
        } else {
          m3.leftCols(start) = m1.leftCols(start);
          m3.middleCols(start, num).setZero();
          m3.rightCols(cols - start) = m1.rightCols(cols - start);
        }

        SparseMatrixType m4 = m2;
        m4.insertEmptyOuterVectors(start, num);

        VERIFY_IS_CWISE_EQUAL(m3, m4.toDense());
      }
    }
  }

  // test sort
  if (inner > 1) {
    bool StorageOrdersMatch = int(DenseMatrix::IsRowMajor) == int(SparseMatrixType::IsRowMajor);
    DenseMatrix m1(rows, cols);
    m1.setZero();
    SparseMatrixType m2(rows, cols);
    // generate random inner indices with no repeats
    Vector<Index, Dynamic> innerIndices(inner);
    innerIndices.setLinSpaced(inner, 0, inner - 1);
    std::random_device rd;
    std::mt19937 g(rd());
    for (Index j = 0; j < outer; j++) {
      std::shuffle(innerIndices.begin(), innerIndices.end(), g);
      Index nzj = internal::random<Index>(2, inner / 2);
      for (Index k = 0; k < nzj; k++) {
        Index i = innerIndices[k];
        Scalar val = internal::random<Scalar>();
        m1.coeffRefByOuterInner(StorageOrdersMatch ? j : i, StorageOrdersMatch ? i : j) = val;
        m2.insertByOuterInner(j, i) = val;
      }
    }

    VERIFY_IS_APPROX(m2, m1);
    // sort wrt greater
    m2.template sortInnerIndices<std::greater<>>();
    // verify that all inner vectors are not sorted wrt less
    VERIFY_IS_EQUAL(m2.template innerIndicesAreSorted<std::less<>>(), 0);
    // verify that all inner vectors are sorted wrt greater
    VERIFY_IS_EQUAL(m2.template innerIndicesAreSorted<std::greater<>>(), m2.outerSize());
    // verify that sort does not change evaluation
    VERIFY_IS_APPROX(m2, m1);
    // sort wrt less
    m2.template sortInnerIndices<std::less<>>();
    // verify that all inner vectors are sorted wrt less
    VERIFY_IS_EQUAL(m2.template innerIndicesAreSorted<std::less<>>(), m2.outerSize());
    // verify that all inner vectors are not sorted wrt greater
    VERIFY_IS_EQUAL(m2.template innerIndicesAreSorted<std::greater<>>(), 0);
    // verify that sort does not change evaluation
    VERIFY_IS_APPROX(m2, m1);

    m2.makeCompressed();
    // sort wrt greater
    m2.template sortInnerIndices<std::greater<>>();
    // verify that all inner vectors are not sorted wrt less
    VERIFY_IS_EQUAL(m2.template innerIndicesAreSorted<std::less<>>(), 0);
    // verify that all inner vectors are sorted wrt greater
    VERIFY_IS_EQUAL(m2.template innerIndicesAreSorted<std::greater<>>(), m2.outerSize());
    // verify that sort does not change evaluation
    VERIFY_IS_APPROX(m2, m1);
    // sort wrt less
    m2.template sortInnerIndices<std::less<>>();
    // verify that all inner vectors are sorted wrt less
    VERIFY_IS_EQUAL(m2.template innerIndicesAreSorted<std::less<>>(), m2.outerSize());
    // verify that all inner vectors are not sorted wrt greater
    VERIFY_IS_EQUAL(m2.template innerIndicesAreSorted<std::greater<>>(), 0);
    // verify that sort does not change evaluation
    VERIFY_IS_APPROX(m2, m1);
  }

  // test basic computations
  {
    DenseMatrix refM1 = DenseMatrix::Zero(rows, cols);
    DenseMatrix refM2 = DenseMatrix::Zero(rows, cols);
    DenseMatrix refM3 = DenseMatrix::Zero(rows, cols);
    DenseMatrix refM4 = DenseMatrix::Zero(rows, cols);
    SparseMatrixType m1(rows, cols);
    SparseMatrixType m2(rows, cols);
    SparseMatrixType m3(rows, cols);
    SparseMatrixType m4(rows, cols);
    initSparse<Scalar>(density, refM1, m1);
    initSparse<Scalar>(density, refM2, m2);
    initSparse<Scalar>(density, refM3, m3);
    initSparse<Scalar>(density, refM4, m4);

    if (internal::random<bool>()) m1.makeCompressed();

    Index m1_nnz = m1.nonZeros();

    VERIFY_IS_APPROX(m1 * s1, refM1 * s1);
    VERIFY_IS_APPROX(m1 + m2, refM1 + refM2);
    VERIFY_IS_APPROX(m1 + m2 + m3, refM1 + refM2 + refM3);
    VERIFY_IS_APPROX(m3.cwiseProduct(m1 + m2), refM3.cwiseProduct(refM1 + refM2));
    VERIFY_IS_APPROX(m1 * s1 - m2, refM1 * s1 - refM2);
    VERIFY_IS_APPROX(m4 = m1 / s1, refM1 / s1);
    VERIFY_IS_EQUAL(m4.nonZeros(), m1_nnz);

    if (SparseMatrixType::IsRowMajor)
      VERIFY_IS_APPROX(m1.innerVector(0).dot(refM2.row(0)), refM1.row(0).dot(refM2.row(0)));
    else
      VERIFY_IS_APPROX(m1.innerVector(0).dot(refM2.col(0)), refM1.col(0).dot(refM2.col(0)));

    DenseVector rv = DenseVector::Random(m1.cols());
    DenseVector cv = DenseVector::Random(m1.rows());
    Index r = internal::random<Index>(0, m1.rows() - 2);
    Index c = internal::random<Index>(0, m1.cols() - 1);
    VERIFY_IS_APPROX((m1.template block<1, Dynamic>(r, 0, 1, m1.cols()).dot(rv)), refM1.row(r).dot(rv));
    VERIFY_IS_APPROX(m1.row(r).dot(rv), refM1.row(r).dot(rv));
    VERIFY_IS_APPROX(m1.col(c).dot(cv), refM1.col(c).dot(cv));

    VERIFY_IS_APPROX(m1.conjugate(), refM1.conjugate());
    VERIFY_IS_APPROX(m1.real(), refM1.real());

    refM4.setRandom();
    // sparse cwise* dense
    VERIFY_IS_APPROX(m3.cwiseProduct(refM4), refM3.cwiseProduct(refM4));
    // dense cwise* sparse
    VERIFY_IS_APPROX(refM4.cwiseProduct(m3), refM4.cwiseProduct(refM3));
    //     VERIFY_IS_APPROX(m3.cwise()/refM4, refM3.cwise()/refM4);

    // mixed sparse-dense
    VERIFY_IS_APPROX(refM4 + m3, refM4 + refM3);
    VERIFY_IS_APPROX(m3 + refM4, refM3 + refM4);
    VERIFY_IS_APPROX(refM4 - m3, refM4 - refM3);
    VERIFY_IS_APPROX(m3 - refM4, refM3 - refM4);
    VERIFY_IS_APPROX((RealScalar(0.5) * refM4 + RealScalar(0.5) * m3).eval(),
                     RealScalar(0.5) * refM4 + RealScalar(0.5) * refM3);
    VERIFY_IS_APPROX((RealScalar(0.5) * refM4 + m3 * RealScalar(0.5)).eval(),
                     RealScalar(0.5) * refM4 + RealScalar(0.5) * refM3);
    VERIFY_IS_APPROX((RealScalar(0.5) * refM4 + m3.cwiseProduct(m3)).eval(),
                     RealScalar(0.5) * refM4 + refM3.cwiseProduct(refM3));

    VERIFY_IS_APPROX((RealScalar(0.5) * refM4 + RealScalar(0.5) * m3).eval(),
                     RealScalar(0.5) * refM4 + RealScalar(0.5) * refM3);
    VERIFY_IS_APPROX((RealScalar(0.5) * refM4 + m3 * RealScalar(0.5)).eval(),
                     RealScalar(0.5) * refM4 + RealScalar(0.5) * refM3);
    VERIFY_IS_APPROX((RealScalar(0.5) * refM4 + (m3 + m3)).eval(), RealScalar(0.5) * refM4 + (refM3 + refM3));
    VERIFY_IS_APPROX(((refM3 + m3) + RealScalar(0.5) * m3).eval(), RealScalar(0.5) * refM3 + (refM3 + refM3));
    VERIFY_IS_APPROX((RealScalar(0.5) * refM4 + (refM3 + m3)).eval(), RealScalar(0.5) * refM4 + (refM3 + refM3));
    VERIFY_IS_APPROX((RealScalar(0.5) * refM4 + (m3 + refM3)).eval(), RealScalar(0.5) * refM4 + (refM3 + refM3));

    VERIFY_IS_APPROX(m1.sum(), refM1.sum());

    m4 = m1;
    refM4 = m4;

    VERIFY_IS_APPROX(m1 *= s1, refM1 *= s1);
    VERIFY_IS_EQUAL(m1.nonZeros(), m1_nnz);
    VERIFY_IS_APPROX(m1 /= s1, refM1 /= s1);
    VERIFY_IS_EQUAL(m1.nonZeros(), m1_nnz);

    VERIFY_IS_APPROX(m1 += m2, refM1 += refM2);
    VERIFY_IS_APPROX(m1 -= m2, refM1 -= refM2);

    refM3 = refM1;

    VERIFY_IS_APPROX(refM1 += m2, refM3 += refM2);
    VERIFY_IS_APPROX(refM1 -= m2, refM3 -= refM2);

    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 = m2 + refM4, refM3 = refM2 + refM4);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 10);
    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 += m2 + refM4, refM3 += refM2 + refM4);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 1);
    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 -= m2 + refM4, refM3 -= refM2 + refM4);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 1);
    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 = refM4 + m2, refM3 = refM2 + refM4);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 1);
    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 += refM4 + m2, refM3 += refM2 + refM4);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 1);
    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 -= refM4 + m2, refM3 -= refM2 + refM4);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 1);

    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 = m2 - refM4, refM3 = refM2 - refM4);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 20);
    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 += m2 - refM4, refM3 += refM2 - refM4);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 1);
    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 -= m2 - refM4, refM3 -= refM2 - refM4);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 1);
    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 = refM4 - m2, refM3 = refM4 - refM2);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 1);
    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 += refM4 - m2, refM3 += refM4 - refM2);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 1);
    g_dense_op_sparse_count = 0;
    VERIFY_IS_APPROX(refM1 -= refM4 - m2, refM3 -= refM4 - refM2);
    VERIFY_IS_EQUAL(g_dense_op_sparse_count, 1);
    refM3 = m3;

    if (rows >= 2 && cols >= 2) {
      VERIFY_RAISES_ASSERT(m1 += m1.innerVector(0));
      VERIFY_RAISES_ASSERT(m1 -= m1.innerVector(0));
      VERIFY_RAISES_ASSERT(refM1 -= m1.innerVector(0));
      VERIFY_RAISES_ASSERT(refM1 += m1.innerVector(0));
    }
    m1 = m4;
    refM1 = refM4;

    // test aliasing
    VERIFY_IS_APPROX((m1 = -m1), (refM1 = -refM1));
    VERIFY_IS_EQUAL(m1.nonZeros(), m1_nnz);
    m1 = m4;
    refM1 = refM4;
    VERIFY_IS_APPROX((m1 = m1.transpose()), (refM1 = refM1.transpose().eval()));
    VERIFY_IS_EQUAL(m1.nonZeros(), m1_nnz);
    m1 = m4;
    refM1 = refM4;
    VERIFY_IS_APPROX((m1 = -m1.transpose()), (refM1 = -refM1.transpose().eval()));
    VERIFY_IS_EQUAL(m1.nonZeros(), m1_nnz);
    m1 = m4;
    refM1 = refM4;
    VERIFY_IS_APPROX((m1 += -m1), (refM1 += -refM1));
    VERIFY_IS_EQUAL(m1.nonZeros(), m1_nnz);
    m1 = m4;
    refM1 = refM4;

    if (m1.isCompressed()) {
      VERIFY_IS_APPROX(m1.coeffs().sum(), m1.sum());
      m1.coeffs() += s1;
      for (Index j = 0; j < m1.outerSize(); ++j)
        for (typename SparseMatrixType::InnerIterator it(m1, j); it; ++it) refM1(it.row(), it.col()) += s1;
      VERIFY_IS_APPROX(m1, refM1);
    }

    // and/or
    {
      typedef SparseMatrix<bool, SparseMatrixType::Options, typename SparseMatrixType::StorageIndex> SpBool;
      SpBool mb1 = m1.real().template cast<bool>();
      SpBool mb2 = m2.real().template cast<bool>();
      VERIFY_IS_EQUAL(mb1.template cast<int>().sum(), refM1.real().template cast<bool>().count());
      VERIFY_IS_EQUAL((mb1 && mb2).template cast<int>().sum(),
                      (refM1.real().template cast<bool>() && refM2.real().template cast<bool>()).count());
      VERIFY_IS_EQUAL((mb1 || mb2).template cast<int>().sum(),
                      (refM1.real().template cast<bool>() || refM2.real().template cast<bool>()).count());
      SpBool mb3 = mb1 && mb2;
      if (mb1.coeffs().all() && mb2.coeffs().all()) {
        VERIFY_IS_EQUAL(mb3.nonZeros(),
                        (refM1.real().template cast<bool>() && refM2.real().template cast<bool>()).count());
      }
    }
  }

  // test reverse iterators
  {
    DenseMatrix refMat2 = DenseMatrix::Zero(rows, cols);
    SparseMatrixType m2(rows, cols);
    initSparse<Scalar>(density, refMat2, m2);
    std::vector<Scalar> ref_value(m2.innerSize());
    std::vector<Index> ref_index(m2.innerSize());
    if (internal::random<bool>()) m2.makeCompressed();
    for (Index j = 0; j < m2.outerSize(); ++j) {
      Index count_forward = 0;

      for (typename SparseMatrixType::InnerIterator it(m2, j); it; ++it) {
        ref_value[ref_value.size() - 1 - count_forward] = it.value();
        ref_index[ref_index.size() - 1 - count_forward] = it.index();
        count_forward++;
      }
      Index count_reverse = 0;
      for (typename SparseMatrixType::ReverseInnerIterator it(m2, j); it; --it) {
        VERIFY_IS_APPROX(std::abs(ref_value[ref_value.size() - count_forward + count_reverse]) + 1,
                         std::abs(it.value()) + 1);
        VERIFY_IS_EQUAL(ref_index[ref_index.size() - count_forward + count_reverse], it.index());
        count_reverse++;
      }
      VERIFY_IS_EQUAL(count_forward, count_reverse);
    }
  }

  // test transpose
  {
    DenseMatrix refMat2 = DenseMatrix::Zero(rows, cols);
    SparseMatrixType m2(rows, cols);
    initSparse<Scalar>(density, refMat2, m2);
    VERIFY_IS_APPROX(m2.transpose().eval(), refMat2.transpose().eval());
    VERIFY_IS_APPROX(m2.transpose(), refMat2.transpose());

    VERIFY_IS_APPROX(SparseMatrixType(m2.adjoint()), refMat2.adjoint());

    // check isApprox handles opposite storage order
    typename Transpose<SparseMatrixType>::PlainObject m3(m2);
    VERIFY(m2.isApprox(m3));
  }

  // test prune
  {
    SparseMatrixType m2(rows, cols);
    DenseMatrix refM2(rows, cols);
    refM2.setZero();
    int countFalseNonZero = 0;
    int countTrueNonZero = 0;
    m2.reserve(VectorXi::Constant(m2.outerSize(), int(m2.innerSize())));
    for (Index j = 0; j < m2.cols(); ++j) {
      for (Index i = 0; i < m2.rows(); ++i) {
        float x = internal::random<float>(0, 1);
        if (x < 0.1f) {
          // do nothing
        } else if (x < 0.5f) {
          countFalseNonZero++;
          m2.insert(i, j) = Scalar(0);
        } else {
          countTrueNonZero++;
          m2.insert(i, j) = Scalar(1);
          refM2(i, j) = Scalar(1);
        }
      }
    }
    if (internal::random<bool>()) m2.makeCompressed();
    VERIFY(countFalseNonZero + countTrueNonZero == m2.nonZeros());
    if (countTrueNonZero > 0) VERIFY_IS_APPROX(m2, refM2);
    m2.prune(Scalar(1));
    VERIFY(countTrueNonZero == m2.nonZeros());
    VERIFY_IS_APPROX(m2, refM2);
  }

  // test setFromTriplets / insertFromTriplets
  {
    typedef Triplet<Scalar, StorageIndex> TripletType;
    Index ntriplets = rows * cols;

    std::vector<TripletType> triplets;

    triplets.reserve(ntriplets);
    DenseMatrix refMat_sum = DenseMatrix::Zero(rows, cols);

    for (Index i = 0; i < ntriplets; ++i) {
      StorageIndex r = internal::random<StorageIndex>(0, StorageIndex(rows - 1));
      StorageIndex c = internal::random<StorageIndex>(0, StorageIndex(cols - 1));
      Scalar v = internal::random<Scalar>();
      triplets.push_back(TripletType(r, c, v));
      refMat_sum(r, c) += v;
    }

    std::vector<TripletType> moreTriplets;
    moreTriplets.reserve(ntriplets);
    DenseMatrix refMat_sum_more = refMat_sum;

    for (Index i = 0; i < ntriplets; ++i) {
      StorageIndex r = internal::random<StorageIndex>(0, StorageIndex(rows - 1));
      StorageIndex c = internal::random<StorageIndex>(0, StorageIndex(cols - 1));
      Scalar v = internal::random<Scalar>();
      moreTriplets.push_back(TripletType(r, c, v));
      refMat_sum_more(r, c) += v;
    }

    // setFromTriplets sorts internally by (outer, inner), so non-commutative
    // reductions (std::multiplies, "last wins") depend on sorted order.
    // Compute these references from sorted triplets to match.
    struct triplet_comp {
      inline bool operator()(const TripletType& a, const TripletType& b) {
        return SparseMatrixType::IsRowMajor ? ((a.row() != b.row()) ? (a.row() < b.row()) : (a.col() < b.col()))
                                            : ((a.col() != b.col()) ? (a.col() < b.col()) : (a.row() < b.row()));
      }
    };

    DenseMatrix refMat_prod = DenseMatrix::Zero(rows, cols);
    DenseMatrix refMat_last = DenseMatrix::Zero(rows, cols);
    {
      auto sorted = triplets;
      std::stable_sort(sorted.begin(), sorted.end(), triplet_comp());
      for (const auto& t : sorted) {
        if (std::abs(refMat_prod(t.row(), t.col())) == 0)
          refMat_prod(t.row(), t.col()) = t.value();
        else
          refMat_prod(t.row(), t.col()) *= t.value();
        refMat_last(t.row(), t.col()) = t.value();
      }
    }

    DenseMatrix refMat_prod_more = refMat_prod;
    DenseMatrix refMat_last_more = refMat_last;
    {
      auto sorted = moreTriplets;
      std::stable_sort(sorted.begin(), sorted.end(), triplet_comp());
      for (const auto& t : sorted) {
        if (std::abs(refMat_prod_more(t.row(), t.col())) == 0)
          refMat_prod_more(t.row(), t.col()) = t.value();
        else
          refMat_prod_more(t.row(), t.col()) *= t.value();
        refMat_last_more(t.row(), t.col()) = t.value();
      }
    }

    SparseMatrixType m(rows, cols);

    // test setFromTriplets / insertFromTriplets

    m.setFromTriplets(triplets.begin(), triplets.end());
    VERIFY_IS_APPROX(m, refMat_sum);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());
    VERIFY(m.isCompressed());
    m.insertFromTriplets(moreTriplets.begin(), moreTriplets.end());
    VERIFY_IS_APPROX(m, refMat_sum_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());

    m.setFromTriplets(triplets.begin(), triplets.end(), std::multiplies<Scalar>());
    VERIFY_IS_APPROX(m, refMat_prod);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());
    VERIFY(m.isCompressed());
    m.insertFromTriplets(moreTriplets.begin(), moreTriplets.end(), std::multiplies<Scalar>());
    VERIFY_IS_APPROX(m, refMat_prod_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());

    m.setFromTriplets(triplets.begin(), triplets.end(), [](Scalar, Scalar b) { return b; });
    VERIFY_IS_APPROX(m, refMat_last);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());
    m.setFromTriplets(triplets.begin(), triplets.end(), [](Scalar, Scalar b) { return b; });
    VERIFY(m.isCompressed());
    m.insertFromTriplets(moreTriplets.begin(), moreTriplets.end(), [](Scalar, Scalar b) { return b; });
    VERIFY_IS_APPROX(m, refMat_last_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());

    // insert into an uncompressed matrix

    VectorXi reserveSizes(m.outerSize());
    for (Index i = 0; i < m.outerSize(); i++) reserveSizes[i] = internal::random<int>(1, 7);

    m.setFromTriplets(triplets.begin(), triplets.end());
    m.reserve(reserveSizes);
    VERIFY(!m.isCompressed());
    m.insertFromTriplets(moreTriplets.begin(), moreTriplets.end());
    VERIFY_IS_APPROX(m, refMat_sum_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());

    m.setFromTriplets(triplets.begin(), triplets.end(), std::multiplies<Scalar>());
    m.reserve(reserveSizes);
    VERIFY(!m.isCompressed());
    m.insertFromTriplets(moreTriplets.begin(), moreTriplets.end(), std::multiplies<Scalar>());
    VERIFY_IS_APPROX(m, refMat_prod_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());

    m.setFromTriplets(triplets.begin(), triplets.end(), [](Scalar, Scalar b) { return b; });
    m.reserve(reserveSizes);
    VERIFY(!m.isCompressed());
    m.insertFromTriplets(moreTriplets.begin(), moreTriplets.end(), [](Scalar, Scalar b) { return b; });
    VERIFY_IS_APPROX(m, refMat_last_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());

    // test setFromSortedTriplets / insertFromSortedTriplets

    // stable_sort is only necessary when the reduction functor is dependent on the order of the triplets
    // this is the case with refMat_last
    // for most cases, std::sort is sufficient and preferred

    std::stable_sort(triplets.begin(), triplets.end(), triplet_comp());
    std::stable_sort(moreTriplets.begin(), moreTriplets.end(), triplet_comp());

    m.setFromSortedTriplets(triplets.begin(), triplets.end());
    VERIFY_IS_APPROX(m, refMat_sum);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());
    VERIFY(m.isCompressed());
    m.insertFromSortedTriplets(moreTriplets.begin(), moreTriplets.end());
    VERIFY_IS_APPROX(m, refMat_sum_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());

    m.setFromSortedTriplets(triplets.begin(), triplets.end(), std::multiplies<Scalar>());
    VERIFY_IS_APPROX(m, refMat_prod);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());
    VERIFY(m.isCompressed());
    m.insertFromSortedTriplets(moreTriplets.begin(), moreTriplets.end(), std::multiplies<Scalar>());
    VERIFY_IS_APPROX(m, refMat_prod_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());

    m.setFromSortedTriplets(triplets.begin(), triplets.end(), [](Scalar, Scalar b) { return b; });
    VERIFY_IS_APPROX(m, refMat_last);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());
    VERIFY(m.isCompressed());
    m.insertFromSortedTriplets(moreTriplets.begin(), moreTriplets.end(), [](Scalar, Scalar b) { return b; });
    VERIFY_IS_APPROX(m, refMat_last_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());

    // insert into an uncompressed matrix

    m.setFromSortedTriplets(triplets.begin(), triplets.end());
    m.reserve(reserveSizes);
    VERIFY(!m.isCompressed());
    m.insertFromSortedTriplets(moreTriplets.begin(), moreTriplets.end());
    VERIFY_IS_APPROX(m, refMat_sum_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());

    m.setFromSortedTriplets(triplets.begin(), triplets.end(), std::multiplies<Scalar>());
    m.reserve(reserveSizes);
    VERIFY(!m.isCompressed());
    m.insertFromSortedTriplets(moreTriplets.begin(), moreTriplets.end(), std::multiplies<Scalar>());
    VERIFY_IS_APPROX(m, refMat_prod_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());

    m.setFromSortedTriplets(triplets.begin(), triplets.end(), [](Scalar, Scalar b) { return b; });
    m.reserve(reserveSizes);
    VERIFY(!m.isCompressed());
    m.insertFromSortedTriplets(moreTriplets.begin(), moreTriplets.end(), [](Scalar, Scalar b) { return b; });
    VERIFY_IS_APPROX(m, refMat_last_more);
    VERIFY_IS_EQUAL(m.innerIndicesAreSorted(), m.outerSize());
  }

  // test Map
  {
    DenseMatrix refMat2(rows, cols), refMat3(rows, cols);
    SparseMatrixType m2(rows, cols), m3(rows, cols);
    initSparse<Scalar>(density, refMat2, m2);
    initSparse<Scalar>(density, refMat3, m3);
    {
      Map<SparseMatrixType> mapMat2(m2.rows(), m2.cols(), m2.nonZeros(), m2.outerIndexPtr(), m2.innerIndexPtr(),
                                    m2.valuePtr(), m2.innerNonZeroPtr());
      Map<SparseMatrixType> mapMat3(m3.rows(), m3.cols(), m3.nonZeros(), m3.outerIndexPtr(), m3.innerIndexPtr(),
                                    m3.valuePtr(), m3.innerNonZeroPtr());
      VERIFY_IS_APPROX(mapMat2 + mapMat3, refMat2 + refMat3);
      VERIFY_IS_APPROX(mapMat2 + mapMat3, refMat2 + refMat3);
    }

    Index i = internal::random<Index>(0, rows - 1);
    Index j = internal::random<Index>(0, cols - 1);
    m2.coeffRef(i, j) = 123;
    if (internal::random<bool>()) m2.makeCompressed();
    Map<SparseMatrixType> mapMat2(rows, cols, m2.nonZeros(), m2.outerIndexPtr(), m2.innerIndexPtr(), m2.valuePtr(),
                                  m2.innerNonZeroPtr());
    VERIFY_IS_EQUAL(m2.coeff(i, j), Scalar(123));
    VERIFY_IS_EQUAL(mapMat2.coeff(i, j), Scalar(123));
    mapMat2.coeffRef(i, j) = -123;
    VERIFY_IS_EQUAL(m2.coeff(i, j), Scalar(-123));
  }

  // test triangularView
  {
    DenseMatrix refMat2(rows, cols), refMat3(rows, cols);
    SparseMatrixType m2(rows, cols), m3(rows, cols);
    initSparse<Scalar>(density, refMat2, m2);
    refMat3 = refMat2.template triangularView<Lower>();
    m3 = m2.template triangularView<Lower>();
    VERIFY_IS_APPROX(m3, refMat3);

    refMat3 = refMat2.template triangularView<Upper>();
    m3 = m2.template triangularView<Upper>();
    VERIFY_IS_APPROX(m3, refMat3);

    {
      refMat3 = refMat2.template triangularView<UnitUpper>();
      m3 = m2.template triangularView<UnitUpper>();
      VERIFY_IS_APPROX(m3, refMat3);

      refMat3 = refMat2.template triangularView<UnitLower>();
      m3 = m2.template triangularView<UnitLower>();
      VERIFY_IS_APPROX(m3, refMat3);
    }

    refMat3 = refMat2.template triangularView<StrictlyUpper>();
    m3 = m2.template triangularView<StrictlyUpper>();
    VERIFY_IS_APPROX(m3, refMat3);

    refMat3 = refMat2.template triangularView<StrictlyLower>();
    m3 = m2.template triangularView<StrictlyLower>();
    VERIFY_IS_APPROX(m3, refMat3);

    // check sparse-triangular to dense
    refMat3 = m2.template triangularView<StrictlyUpper>();
    VERIFY_IS_APPROX(refMat3, DenseMatrix(refMat2.template triangularView<StrictlyUpper>()));

    // check sparse triangular view iteration-based evaluation
    m2.setZero();
    VERIFY_IS_CWISE_EQUAL(m2.template triangularView<UnitLower>().toDense(), DenseMatrix::Identity(rows, cols));
    VERIFY_IS_CWISE_EQUAL(m2.template triangularView<UnitUpper>().toDense(), DenseMatrix::Identity(rows, cols));
  }

  // test selfadjointView
  if (!SparseMatrixType::IsRowMajor) {
    DenseMatrix refMat2(rows, rows), refMat3(rows, rows);
    SparseMatrixType m2(rows, rows), m3(rows, rows);
    initSparse<Scalar>(density, refMat2, m2);
    refMat3 = refMat2.template selfadjointView<Lower>();
    m3 = m2.template selfadjointView<Lower>();
    VERIFY_IS_APPROX(m3, refMat3);

    refMat3 += refMat2.template selfadjointView<Lower>();
    m3 += m2.template selfadjointView<Lower>();
    VERIFY_IS_APPROX(m3, refMat3);

    refMat3 -= refMat2.template selfadjointView<Lower>();
    m3 -= m2.template selfadjointView<Lower>();
    VERIFY_IS_APPROX(m3, refMat3);

    // selfadjointView only works for square matrices:
    SparseMatrixType m4(rows, rows + 1);
    VERIFY_RAISES_ASSERT(m4.template selfadjointView<Lower>());
    VERIFY_RAISES_ASSERT(m4.template selfadjointView<Upper>());
  }

  // test sparseView
  {
    DenseMatrix refMat2 = DenseMatrix::Zero(rows, rows);
    SparseMatrixType m2(rows, rows);
    initSparse<Scalar>(density, refMat2, m2);
    VERIFY_IS_APPROX(m2.eval(), refMat2.sparseView().eval());

    // sparse view on expressions:
    VERIFY_IS_APPROX((s1 * m2).eval(), (s1 * refMat2).sparseView().eval());
    VERIFY_IS_APPROX((m2 + m2).eval(), (refMat2 + refMat2).sparseView().eval());
    VERIFY_IS_APPROX((m2 * m2).eval(), (refMat2.lazyProduct(refMat2)).sparseView().eval());
    VERIFY_IS_APPROX((m2 * m2).eval(), (refMat2 * refMat2).sparseView().eval());
  }

  // test diagonal
  {
    DenseMatrix refMat2 = DenseMatrix::Zero(rows, cols);
    SparseMatrixType m2(rows, cols);
    initSparse<Scalar>(density, refMat2, m2);
    VERIFY_IS_APPROX(m2.diagonal(), refMat2.diagonal().eval());
    DenseVector d = m2.diagonal();
    VERIFY_IS_APPROX(d, refMat2.diagonal().eval());
    d = m2.diagonal().array();
    VERIFY_IS_APPROX(d, refMat2.diagonal().eval());
    VERIFY_IS_APPROX(const_cast<const SparseMatrixType&>(m2).diagonal(), refMat2.diagonal().eval());

    initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag);
    m2.diagonal() += refMat2.diagonal();
    refMat2.diagonal() += refMat2.diagonal();
    VERIFY_IS_APPROX(m2, refMat2);
  }

  // test diagonal to sparse
  {
    DenseVector d = DenseVector::Random(rows);
    DenseMatrix refMat2 = d.asDiagonal();
    SparseMatrixType m2;
    m2 = d.asDiagonal();
    VERIFY_IS_APPROX(m2, refMat2);
    SparseMatrixType m3(d.asDiagonal());
    VERIFY_IS_APPROX(m3, refMat2);
    refMat2 += d.asDiagonal();
    m2 += d.asDiagonal();
    VERIFY_IS_APPROX(m2, refMat2);
    m2.setZero();
    m2 += d.asDiagonal();
    refMat2.setZero();
    refMat2 += d.asDiagonal();
    VERIFY_IS_APPROX(m2, refMat2);
    m2.setZero();
    m2 -= d.asDiagonal();
    refMat2.setZero();
    refMat2 -= d.asDiagonal();
    VERIFY_IS_APPROX(m2, refMat2);

    initSparse<Scalar>(density, refMat2, m2);
    m2.makeCompressed();
    m2 += d.asDiagonal();
    refMat2 += d.asDiagonal();
    VERIFY_IS_APPROX(m2, refMat2);

    initSparse<Scalar>(density, refMat2, m2);
    m2.makeCompressed();
    VectorXi res(rows);
    for (Index i = 0; i < rows; ++i) res(i) = internal::random<int>(0, 3);
    m2.reserve(res);
    m2 -= d.asDiagonal();
    refMat2 -= d.asDiagonal();
    VERIFY_IS_APPROX(m2, refMat2);

    // zero-length diagonal
    d = DenseVector(0);
    SparseMatrixType m4(d.asDiagonal());
  }

  // test conservative resize
  {
    std::vector<std::pair<StorageIndex, StorageIndex>> inc;
    if (rows > 3 && cols > 2) inc.push_back(std::pair<StorageIndex, StorageIndex>(-3, -2));
    inc.push_back(std::pair<StorageIndex, StorageIndex>(0, 0));
    inc.push_back(std::pair<StorageIndex, StorageIndex>(3, 2));
    inc.push_back(std::pair<StorageIndex, StorageIndex>(3, 0));
    inc.push_back(std::pair<StorageIndex, StorageIndex>(0, 3));
    inc.push_back(std::pair<StorageIndex, StorageIndex>(0, -1));
    inc.push_back(std::pair<StorageIndex, StorageIndex>(-1, 0));
    inc.push_back(std::pair<StorageIndex, StorageIndex>(-1, -1));

    for (size_t i = 0; i < inc.size(); i++) {
      StorageIndex incRows = inc[i].first;
      StorageIndex incCols = inc[i].second;
      SparseMatrixType m1(rows, cols);
      DenseMatrix refMat1 = DenseMatrix::Zero(rows, cols);
      initSparse<Scalar>(density, refMat1, m1);

      SparseMatrixType m2 = m1;
      m2.makeCompressed();

      m1.conservativeResize(rows + incRows, cols + incCols);
      m2.conservativeResize(rows + incRows, cols + incCols);
      refMat1.conservativeResize(rows + incRows, cols + incCols);
      if (incRows > 0) refMat1.bottomRows(incRows).setZero();
      if (incCols > 0) refMat1.rightCols(incCols).setZero();

      VERIFY_IS_APPROX(m1, refMat1);
      VERIFY_IS_APPROX(m2, refMat1);

      // Insert new values
      if (incRows > 0) m1.insert(m1.rows() - 1, 0) = refMat1(refMat1.rows() - 1, 0) = 1;
      if (incCols > 0) m1.insert(0, m1.cols() - 1) = refMat1(0, refMat1.cols() - 1) = 1;

      VERIFY_IS_APPROX(m1, refMat1);
    }
  }

  // test Identity matrix
  {
    DenseMatrix refMat1 = DenseMatrix::Identity(rows, rows);
    SparseMatrixType m1(rows, rows);
    m1.setIdentity();
    VERIFY_IS_APPROX(m1, refMat1);
    for (int k = 0; k < rows * rows / 4; ++k) {
      Index i = internal::random<Index>(0, rows - 1);
      Index j = internal::random<Index>(0, rows - 1);
      Scalar v = internal::random<Scalar>();
      m1.coeffRef(i, j) = v;
      refMat1.coeffRef(i, j) = v;
      VERIFY_IS_APPROX(m1, refMat1);
      if (internal::random<Index>(0, 10) < 2) m1.makeCompressed();
    }
    m1.setIdentity();
    refMat1.setIdentity();
    VERIFY_IS_APPROX(m1, refMat1);
  }

  // test array/vector of InnerIterator
  {
    typedef typename SparseMatrixType::InnerIterator IteratorType;

    DenseMatrix refMat2 = DenseMatrix::Zero(rows, cols);
    SparseMatrixType m2(rows, cols);
    initSparse<Scalar>(density, refMat2, m2);
    IteratorType static_array[2];
    static_array[0] = IteratorType(m2, 0);
    static_array[1] = IteratorType(m2, m2.outerSize() - 1);
    VERIFY(static_array[0] || m2.innerVector(static_array[0].outer()).nonZeros() == 0);
    VERIFY(static_array[1] || m2.innerVector(static_array[1].outer()).nonZeros() == 0);
    if (static_array[0] && static_array[1]) {
      ++(static_array[1]);
      static_array[1] = IteratorType(m2, 0);
      VERIFY(static_array[1]);
      VERIFY(static_array[1].index() == static_array[0].index());
      VERIFY(static_array[1].outer() == static_array[0].outer());
      VERIFY(static_array[1].value() == static_array[0].value());
    }

    std::vector<IteratorType> iters(2);
    iters[0] = IteratorType(m2, 0);
    iters[1] = IteratorType(m2, m2.outerSize() - 1);
  }

  // test reserve with empty rows/columns
  {
    SparseMatrixType m1(0, cols);
    m1.reserve(ArrayXi::Constant(m1.outerSize(), 1));
    SparseMatrixType m2(rows, 0);
    m2.reserve(ArrayXi::Constant(m2.outerSize(), 1));
  }

  // test move
  {
    using TransposedType = SparseMatrix<Scalar, SparseMatrixType::IsRowMajor ? ColMajor : RowMajor,
                                        typename SparseMatrixType::StorageIndex>;
    DenseMatrix refMat1 = DenseMatrix::Random(rows, cols);
    SparseMatrixType m1(rows, cols);
    initSparse<Scalar>(density, refMat1, m1);
    // test move ctor
    SparseMatrixType m2(std::move(m1));
    VERIFY_IS_APPROX(m2, refMat1);
    // test move assignment
    m1 = std::move(m2);
    VERIFY_IS_APPROX(m1, refMat1);
    // test move ctor (SparseMatrixBase)
    TransposedType m3(std::move(m1.transpose()));
    VERIFY_IS_APPROX(m3, refMat1.transpose());
    // test move assignment (SparseMatrixBase)
    m2 = std::move(m3.transpose());
    VERIFY_IS_APPROX(m2, refMat1);
  }
}

template <typename SparseMatrixType>
void big_sparse_triplet(Index rows, Index cols, double density) {
  typedef typename SparseMatrixType::StorageIndex StorageIndex;
  typedef typename SparseMatrixType::Scalar Scalar;
  typedef Triplet<Scalar, Index> TripletType;
  std::vector<TripletType> triplets;
  double nelements = density * static_cast<double>(rows * cols);
  VERIFY(nelements >= 0 && nelements < static_cast<double>(NumTraits<StorageIndex>::highest()));
  Index ntriplets = Index(nelements);
  triplets.reserve(ntriplets);
  Scalar sum = Scalar(0);
  for (Index i = 0; i < ntriplets; ++i) {
    Index r = internal::random<Index>(0, rows - 1);
    Index c = internal::random<Index>(0, cols - 1);
    // use positive values to prevent numerical cancellation errors in sum
    Scalar v = numext::abs(internal::random<Scalar>());
    triplets.push_back(TripletType(r, c, v));
    sum += v;
  }
  SparseMatrixType m(rows, cols);
  m.setFromTriplets(triplets.begin(), triplets.end());
  VERIFY(m.nonZeros() <= ntriplets);
  VERIFY_IS_APPROX(sum, m.sum());
}

template <int>
void bug1105() {
  // Regression test for bug 1105
  int n = Eigen::internal::random<int>(200, 600);
  SparseMatrix<std::complex<double>, 0, long> mat(n, n);
  std::complex<double> val;

  for (int i = 0; i < n; ++i) {
    mat.coeffRef(i, i % (n / 10)) = val;
    VERIFY(mat.data().allocatedSize() < 20 * n);
  }
}

#ifndef EIGEN_SPARSE_TEST_INCLUDED_FROM_SPARSE_EXTRA

EIGEN_DECLARE_TEST(sparse_basic) {
  g_dense_op_sparse_count = 0;  // Suppresses compiler warning.
  for (int i = 0; i < g_repeat; i++) {
    int r = Eigen::internal::random<int>(1, 200), c = Eigen::internal::random<int>(1, 200);
    if (Eigen::internal::random<int>(0, 3) == 0) {
      r = c;  // check square matrices in 25% of tries
    }
    EIGEN_UNUSED_VARIABLE(r + c);
    CALL_SUBTEST_1((sparse_basic(SparseMatrix<double>(1, 1))));
    CALL_SUBTEST_1((sparse_basic(SparseMatrix<double>(8, 8))));
    CALL_SUBTEST_2((sparse_basic(SparseMatrix<std::complex<double>, ColMajor>(r, c))));
    CALL_SUBTEST_2((sparse_basic(SparseMatrix<std::complex<double>, RowMajor>(r, c))));
    CALL_SUBTEST_2((sparse_basic(SparseMatrix<float, RowMajor>(r, c))));
    CALL_SUBTEST_2((sparse_basic(SparseMatrix<float, ColMajor>(r, c))));
    CALL_SUBTEST_3((sparse_basic(SparseMatrix<double, ColMajor>(r, c))));
    CALL_SUBTEST_3((sparse_basic(SparseMatrix<double, RowMajor>(r, c))));
    CALL_SUBTEST_4((sparse_basic(SparseMatrix<double, ColMajor, long int>(r, c))));
    CALL_SUBTEST_4((sparse_basic(SparseMatrix<double, RowMajor, long int>(r, c))));

    r = Eigen::internal::random<int>(1, 100);
    c = Eigen::internal::random<int>(1, 100);
    if (Eigen::internal::random<int>(0, 3) == 0) {
      r = c;  // check square matrices in 25% of tries
    }

    CALL_SUBTEST_5((sparse_basic(SparseMatrix<double, ColMajor, short int>(short(r), short(c)))));
    CALL_SUBTEST_5((sparse_basic(SparseMatrix<double, RowMajor, short int>(short(r), short(c)))));
  }

  // Regression test for bug 900: (manually insert higher values here, if you have enough RAM):
  CALL_SUBTEST_5((big_sparse_triplet<SparseMatrix<float, RowMajor, int>>(10000, 10000, 0.125)));
  CALL_SUBTEST_5((big_sparse_triplet<SparseMatrix<double, ColMajor, long int>>(10000, 10000, 0.125)));

  CALL_SUBTEST_5(bug1105<0>());
}
#endif