template<typename Scalar> void sparse_llt(int rows, int cols)
{
double density = std::max(8./(rows*cols), 0.01);
typedef Matrix<Scalar,Dynamic,Dynamic> DenseMatrix;
typedef Matrix<Scalar,Dynamic,1> DenseVector;
// TODO fix the issue with complex (see SparseLLT::solveInPlace)
SparseMatrix<Scalar> m2(rows, cols);
DenseMatrix refMat2(rows, cols);
DenseVector b = DenseVector::Random(cols);
DenseVector refX(cols), x(cols);
initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag|MakeLowerTriangular, 0, 0);
for(int i=0; i<rows; ++i)
m2.coeffRef(i,i) = refMat2(i,i) = ei_abs(ei_real(refMat2(i,i)));
refX = refMat2.template selfadjointView<Lower>().llt().solve(b);
if (!NumTraits<Scalar>::IsComplex)
{
x = b;
SparseLLT<SparseMatrix<Scalar> > (m2).solveInPlace(x);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LLT: default");
}
#ifdef EIGEN_CHOLMOD_SUPPORT
x = b;
SparseLLT<SparseMatrix<Scalar> ,Cholmod>(m2).solveInPlace(x);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LLT: cholmod");
#endif
#ifdef EIGEN_TAUCS_SUPPORT
// TODO fix TAUCS with complexes
if (!NumTraits<Scalar>::IsComplex)
{
x = b;
// SparseLLT<SparseMatrix<Scalar> ,Taucs>(m2,IncompleteFactorization).solveInPlace(x);
// VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LLT: taucs (IncompleteFactorization)");
x = b;
SparseLLT<SparseMatrix<Scalar> ,Taucs>(m2,SupernodalMultifrontal).solveInPlace(x);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LLT: taucs (SupernodalMultifrontal)");
x = b;
SparseLLT<SparseMatrix<Scalar> ,Taucs>(m2,SupernodalLeftLooking).solveInPlace(x);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LLT: taucs (SupernodalLeftLooking)");
}
#endif
}
template<typename Scalar> void sparse_lu(int rows, int cols)
{
double density = std::max(8./(rows*cols), 0.01);
typedef Matrix<Scalar,Dynamic,Dynamic> DenseMatrix;
typedef Matrix<Scalar,Dynamic,1> DenseVector;
DenseVector vec1 = DenseVector::Random(rows);
std::vector<Vector2i> zeroCoords;
std::vector<Vector2i> nonzeroCoords;
SparseMatrix<Scalar> m2(rows, cols);
DenseMatrix refMat2(rows, cols);
DenseVector b = DenseVector::Random(cols);
DenseVector refX(cols), x(cols);
initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag, &zeroCoords, &nonzeroCoords);
FullPivLU<DenseMatrix> refLu(refMat2);
refX = refLu.solve(b);
#if defined(EIGEN_SUPERLU_SUPPORT) || defined(EIGEN_UMFPACK_SUPPORT)
Scalar refDet = refLu.determinant();
#endif
x.setZero();
// // SparseLU<SparseMatrix<Scalar> > (m2).solve(b,&x);
// // VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LU: default");
#ifdef EIGEN_UMFPACK_SUPPORT
{
// check solve
x.setZero();
SparseLU<SparseMatrix<Scalar>,UmfPack> lu(m2);
VERIFY(lu.succeeded() && "umfpack LU decomposition failed");
VERIFY(lu.solve(b,&x) && "umfpack LU solving failed");
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LU: umfpack");
VERIFY_IS_APPROX(refDet,lu.determinant());
// TODO check the extracted data
//std::cerr << slu.matrixL() << "\n";
}
#endif
#ifdef EIGEN_SUPERLU_SUPPORT
{
x.setZero();
SparseLU<SparseMatrix<Scalar>,SuperLU> slu(m2);
if (slu.succeeded())
{
if (slu.solve(b,&x)) {
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LU: SuperLU");
}
// std::cerr << refDet << " == " << slu.determinant() << "\n";
if (slu.solve(b, &x, SvTranspose)) {
VERIFY(b.isApprox(m2.transpose() * x, test_precision<Scalar>()));
}
if (slu.solve(b, &x, SvAdjoint)) {
VERIFY(b.isApprox(m2.adjoint() * x, test_precision<Scalar>()));
}
if (!NumTraits<Scalar>::IsComplex) {
VERIFY_IS_APPROX(refDet,slu.determinant()); // FIXME det is not very stable for complex
}
}
}
#endif
}
template<typename SparseMatrixType> void sparse_basic(const SparseMatrixType& ref)
{
typedef typename SparseMatrixType::Index Index;
const Index rows = ref.rows();
const Index cols = ref.cols();
typedef typename SparseMatrixType::Scalar Scalar;
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;
Scalar eps = 1e-6;
SparseMatrixType m(rows, cols);
DenseMatrix refMat = DenseMatrix::Zero(rows, cols);
DenseVector vec1 = DenseVector::Random(rows);
Scalar s1 = internal::random<Scalar>();
std::vector<Vector2i> zeroCoords;
std::vector<Vector2i> nonzeroCoords;
initSparse<Scalar>(density, refMat, m, 0, &zeroCoords, &nonzeroCoords);
if (zeroCoords.size()==0 || nonzeroCoords.size()==0)
return;
// test coeff and coeffRef
for (int i=0; i<(int)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[0].x(),zeroCoords[0].y()) = 5 );
}
VERIFY_IS_APPROX(m, refMat);
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 InnerIterators and Block expressions
for (int t=0; t<10; ++t)
{
int j = internal::random<int>(0,cols-1);
int i = internal::random<int>(0,rows-1);
int w = internal::random<int>(1,cols-j-1);
int h = internal::random<int>(1,rows-i-1);
// VERIFY_IS_APPROX(m.block(i,j,h,w), refMat.block(i,j,h,w));
for(int c=0; c<w; c++)
{
VERIFY_IS_APPROX(m.block(i,j,h,w).col(c), refMat.block(i,j,h,w).col(c));
for(int r=0; r<h; r++)
{
// VERIFY_IS_APPROX(m.block(i,j,h,w).col(c).coeff(r), refMat.block(i,j,h,w).col(c).coeff(r));
}
}
// for(int r=0; r<h; r++)
// {
// VERIFY_IS_APPROX(m.block(i,j,h,w).row(r), refMat.block(i,j,h,w).row(r));
// for(int c=0; c<w; c++)
// {
// VERIFY_IS_APPROX(m.block(i,j,h,w).row(r).coeff(c), refMat.block(i,j,h,w).row(r).coeff(c));
// }
// }
}
for(int c=0; c<cols; c++)
{
VERIFY_IS_APPROX(m.col(c) + m.col(c), (m + m).col(c));
VERIFY_IS_APPROX(m.col(c) + m.col(c), refMat.col(c) + refMat.col(c));
}
for(int r=0; r<rows; r++)
{
VERIFY_IS_APPROX(m.row(r) + m.row(r), (m + m).row(r));
VERIFY_IS_APPROX(m.row(r) + m.row(r), refMat.row(r) + refMat.row(r));
}
*/
// test insert (inner random)
{
DenseMatrix m1(rows,cols);
m1.setZero();
SparseMatrixType m2(rows,cols);
m2.reserve(10);
for (int j=0; j<cols; ++j)
{
for (int k=0; k<rows/2; ++k)
{
int i = internal::random<int>(0,rows-1);
if (m1.coeff(i,j)==Scalar(0))
m2.insert(i,j) = m1(i,j) = internal::random<Scalar>();
}
}
m2.finalize();
VERIFY_IS_APPROX(m2,m1);
}
// test insert (fully random)
{
DenseMatrix m1(rows,cols);
m1.setZero();
SparseMatrixType m2(rows,cols);
m2.reserve(10);
for (int k=0; k<rows*cols; ++k)
{
int i = internal::random<int>(0,rows-1);
int j = internal::random<int>(0,cols-1);
if (m1.coeff(i,j)==Scalar(0))
m2.insert(i,j) = m1(i,j) = internal::random<Scalar>();
}
m2.finalize();
VERIFY_IS_APPROX(m2,m1);
}
// test basic computations
{
DenseMatrix refM1 = DenseMatrix::Zero(rows, rows);
DenseMatrix refM2 = DenseMatrix::Zero(rows, rows);
DenseMatrix refM3 = DenseMatrix::Zero(rows, rows);
DenseMatrix refM4 = DenseMatrix::Zero(rows, rows);
SparseMatrixType m1(rows, rows);
SparseMatrixType m2(rows, rows);
SparseMatrixType m3(rows, rows);
SparseMatrixType m4(rows, rows);
initSparse<Scalar>(density, refM1, m1);
initSparse<Scalar>(density, refM2, m2);
initSparse<Scalar>(density, refM3, m3);
initSparse<Scalar>(density, refM4, m4);
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(m1*=s1, refM1*=s1);
VERIFY_IS_APPROX(m1/=s1, refM1/=s1);
VERIFY_IS_APPROX(m1+=m2, refM1+=refM2);
VERIFY_IS_APPROX(m1-=m2, refM1-=refM2);
VERIFY_IS_APPROX(m1.col(0).dot(refM2.row(0)), refM1.col(0).dot(refM2.row(0)));
refM4.setRandom();
// sparse cwise* dense
VERIFY_IS_APPROX(m3.cwiseProduct(refM4), refM3.cwiseProduct(refM4));
// VERIFY_IS_APPROX(m3.cwise()/refM4, refM3.cwise()/refM4);
}
// test transpose
{
DenseMatrix refMat2 = DenseMatrix::Zero(rows, rows);
SparseMatrixType m2(rows, rows);
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());
}
// test innerVector()
{
DenseMatrix refMat2 = DenseMatrix::Zero(rows, rows);
SparseMatrixType m2(rows, rows);
initSparse<Scalar>(density, refMat2, m2);
int j0 = internal::random(0,rows-1);
int j1 = internal::random(0,rows-1);
VERIFY_IS_APPROX(m2.innerVector(j0), refMat2.col(j0));
VERIFY_IS_APPROX(m2.innerVector(j0)+m2.innerVector(j1), refMat2.col(j0)+refMat2.col(j1));
//m2.innerVector(j0) = 2*m2.innerVector(j1);
//refMat2.col(j0) = 2*refMat2.col(j1);
//VERIFY_IS_APPROX(m2, refMat2);
}
// test innerVectors()
{
DenseMatrix refMat2 = DenseMatrix::Zero(rows, rows);
SparseMatrixType m2(rows, rows);
initSparse<Scalar>(density, refMat2, m2);
int j0 = internal::random(0,rows-2);
int j1 = internal::random(0,rows-2);
int n0 = internal::random<int>(1,rows-(std::max)(j0,j1));
VERIFY_IS_APPROX(m2.innerVectors(j0,n0), refMat2.block(0,j0,rows,n0));
VERIFY_IS_APPROX(m2.innerVectors(j0,n0)+m2.innerVectors(j1,n0),
refMat2.block(0,j0,rows,n0)+refMat2.block(0,j1,rows,n0));
//m2.innerVectors(j0,n0) = m2.innerVectors(j0,n0) + m2.innerVectors(j1,n0);
//refMat2.block(0,j0,rows,n0) = refMat2.block(0,j0,rows,n0) + refMat2.block(0,j1,rows,n0);
}
// test prune
{
SparseMatrixType m2(rows, rows);
DenseMatrix refM2(rows, rows);
refM2.setZero();
int countFalseNonZero = 0;
int countTrueNonZero = 0;
for (int j=0; j<m2.outerSize(); ++j)
{
m2.startVec(j);
for (int i=0; i<m2.innerSize(); ++i)
{
float x = internal::random<float>(0,1);
if (x<0.1)
{
// do nothing
}
else if (x<0.5)
{
countFalseNonZero++;
m2.insertBackByOuterInner(j,i) = Scalar(0);
}
else
{
countTrueNonZero++;
m2.insertBackByOuterInner(j,i) = refM2(i,j) = Scalar(1);
}
}
}
m2.finalize();
VERIFY(countFalseNonZero+countTrueNonZero == m2.nonZeros());
VERIFY_IS_APPROX(m2, refM2);
m2.prune(Scalar(1));
VERIFY(countTrueNonZero==m2.nonZeros());
VERIFY_IS_APPROX(m2, refM2);
}
// test selfadjointView
{
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);
}
// 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());
}
}
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;
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;
Scalar eps = 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))
m2.insert(i,j) = m1(i,j) = internal::random<Scalar>();
}
}
if(call_reserve && !SparseMatrixType::IsRowMajor)
{
VERIFY(g_realloc_count==0);
}
m2.finalize();
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))
m2.insert(i,j) = m1(i,j) = internal::random<Scalar>();
else
{
Scalar v = internal::random<Scalar>();
m2.coeffRef(i,j) += v;
m1(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))
m2.insert(i,j) = m1(i,j) = internal::random<Scalar>();
if(mode==3)
m2.reserve(r);
}
if(internal::random<int>()%2)
m2.makeCompressed();
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();
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);
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);
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(m1.sum(), refM1.sum());
VERIFY_IS_APPROX(m1*=s1, refM1*=s1);
VERIFY_IS_APPROX(m1/=s1, refM1/=s1);
VERIFY_IS_APPROX(m1+=m2, refM1+=refM2);
VERIFY_IS_APPROX(m1-=m2, refM1-=refM2);
// test aliasing
VERIFY_IS_APPROX((m1 = -m1), (refM1 = -refM1));
VERIFY_IS_APPROX((m1 = m1.transpose()), (refM1 = refM1.transpose().eval()));
VERIFY_IS_APPROX((m1 = -m1.transpose()), (refM1 = -refM1.transpose().eval()));
VERIFY_IS_APPROX((m1 += -m1), (refM1 += -refM1));
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
{
typedef Triplet<Scalar,StorageIndex> TripletType;
std::vector<TripletType> triplets;
Index ntriplets = rows*cols;
triplets.reserve(ntriplets);
DenseMatrix refMat_sum = DenseMatrix::Zero(rows,cols);
DenseMatrix refMat_prod = DenseMatrix::Zero(rows,cols);
DenseMatrix refMat_last = 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;
if(std::abs(refMat_prod(r,c))==0)
refMat_prod(r,c) = v;
else
refMat_prod(r,c) *= v;
refMat_last(r,c) = v;
}
SparseMatrixType m(rows,cols);
m.setFromTriplets(triplets.begin(), triplets.end());
VERIFY_IS_APPROX(m, refMat_sum);
m.setFromTriplets(triplets.begin(), triplets.end(), std::multiplies<Scalar>());
VERIFY_IS_APPROX(m, refMat_prod);
#if (defined(__cplusplus) && __cplusplus >= 201103L)
m.setFromTriplets(triplets.begin(), triplets.end(), [] (Scalar,Scalar b) { return b; });
VERIFY_IS_APPROX(m, refMat_last);
#endif
}
// 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);
}
{
MappedSparseMatrix<Scalar,SparseMatrixType::Options,StorageIndex> mapMat2(m2.rows(), m2.cols(), m2.nonZeros(), m2.outerIndexPtr(), m2.innerIndexPtr(), m2.valuePtr(), m2.innerNonZeroPtr());
MappedSparseMatrix<Scalar,SparseMatrixType::Options,StorageIndex> 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-traingular to dense
refMat3 = m2.template triangularView<StrictlyUpper>();
VERIFY_IS_APPROX(refMat3, DenseMatrix(refMat2.template triangularView<StrictlyUpper>()));
}
// 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());
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(rows, rows);
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);
}
// 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));
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);
m1.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);
// 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);
}
}
template<typename Scalar> void sparse_solvers(int rows, int cols)
{
double density = std::max(8./(rows*cols), 0.01);
typedef Matrix<Scalar,Dynamic,Dynamic> DenseMatrix;
typedef Matrix<Scalar,Dynamic,1> DenseVector;
// Scalar eps = 1e-6;
DenseVector vec1 = DenseVector::Random(rows);
std::vector<Vector2i> zeroCoords;
std::vector<Vector2i> nonzeroCoords;
// test triangular solver
{
DenseVector vec2 = vec1, vec3 = vec1;
SparseMatrix<Scalar> m2(rows, cols);
DenseMatrix refMat2 = DenseMatrix::Zero(rows, cols);
// lower
initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag|MakeLowerTriangular, &zeroCoords, &nonzeroCoords);
VERIFY_IS_APPROX(refMat2.template marked<LowerTriangular>().solveTriangular(vec2),
m2.template marked<LowerTriangular>().solveTriangular(vec3));
// lower - transpose
initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag|MakeLowerTriangular, &zeroCoords, &nonzeroCoords);
VERIFY_IS_APPROX(refMat2.template marked<LowerTriangular>().transpose().solveTriangular(vec2),
m2.template marked<LowerTriangular>().transpose().solveTriangular(vec3));
// upper
initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag|MakeUpperTriangular, &zeroCoords, &nonzeroCoords);
VERIFY_IS_APPROX(refMat2.template marked<UpperTriangular>().solveTriangular(vec2),
m2.template marked<UpperTriangular>().solveTriangular(vec3));
// upper - transpose
initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag|MakeUpperTriangular, &zeroCoords, &nonzeroCoords);
VERIFY_IS_APPROX(refMat2.template marked<UpperTriangular>().transpose().solveTriangular(vec2),
m2.template marked<UpperTriangular>().transpose().solveTriangular(vec3));
}
// test LLT
{
// TODO fix the issue with complex (see SparseLLT::solveInPlace)
SparseMatrix<Scalar> m2(rows, cols);
DenseMatrix refMat2(rows, cols);
DenseVector b = DenseVector::Random(cols);
DenseVector refX(cols), x(cols);
initSPD(density, refMat2, m2);
refMat2.llt().solve(b, &refX);
typedef SparseMatrix<Scalar,LowerTriangular|SelfAdjoint> SparseSelfAdjointMatrix;
if (!NumTraits<Scalar>::IsComplex)
{
x = b;
SparseLLT<SparseSelfAdjointMatrix> (m2).solveInPlace(x);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LLT: default");
}
#ifdef EIGEN_CHOLMOD_SUPPORT
x = b;
SparseLLT<SparseSelfAdjointMatrix,Cholmod>(m2).solveInPlace(x);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LLT: cholmod");
#endif
if (!NumTraits<Scalar>::IsComplex)
{
#ifdef EIGEN_TAUCS_SUPPORT
x = b;
SparseLLT<SparseSelfAdjointMatrix,Taucs>(m2,IncompleteFactorization).solveInPlace(x);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LLT: taucs (IncompleteFactorization)");
x = b;
SparseLLT<SparseSelfAdjointMatrix,Taucs>(m2,SupernodalMultifrontal).solveInPlace(x);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LLT: taucs (SupernodalMultifrontal)");
x = b;
SparseLLT<SparseSelfAdjointMatrix,Taucs>(m2,SupernodalLeftLooking).solveInPlace(x);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LLT: taucs (SupernodalLeftLooking)");
#endif
}
}
// test LDLT
if (!NumTraits<Scalar>::IsComplex)
{
// TODO fix the issue with complex (see SparseLDLT::solveInPlace)
SparseMatrix<Scalar> m2(rows, cols);
DenseMatrix refMat2(rows, cols);
DenseVector b = DenseVector::Random(cols);
DenseVector refX(cols), x(cols);
//initSPD(density, refMat2, m2);
initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag|MakeUpperTriangular, 0, 0);
refMat2 += refMat2.adjoint();
refMat2.diagonal() *= 0.5;
refMat2.ldlt().solve(b, &refX);
typedef SparseMatrix<Scalar,UpperTriangular|SelfAdjoint> SparseSelfAdjointMatrix;
x = b;
SparseLDLT<SparseSelfAdjointMatrix> ldlt(m2);
if (ldlt.succeeded())
ldlt.solveInPlace(x);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LDLT: default");
}
// test LU
{
static int count = 0;
SparseMatrix<Scalar> m2(rows, cols);
DenseMatrix refMat2(rows, cols);
DenseVector b = DenseVector::Random(cols);
DenseVector refX(cols), x(cols);
initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag, &zeroCoords, &nonzeroCoords);
LU<DenseMatrix> refLu(refMat2);
refLu.solve(b, &refX);
#if defined(EIGEN_SUPERLU_SUPPORT) || defined(EIGEN_UMFPACK_SUPPORT)
Scalar refDet = refLu.determinant();
#endif
x.setZero();
// // SparseLU<SparseMatrix<Scalar> > (m2).solve(b,&x);
// // VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LU: default");
#ifdef EIGEN_SUPERLU_SUPPORT
{
x.setZero();
SparseLU<SparseMatrix<Scalar>,SuperLU> slu(m2);
if (slu.succeeded())
{
if (slu.solve(b,&x)) {
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LU: SuperLU");
}
// std::cerr << refDet << " == " << slu.determinant() << "\n";
if (count==0) {
VERIFY_IS_APPROX(refDet,slu.determinant()); // FIXME det is not very stable for complex
}
}
}
#endif
#ifdef EIGEN_UMFPACK_SUPPORT
{
// check solve
x.setZero();
SparseLU<SparseMatrix<Scalar>,UmfPack> slu(m2);
if (slu.succeeded()) {
if (slu.solve(b,&x)) {
if (count==0) {
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LU: umfpack"); // FIXME solve is not very stable for complex
}
}
VERIFY_IS_APPROX(refDet,slu.determinant());
// TODO check the extracted data
//std::cerr << slu.matrixL() << "\n";
}
}
#endif
count++;
}
}
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;
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;
Scalar eps = 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))
m2.insert(i,j) = m1(i,j) = internal::random<Scalar>();
}
}
if(call_reserve && !SparseMatrixType::IsRowMajor)
{
VERIFY(g_realloc_count==0);
}
m2.finalize();
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))
m2.insert(i,j) = m1(i,j) = internal::random<Scalar>();
else
{
Scalar v = internal::random<Scalar>();
m2.coeffRef(i,j) += v;
m1(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))
m2.insert(i,j) = m1(i,j) = internal::random<Scalar>();
if(mode==3)
m2.reserve(r);
}
if(internal::random<int>()%2)
m2.makeCompressed();
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);
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(m1*=s1, refM1*=s1);
VERIFY_IS_APPROX(m1/=s1, refM1/=s1);
VERIFY_IS_APPROX(m1+=m2, refM1+=refM2);
VERIFY_IS_APPROX(m1-=m2, refM1-=refM2);
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));
// VERIFY_IS_APPROX(m3.cwise()/refM4, refM3.cwise()/refM4);
// test aliasing
VERIFY_IS_APPROX((m1 = -m1), (refM1 = -refM1));
VERIFY_IS_APPROX((m1 = m1.transpose()), (refM1 = refM1.transpose().eval()));
VERIFY_IS_APPROX((m1 = -m1.transpose()), (refM1 = -refM1.transpose().eval()));
VERIFY_IS_APPROX((m1 += -m1), (refM1 += -refM1));
}
// 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;
for (Index j=0; j<m2.outerSize(); ++j)
{
m2.startVec(j);
for (Index i=0; i<m2.innerSize(); ++i)
{
float x = internal::random<float>(0,1);
if (x<0.1)
{
// do nothing
}
else if (x<0.5)
{
countFalseNonZero++;
m2.insertBackByOuterInner(j,i) = Scalar(0);
}
else
{
countTrueNonZero++;
m2.insertBackByOuterInner(j,i) = Scalar(1);
if(SparseMatrixType::IsRowMajor)
refM2(j,i) = Scalar(1);
else
refM2(i,j) = Scalar(1);
}
}
}
m2.finalize();
VERIFY(countFalseNonZero+countTrueNonZero == m2.nonZeros());
VERIFY_IS_APPROX(m2, refM2);
m2.prune(Scalar(1));
VERIFY(countTrueNonZero==m2.nonZeros());
VERIFY_IS_APPROX(m2, refM2);
}
// test setFromTriplets
{
typedef Triplet<Scalar,StorageIndex> TripletType;
std::vector<TripletType> triplets;
Index ntriplets = rows*cols;
triplets.reserve(ntriplets);
DenseMatrix refMat(rows,cols);
refMat.setZero();
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(r,c) += v;
}
SparseMatrixType m(rows,cols);
m.setFromTriplets(triplets.begin(), triplets.end());
VERIFY_IS_APPROX(m, refMat);
}
// 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);
}
{
MappedSparseMatrix<Scalar,SparseMatrixType::Options,StorageIndex> mapMat2(m2.rows(), m2.cols(), m2.nonZeros(), m2.outerIndexPtr(), m2.innerIndexPtr(), m2.valuePtr(), m2.innerNonZeroPtr());
MappedSparseMatrix<Scalar,SparseMatrixType::Options,StorageIndex> 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);
}
}
// 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);
if(inner>=outer) // FIXME this should be implemented for outer>inner as well
{
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);
}
// 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);
// 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());
}
// 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());
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(rows, rows);
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);
}
// 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));
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);
m1.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);
// 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);
}
}
template<typename Scalar> void sparse_ldlt(int rows, int cols)
{
static bool odd = true;
odd = !odd;
double density = (std::max)(8./(rows*cols), 0.01);
typedef Matrix<Scalar,Dynamic,Dynamic> DenseMatrix;
typedef Matrix<Scalar,Dynamic,1> DenseVector;
SparseMatrix<Scalar> m2(rows, cols);
DenseMatrix refMat2(rows, cols);
DenseVector b = DenseVector::Random(cols);
DenseVector refX(cols), x(cols);
initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag|MakeUpperTriangular, 0, 0);
SparseMatrix<Scalar> m3 = m2 * m2.adjoint(), m3_lo(rows,rows), m3_up(rows,rows);
DenseMatrix refMat3 = refMat2 * refMat2.adjoint();
refX = refMat3.template selfadjointView<Upper>().ldlt().solve(b);
typedef SparseMatrix<Scalar,Upper|SelfAdjoint> SparseSelfAdjointMatrix;
x = b;
SparseLDLT<SparseSelfAdjointMatrix> ldlt(m3);
if (ldlt.succeeded())
ldlt.solveInPlace(x);
else
std::cerr << "warning LDLT failed\n";
VERIFY_IS_APPROX(refMat3.template selfadjointView<Upper>() * x, b);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LDLT: default");
#ifdef EIGEN_CHOLMOD_SUPPORT
{
x = b;
SparseLDLT<SparseSelfAdjointMatrix, Cholmod> ldlt2(m3);
if (ldlt2.succeeded())
{
ldlt2.solveInPlace(x);
VERIFY_IS_APPROX(refMat3.template selfadjointView<Upper>() * x, b);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LDLT: cholmod solveInPlace");
x = ldlt2.solve(b);
VERIFY_IS_APPROX(refMat3.template selfadjointView<Upper>() * x, b);
VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LDLT: cholmod solve");
}
else
std::cerr << "warning LDLT failed\n";
}
#endif
// new Simplicial LLT
// new API
{
SparseMatrix<Scalar> m2(rows, cols);
DenseMatrix refMat2(rows, cols);
DenseVector b = DenseVector::Random(cols);
DenseVector ref_x(cols), x(cols);
DenseMatrix B = DenseMatrix::Random(rows,cols);
DenseMatrix ref_X(rows,cols), X(rows,cols);
initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag|MakeLowerTriangular, 0, 0);
for(int i=0; i<rows; ++i)
m2.coeffRef(i,i) = refMat2(i,i) = internal::abs(internal::real(refMat2(i,i)));
SparseMatrix<Scalar> m3 = m2 * m2.adjoint(), m3_lo(rows,rows), m3_up(rows,rows);
DenseMatrix refMat3 = refMat2 * refMat2.adjoint();
m3_lo.template selfadjointView<Lower>().rankUpdate(m2,0);
m3_up.template selfadjointView<Upper>().rankUpdate(m2,0);
// with a single vector as the rhs
ref_x = refMat3.template selfadjointView<Lower>().llt().solve(b);
x = SimplicialCholesky<SparseMatrix<Scalar>, Lower>().setMode(odd ? SimplicialCholeskyLLt : SimplicialCholeskyLDLt).compute(m3).solve(b);
VERIFY(ref_x.isApprox(x,test_precision<Scalar>()) && "SimplicialCholesky: solve, full storage, lower, single dense rhs");
x = SimplicialCholesky<SparseMatrix<Scalar>, Upper>().setMode(odd ? SimplicialCholeskyLLt : SimplicialCholeskyLDLt).compute(m3).solve(b);
VERIFY(ref_x.isApprox(x,test_precision<Scalar>()) && "SimplicialCholesky: solve, full storage, upper, single dense rhs");
x = SimplicialCholesky<SparseMatrix<Scalar>, Lower>(m3_lo).solve(b);
VERIFY(ref_x.isApprox(x,test_precision<Scalar>()) && "SimplicialCholesky: solve, lower only, single dense rhs");
x = SimplicialCholesky<SparseMatrix<Scalar>, Upper>(m3_up).solve(b);
VERIFY(ref_x.isApprox(x,test_precision<Scalar>()) && "SimplicialCholesky: solve, upper only, single dense rhs");
// with multiple rhs
ref_X = refMat3.template selfadjointView<Lower>().llt().solve(B);
X = SimplicialCholesky<SparseMatrix<Scalar>, Lower>().setMode(odd ? SimplicialCholeskyLLt : SimplicialCholeskyLDLt).compute(m3).solve(B);
VERIFY(ref_X.isApprox(X,test_precision<Scalar>()) && "SimplicialCholesky: solve, full storage, lower, multiple dense rhs");
X = SimplicialCholesky<SparseMatrix<Scalar>, Upper>().setMode(odd ? SimplicialCholeskyLLt : SimplicialCholeskyLDLt).compute(m3).solve(B);
VERIFY(ref_X.isApprox(X,test_precision<Scalar>()) && "SimplicialCholesky: solve, full storage, upper, multiple dense rhs");
// with a sparse rhs
// SparseMatrix<Scalar> spB(rows,cols), spX(rows,cols);
// B.diagonal().array() += 1;
// spB = B.sparseView(0.5,1);
//
// ref_X = refMat3.template selfadjointView<Lower>().llt().solve(DenseMatrix(spB));
//
// spX = SimplicialCholesky<SparseMatrix<Scalar>, Lower>(m3).solve(spB);
// VERIFY(ref_X.isApprox(spX.toDense(),test_precision<Scalar>()) && "LLT: cholmod solve, multiple sparse rhs");
//
// spX = SimplicialCholesky<SparseMatrix<Scalar>, Upper>(m3).solve(spB);
// VERIFY(ref_X.isApprox(spX.toDense(),test_precision<Scalar>()) && "LLT: cholmod solve, multiple sparse rhs");
}
// for(int i=0; i<rows; ++i)
// m2.coeffRef(i,i) = refMat2(i,i) = internal::abs(internal::real(refMat2(i,i)));
//
// refX = refMat2.template selfadjointView<Upper>().ldlt().solve(b);
// typedef SparseMatrix<Scalar,Upper|SelfAdjoint> SparseSelfAdjointMatrix;
// x = b;
// SparseLDLT<SparseSelfAdjointMatrix> ldlt(m2);
// if (ldlt.succeeded())
// ldlt.solveInPlace(x);
// else
// std::cerr << "warning LDLT failed\n";
//
// VERIFY_IS_APPROX(refMat2.template selfadjointView<Upper>() * x, b);
// VERIFY(refX.isApprox(x,test_precision<Scalar>()) && "LDLT: default");
}
template<typename Scalar> void sparse_llt(int rows, int cols)
{
double density = std::max(8./(rows*cols), 0.01);
typedef Matrix<Scalar,Dynamic,Dynamic> DenseMatrix;
typedef Matrix<Scalar,Dynamic,1> DenseVector;
// TODO fix the issue with complex (see SparseLLT::solveInPlace)
SparseMatrix<Scalar> m2(rows, cols);
DenseMatrix refMat2(rows, cols);
DenseVector b = DenseVector::Random(cols);
DenseVector ref_x(cols), x(cols);
DenseMatrix B = DenseMatrix::Random(rows,cols);
DenseMatrix ref_X(rows,cols), X(rows,cols);
initSparse<Scalar>(density, refMat2, m2, ForceNonZeroDiag|MakeLowerTriangular, 0, 0);
for(int i=0; i<rows; ++i)
m2.coeffRef(i,i) = refMat2(i,i) = internal::abs(internal::real(refMat2(i,i)));
ref_x = refMat2.template selfadjointView<Lower>().llt().solve(b);
if (!NumTraits<Scalar>::IsComplex)
{
x = b;
SparseLLT<SparseMatrix<Scalar> > (m2).solveInPlace(x);
VERIFY(ref_x.isApprox(x,test_precision<Scalar>()) && "LLT: default");
}
#ifdef EIGEN_CHOLMOD_SUPPORT
// legacy API
{
// Cholmod, as configured in CholmodSupport.h, only supports self-adjoint matrices
SparseMatrix<Scalar> m3 = m2.adjoint()*m2;
DenseMatrix refMat3 = refMat2.adjoint()*refMat2;
ref_x = refMat3.template selfadjointView<Lower>().llt().solve(b);
x = b;
SparseLLT<SparseMatrix<Scalar>, Cholmod>(m3).solveInPlace(x);
VERIFY((m3*x).isApprox(b,test_precision<Scalar>()) && "LLT legacy: cholmod solveInPlace");
x = SparseLLT<SparseMatrix<Scalar>, Cholmod>(m3).solve(b);
VERIFY(ref_x.isApprox(x,test_precision<Scalar>()) && "LLT legacy: cholmod solve");
}
// new API
{
// Cholmod, as configured in CholmodSupport.h, only supports self-adjoint matrices
SparseMatrix<Scalar> m3 = m2 * m2.adjoint(), m3_lo(rows,rows), m3_up(rows,rows);
DenseMatrix refMat3 = refMat2 * refMat2.adjoint();
m3_lo.template selfadjointView<Lower>().rankUpdate(m2,0);
m3_up.template selfadjointView<Upper>().rankUpdate(m2,0);
// with a single vector as the rhs
ref_x = refMat3.template selfadjointView<Lower>().llt().solve(b);
x = CholmodDecomposition<SparseMatrix<Scalar>, Lower>(m3).solve(b);
VERIFY(ref_x.isApprox(x,test_precision<Scalar>()) && "LLT: cholmod solve, single dense rhs");
x = CholmodDecomposition<SparseMatrix<Scalar>, Upper>(m3).solve(b);
VERIFY(ref_x.isApprox(x,test_precision<Scalar>()) && "LLT: cholmod solve, single dense rhs");
x = CholmodDecomposition<SparseMatrix<Scalar>, Lower>(m3_lo).solve(b);
VERIFY(ref_x.isApprox(x,test_precision<Scalar>()) && "LLT: cholmod solve, single dense rhs");
x = CholmodDecomposition<SparseMatrix<Scalar>, Upper>(m3_up).solve(b);
VERIFY(ref_x.isApprox(x,test_precision<Scalar>()) && "LLT: cholmod solve, single dense rhs");
// with multiple rhs
ref_X = refMat3.template selfadjointView<Lower>().llt().solve(B);
#ifndef EIGEN_DEFAULT_TO_ROW_MAJOR
// TODO make sure the API is properly documented about this fact
X = CholmodDecomposition<SparseMatrix<Scalar>, Lower>(m3).solve(B);
VERIFY(ref_X.isApprox(X,test_precision<Scalar>()) && "LLT: cholmod solve, multiple dense rhs");
X = CholmodDecomposition<SparseMatrix<Scalar>, Upper>(m3).solve(B);
VERIFY(ref_X.isApprox(X,test_precision<Scalar>()) && "LLT: cholmod solve, multiple dense rhs");
#endif
// with a sparse rhs
SparseMatrix<Scalar> spB(rows,cols), spX(rows,cols);
B.diagonal().array() += 1;
spB = B.sparseView(0.5,1);
ref_X = refMat3.template selfadjointView<Lower>().llt().solve(DenseMatrix(spB));
spX = CholmodDecomposition<SparseMatrix<Scalar>, Lower>(m3).solve(spB);
VERIFY(ref_X.isApprox(spX.toDense(),test_precision<Scalar>()) && "LLT: cholmod solve, multiple sparse rhs");
spX = CholmodDecomposition<SparseMatrix<Scalar>, Upper>(m3).solve(spB);
VERIFY(ref_X.isApprox(spX.toDense(),test_precision<Scalar>()) && "LLT: cholmod solve, multiple sparse rhs");
}
#endif
}
