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* add cmake files to find (optional) supported libraries

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* add unit tests for sparse cholesky
This commit is contained in:
Gael Guennebaud 2008-10-20 10:43:11 +00:00
parent f44316e5f8
commit fa27cd1ed0
10 changed files with 331 additions and 97 deletions
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7
Eigen/Sparse
View File

@ -16,9 +16,16 @@
#ifdef EIGEN_TAUCS_SUPPORT
// taucs.h declares a lot of mess
#define isnan
#define finite
#define isinf
extern "C" {
#include "taucs.h"
}
#undef isnan
#undef finite
#undef isinf
#ifdef min
#undef min

5
Eigen/src/Sparse/CholmodSupport.h
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@ -195,11 +195,12 @@ template<typename MatrixType>
template<typename Derived>
void SparseLLT<MatrixType,Cholmod>::solveInPlace(MatrixBase<Derived> &b) const
{
if (m_status & MatrixLIsDirty)
matrixL();
const int size = m_matrix.rows();
ei_assert(size==b.rows());
if (m_status & MatrixLIsDirty)
matrixL();
Base::solveInPlace(b);
}

4
Eigen/src/Sparse/SparseLLT.h
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@ -139,7 +139,6 @@ void SparseLLT<MatrixType,Backend>::compute(const MatrixType& a)
for (int j = 0; j < size; ++j)
{
Scalar x = ei_real(a.coeff(j,j));
int endSize = size-j-1;
// TODO better estimate of the density !
tempVector.init(density>0.001? IsDense : IsSparse);
@ -191,7 +190,8 @@ bool SparseLLT<MatrixType, Backend>::solveInPlace(MatrixBase<Derived> &b) const
ei_assert(size==b.rows());
m_matrix.solveTriangularInPlace(b);
m_matrix.adjoint().solveTriangularInPlace(b);
// FIXME should be .adjoint() but it fails to compile...
m_matrix.transpose().solveTriangularInPlace(b);
return true;
}

19
Eigen/src/Sparse/TaucsSupport.h
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@ -151,7 +151,7 @@ void SparseLLT<MatrixType,Taucs>::compute(const MatrixType& a)
else
{
// use the faster Multifrontal routine
m_taucsSupernodalFactor = taucs_ccs_factor_llt_ll(&taucsMatA);
m_taucsSupernodalFactor = taucs_ccs_factor_llt_mf(&taucsMatA);
}
m_status = (m_status & ~IncompleteFactorization) | CompleteFactorization | MatrixLIsDirty;
}
@ -177,16 +177,19 @@ template<typename MatrixType>
template<typename Derived>
void SparseLLT<MatrixType,Taucs>::solveInPlace(MatrixBase<Derived> &b) const
{
const int size = m_matrix.rows();
ei_assert(size==b.rows());
if (m_status & MatrixLIsDirty)
{
// ei_assert(!(m_status & SupernodalFactorIsDirty));
// taucs_supernodal_solve_llt(m_taucsSupernodalFactor,double* b);
//matrixL();
// TODO use taucs's supernodal solver, in particular check types, storage order, etc.
// VectorXb x(b.rows());
// for (int j=0; j<b.cols(); ++j)
// {
// taucs_supernodal_solve_llt(m_taucsSupernodalFactor,x.data(),&b.col(j).coeffRef(0));
// b.col(j) = x;
// }
matrixL();
}
else
{
Base::solveInPlace(b);
}

76
cmake/FindCholmod.cmake Normal file
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@ -0,0 +1,76 @@
if (CHOLMOD_INCLUDES AND CHOLMOD_LIBRARIES)
set(CHOLMOD_FIND_QUIETLY TRUE)
endif (CHOLMOD_INCLUDES AND CHOLMOD_LIBRARIES)
find_path(CHOLMOD_INCLUDES
NAMES
cholmod.h
PATHS
$ENV{CHOLMODDIR}
${INCLUDE_INSTALL_DIR}
PATH_SUFFIXES
suitesparse
)
find_library(CHOLMOD_LIBRARIES cholmod PATHS $ENV{CHOLMODDIR} ${LIB_INSTALL_DIR})
if(CHOLMOD_LIBRARIES)
get_filename_component(CHOLMOD_LIBDIR ${CHOLMOD_LIBRARIES} PATH)
message("found CHOLMOD_LIBRARIES " ${CHOLMOD_LIBRARIES} " in " ${CHOLMOD_LIBDIR})
find_library(AMD_LIBRARY amd PATHS ${CHOLMOD_LIBDIR} $ENV{CHOLMODDIR} ${LIB_INSTALL_DIR})
if (AMD_LIBRARY)
set(CHOLMOD_LIBRARIES ${CHOLMOD_LIBRARIES} ${AMD_LIBRARY})
message ("AMD_LIBRARY found")
else (AMD_LIBRARY)
message ("AMD_LIBRARY not found")
set(CHOLMOD_LIBRARIES FALSE)
endif (AMD_LIBRARY)
endif(CHOLMOD_LIBRARIES)
if(CHOLMOD_LIBRARIES)
message("CHOLMOD_LIBRARIES still here")
find_library(COLAMD_LIBRARY colamd PATHS ${CHOLMOD_LIBDIR} $ENV{CHOLMODDIR} ${LIB_INSTALL_DIR})
if (COLAMD_LIBRARY)
set(CHOLMOD_LIBRARIES ${CHOLMOD_LIBRARIES} ${COLAMD_LIBRARY})
else (COLAMD_LIBRARY)
set(CHOLMOD_LIBRARIES FALSE)
endif (COLAMD_LIBRARY)
endif(CHOLMOD_LIBRARIES)
if(CHOLMOD_LIBRARIES)
message("CHOLMOD_LIBRARIES still here")
find_library(CAMD_LIBRARY camd PATHS ${CHOLMOD_LIBDIR} $ENV{CHOLMODDIR} ${LIB_INSTALL_DIR})
if (CAMD_LIBRARY)
set(CHOLMOD_LIBRARIES ${CHOLMOD_LIBRARIES} ${CAMD_LIBRARY})
else (CAMD_LIBRARY)
set(CHOLMOD_LIBRARIES FALSE)
endif (CAMD_LIBRARY)
endif(CHOLMOD_LIBRARIES)
if(CHOLMOD_LIBRARIES)
message("CHOLMOD_LIBRARIES still here (last)")
find_library(CCOLAMD_LIBRARY colamd PATHS ${CHOLMOD_LIBDIR} $ENV{CHOLMODDIR} ${LIB_INSTALL_DIR})
if (CCOLAMD_LIBRARY)
set(CHOLMOD_LIBRARIES ${CHOLMOD_LIBRARIES} ${CCOLAMD_LIBRARY})
else (CCOLAMD_LIBRARY)
set(CHOLMOD_LIBRARIES FALSE)
endif (CCOLAMD_LIBRARY)
endif(CHOLMOD_LIBRARIES)
# if(CHOLMOD_LIBRARIES)
# endif(CHOLMOD_LIBRARIES)
include(FindPackageHandleStandardArgs)
find_package_handle_standard_args(CHOLMOD DEFAULT_MSG
CHOLMOD_INCLUDES CHOLMOD_LIBRARIES)
mark_as_advanced(CHOLMOD_INCLUDES CHOLMOD_LIBRARIES AMD_LIBRARY COLAMD_LIBRARY)

20
cmake/FindSuperLU.cmake Normal file
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@ -0,0 +1,20 @@
if (SUPERLU_INCLUDES AND SUPERLU_LIBRARIES)
set(SUPERLU_FIND_QUIETLY TRUE)
endif (SUPERLU_INCLUDES AND SUPERLU_LIBRARIES)
find_path(SUPERLU_INCLUDES
NAMES
superlu/supermatrix.h
PATHS
$ENV{SUPERLUDIR}
${INCLUDE_INSTALL_DIR}
)
find_library(SUPERLU_LIBRARIES superlu PATHS $ENV{SUPERLUDIR} ${LIB_INSTALL_DIR})
include(FindPackageHandleStandardArgs)
find_package_handle_standard_args(SUPERLU DEFAULT_MSG
SUPERLU_INCLUDES SUPERLU_LIBRARIES)
mark_as_advanced(SUPERLU_INCLUDES SUPERLU_LIBRARIES)

20
cmake/FindTaucs.cmake Normal file
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@ -0,0 +1,20 @@
if (TAUCS_INCLUDES AND TAUCS_LIBRARIES)
set(TAUCS_FIND_QUIETLY TRUE)
endif (TAUCS_INCLUDES AND TAUCS_LIBRARIES)
find_path(TAUCS_INCLUDES
NAMES
taucs.h
PATHS
$ENV{TAUCSDIR}
${INCLUDE_INSTALL_DIR}
)
find_library(TAUCS_LIBRARIES taucs PATHS $ENV{TAUCSDIR} ${LIB_INSTALL_DIR})
include(FindPackageHandleStandardArgs)
find_package_handle_standard_args(TAUCS DEFAULT_MSG
TAUCS_INCLUDES TAUCS_LIBRARIES)
mark_as_advanced(TAUCS_INCLUDES TAUCS_LIBRARIES)

46
cmake/FindUmfpack.cmake Normal file
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@ -0,0 +1,46 @@
if (UMFPACK_INCLUDES AND UMFPACK_LIBRARIES)
set(UMFPACK_FIND_QUIETLY TRUE)
endif (UMFPACK_INCLUDES AND UMFPACK_LIBRARIES)
find_path(UMFPACK_INCLUDES
NAMES
umfpack.h
PATHS
$ENV{UMFPACKDIR}
${INCLUDE_INSTALL_DIR}
PATH_SUFFIXES
suitesparse
)
find_library(UMFPACK_LIBRARIES umfpack PATHS $ENV{UMFPACKDIR} ${LIB_INSTALL_DIR})
if(UMFPACK_LIBRARIES)
get_filename_component(UMFPACK_LIBDIR ${UMFPACK_LIBRARIES} PATH)
find_library(AMD_LIBRARY amd PATHS ${UMFPACK_LIBDIR} $ENV{UMFPACKDIR} ${LIB_INSTALL_DIR})
if (AMD_LIBRARY)
set(UMFPACK_LIBRARIES ${UMFPACK_LIBRARIES} ${AMD_LIBRARY})
else (AMD_LIBRARY)
set(UMFPACK_LIBRARIES FALSE)
endif (AMD_LIBRARY)
endif(UMFPACK_LIBRARIES)
if(UMFPACK_LIBRARIES)
find_library(COLAMD_LIBRARY colamd PATHS ${UMFPACK_LIBDIR} $ENV{UMFPACKDIR} ${LIB_INSTALL_DIR})
if (COLAMD_LIBRARY)
set(UMFPACK_LIBRARIES ${UMFPACK_LIBRARIES} ${COLAMD_LIBRARY})
else (COLAMD_LIBRARY)
set(UMFPACK_LIBRARIES FALSE)
endif (COLAMD_LIBRARY)
endif(UMFPACK_LIBRARIES)
include(FindPackageHandleStandardArgs)
find_package_handle_standard_args(UMFPACK DEFAULT_MSG
UMFPACK_INCLUDES UMFPACK_LIBRARIES)
mark_as_advanced(UMFPACK_INCLUDES UMFPACK_LIBRARIES AMD_LIBRARY COLAMD_LIBRARY)

190
test/CMakeLists.txt
View File

@ -1,31 +1,59 @@
IF(BUILD_TESTS)
if(BUILD_TESTS)
set(EXTERNAL_LIBS "")
find_package(GSL)
if(GSL_FOUND)
add_definitions("-DHAS_GSL")
set(EXTERNAL_LIBS ${EXTERNAL_LIBS} ${GSL_LIBRARIES})
endif(GSL_FOUND)
IF(CMAKE_COMPILER_IS_GNUCXX)
IF(CMAKE_SYSTEM_NAME MATCHES Linux)
SET(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -g2")
SET(CMAKE_CXX_FLAGS_RELWITHDEBINFO "${CMAKE_CXX_FLAGS_RELWITHDEBINFO} -O2 -g2")
SET(CMAKE_CXX_FLAGS_RELEASE "${CMAKE_CXX_FLAGS_RELEASE} -fno-inline-functions")
SET(CMAKE_CXX_FLAGS_DEBUG "${CMAKE_CXX_FLAGS_DEBUG} -O0 -g2")
ENDIF(CMAKE_SYSTEM_NAME MATCHES Linux)
SET(EI_OFLAG "-O2")
ELSE(CMAKE_COMPILER_IS_GNUCXX)
SET(EI_OFLAG "")
ENDIF(CMAKE_COMPILER_IS_GNUCXX)
find_package(Taucs)
if(TAUCS_FOUND)
add_definitions("-DEIGEN_TAUCS_SUPPORT")
include_directories(${TAUCS_INCLUDES})
set(EXTERNAL_LIBS ${EXTERNAL_LIBS} ${TAUCS_LIBRARIES})
endif(TAUCS_FOUND)
OPTION(EIGEN_NO_ASSERTION_CHECKING "Disable checking of assertions" OFF)
find_package(Cholmod)
if(CHOLMOD_FOUND)
message("add EIGEN_CHOLMOD_SUPPORT " ${CHOLMOD_LIBRARIES})
add_definitions("-DEIGEN_CHOLMOD_SUPPORT")
include_directories(${CHOLMOD_INCLUDES})
set(EXTERNAL_LIBS ${EXTERNAL_LIBS} ${CHOLMOD_LIBRARIES})
endif(CHOLMOD_FOUND)
# similar to SET_TARGET_PROPERTIES but append the property instead of overwriting it
MACRO(EI_ADD_TARGET_PROPERTY target prop value)
# find_package(Umfpack)
# if(UMFPACK_FOUND)
# add_definitions("-DEIGEN_UMFPACK_SUPPORT")
# endif(UMFPACK_FOUND)
#
# find_package(SuperLU)
# if(SUPERLU_FOUND)
# add_definitions("-DEIGEN_SUPERLU_SUPPORT")
# endif(SUPERLU_FOUND)
GET_TARGET_PROPERTY(previous ${target} ${prop})
SET_TARGET_PROPERTIES(${target} PROPERTIES ${prop} "${previous} ${value}")
if(CMAKE_COMPILER_IS_GNUCXX)
if(CMAKE_SYSTEM_NAME MATCHES Linux)
set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -g2")
set(CMAKE_CXX_FLAGS_RELWITHDEBINFO "${CMAKE_CXX_FLAGS_RELWITHDEBINFO} -O2 -g2")
set(CMAKE_CXX_FLAGS_RELEASE "${CMAKE_CXX_FLAGS_RELEASE} -fno-inline-functions")
set(CMAKE_CXX_FLAGS_DEBUG "${CMAKE_CXX_FLAGS_DEBUG} -O0 -g2")
endif(CMAKE_SYSTEM_NAME MATCHES Linux)
set(EI_OFLAG "-O2")
else(CMAKE_COMPILER_IS_GNUCXX)
set(EI_OFLAG "")
endif(CMAKE_COMPILER_IS_GNUCXX)
ENDMACRO(EI_ADD_TARGET_PROPERTY)
option(EIGEN_NO_ASSERTION_CHECKING "Disable checking of assertions" OFF)
# similar to set_target_properties but append the property instead of overwriting it
macro(ei_add_target_property target prop value)
get_target_property(previous ${target} ${prop})
set_target_properties(${target} PROPERTIES ${prop} "${previous} ${value}")
endmacro(ei_add_target_property)
# Macro to add a test
#
@ -42,86 +70,84 @@ ENDMACRO(EI_ADD_TARGET_PROPERTY)
# "ctest -V" or "ctest -V -R <testname>"
# On other platform use ctest as usual
#
MACRO(EI_ADD_TEST testname)
macro(ei_add_test testname)
SET(targetname test_${testname})
set(targetname test_${testname})
SET(filename ${testname}.cpp)
ADD_EXECUTABLE(${targetname} ${filename})
set(filename ${testname}.cpp)
add_executable(${targetname} ${filename})
IF(NOT EIGEN_NO_ASSERTION_CHECKING)
if(NOT EIGEN_NO_ASSERTION_CHECKING)
SET_TARGET_PROPERTIES(${targetname} PROPERTIES COMPILE_FLAGS "-fexceptions")
OPTION(EIGEN_DEBUG_ASSERTS "Enable debuging of assertions" OFF)
IF(EIGEN_DEBUG_ASSERTS)
SET_TARGET_PROPERTIES(${targetname} PROPERTIES COMPILE_DEFINITIONS "-DEIGEN_DEBUG_ASSERTS=1")
ENDIF(EIGEN_DEBUG_ASSERTS)
set_target_properties(${targetname} PROPERTIES COMPILE_FLAGS "-fexceptions")
option(EIGEN_DEBUG_ASSERTS "Enable debuging of assertions" OFF)
if(EIGEN_DEBUG_ASSERTS)
set_target_properties(${targetname} PROPERTIES COMPILE_DEFINITIONS "-DEIGEN_DEBUG_ASSERTS=1")
endif(EIGEN_DEBUG_ASSERTS)
ELSE(NOT EIGEN_NO_ASSERTION_CHECKING)
else(NOT EIGEN_NO_ASSERTION_CHECKING)
SET_TARGET_PROPERTIES(${targetname} PROPERTIES COMPILE_DEFINITIONS "-DEIGEN_NO_ASSERTION_CHECKING=1")
set_target_properties(${targetname} PROPERTIES COMPILE_DEFINITIONS "-DEIGEN_NO_ASSERTION_CHECKING=1")
ENDIF(NOT EIGEN_NO_ASSERTION_CHECKING)
endif(NOT EIGEN_NO_ASSERTION_CHECKING)
IF(${ARGC} GREATER 1)
EI_ADD_TARGET_PROPERTY(${targetname} COMPILE_FLAGS "${ARGV1}")
ENDIF(${ARGC} GREATER 1)
if(${ARGC} GREATER 1)
ei_add_target_property(${targetname} COMPILE_FLAGS "${ARGV1}")
endif(${ARGC} GREATER 1)
EI_ADD_TARGET_PROPERTY(${targetname} COMPILE_FLAGS "-DEIGEN_TEST_FUNC=${testname}")
ei_add_target_property(${targetname} COMPILE_FLAGS "-DEIGEN_TEST_FUNC=${testname}")
IF(TEST_LIB)
if(TEST_LIB)
target_link_libraries(${targetname} Eigen2)
ENDIF(TEST_LIB)
endif(TEST_LIB)
if(GSL_FOUND)
target_link_libraries(${targetname} ${GSL_LIBRARIES})
endif(GSL_FOUND)
target_link_libraries(${targetname} ${EXTERNAL_LIBS})
IF(WIN32)
ADD_TEST(${testname} "${targetname}")
ELSE(WIN32)
ADD_TEST(${testname} "${CMAKE_CURRENT_SOURCE_DIR}/runtest.sh" "${testname}")
ENDIF(WIN32)
if(WIN32)
add_test(${testname} "${targetname}")
else(WIN32)
add_test(${testname} "${CMAKE_CURRENT_SOURCE_DIR}/runtest.sh" "${testname}")
endif(WIN32)
ENDMACRO(EI_ADD_TEST)
endmacro(ei_add_test)
ENABLE_TESTING()
enable_testing()
IF(TEST_LIB)
ADD_DEFINITIONS("-DEIGEN_EXTERN_INSTANTIATIONS=1")
ENDIF(TEST_LIB)
if(TEST_LIB)
add_definitions("-DEIGEN_EXTERN_INSTANTIATIONS=1")
endif(TEST_LIB)
EI_ADD_TEST(meta)
EI_ADD_TEST(sizeof)
EI_ADD_TEST(dynalloc)
EI_ADD_TEST(nomalloc)
EI_ADD_TEST(packetmath)
EI_ADD_TEST(basicstuff)
EI_ADD_TEST(linearstructure)
EI_ADD_TEST(cwiseop)
EI_ADD_TEST(sum)
EI_ADD_TEST(product_small)
EI_ADD_TEST(product_large ${EI_OFLAG})
EI_ADD_TEST(adjoint)
EI_ADD_TEST(submatrices)
EI_ADD_TEST(miscmatrices)
EI_ADD_TEST(commainitializer)
EI_ADD_TEST(smallvectors)
EI_ADD_TEST(map)
EI_ADD_TEST(array)
EI_ADD_TEST(triangular)
EI_ADD_TEST(cholesky)
EI_ADD_TEST(lu ${EI_OFLAG})
EI_ADD_TEST(determinant)
EI_ADD_TEST(inverse)
EI_ADD_TEST(qr)
EI_ADD_TEST(eigensolver)
EI_ADD_TEST(svd)
EI_ADD_TEST(geometry)
EI_ADD_TEST(hyperplane)
EI_ADD_TEST(parametrizedline)
EI_ADD_TEST(regression)
EI_ADD_TEST(sparse ${EI_OFLAG})
ei_add_test(meta)
ei_add_test(sizeof)
ei_add_test(dynalloc)
ei_add_test(nomalloc)
ei_add_test(packetmath)
ei_add_test(basicstuff)
ei_add_test(linearstructure)
ei_add_test(cwiseop)
ei_add_test(sum)
ei_add_test(product_small)
ei_add_test(product_large ${EI_OFLAG})
ei_add_test(adjoint)
ei_add_test(submatrices)
ei_add_test(miscmatrices)
ei_add_test(commainitializer)
ei_add_test(smallvectors)
ei_add_test(map)
ei_add_test(array)
ei_add_test(triangular)
ei_add_test(cholesky)
ei_add_test(lu ${EI_OFLAG})
ei_add_test(determinant)
ei_add_test(inverse)
ei_add_test(qr)
ei_add_test(eigensolver)
ei_add_test(svd)
ei_add_test(geometry)
ei_add_test(hyperplane)
ei_add_test(parametrizedline)
ei_add_test(regression)
ei_add_test(sparse )
ENDIF(BUILD_TESTS)
endif(BUILD_TESTS)

41
test/sparse.cpp
View File

@ -23,6 +23,8 @@
// Eigen. If not, see <http://www.gnu.org/licenses/>.
#include "main.h"
#include <Eigen/Cholesky>
#include <Eigen/LU>
#include <Eigen/Sparse>
enum {
@ -46,8 +48,7 @@ initSparse(double density,
{
Scalar v = (ei_random<Scalar>(0,1) < density) ? ei_random<Scalar>() : 0;
if ((flags&ForceNonZeroDiag) && (i==j))
while (ei_abs(v)<1e-2)
v = ei_random<Scalar>();
v = ei_random<Scalar>(Scalar(5.),Scalar(20.));
if ((flags & MakeLowerTriangular) && j>i)
v = 0;
else if ((flags & MakeUpperTriangular) && j<i)
@ -98,7 +99,7 @@ template<typename Scalar> void sparse(int rows, int cols)
refMat.coeffRef(nonzeroCoords[0].x(), nonzeroCoords[0].y()) = Scalar(5);
VERIFY_IS_APPROX(m, refMat);
#if 0
// test InnerIterators and Block expressions
for(int j=0; j<cols; j++)
{
@ -216,9 +217,43 @@ template<typename Scalar> void sparse(int rows, int cols)
// TODO test row major
}
#endif
// test LLT
{
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, &zeroCoords, &nonzeroCoords);
refMat2 += refMat2.adjoint();
refMat2.diagonal() *= 0.5;
refMat2.llt().solve(b, &refX);
// std::cerr << refMat2 << "\n\n" << refMat2.llt().matrixL() << "\n\n";
// std::cerr << m2 << "\n\n";
typedef SparseMatrix<Scalar,Lower|SelfAdjoint> SparseSelfAdjointMatrix;
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
#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
}
}