text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2006-2008 Benoit Jacob <jacob.benoit.1@gmail.com>
  // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
  //
  // This Source Code Form is subject to the terms of the Mozilla
  // Public License v. 2.0. If a copy of the MPL was not distributed
  // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
  
  #include <cstdlib>
  #include <cerrno>
  #include <ctime>
  #include <iostream>
  #include <fstream>
  #include <string>
  #include <sstream>
  #include <vector>
  #include <typeinfo>
  #include <functional>
  
  // The following includes of STL headers have to be done _before_ the
  // definition of macros min() and max().  The reason is that many STL
  // implementations will not work properly as the min and max symbols collide
  // with the STL functions std:min() and std::max().  The STL headers may check
  // for the macro definition of min/max and issue a warning or undefine the
  // macros.
  //
  // Still, Windows defines min() and max() in windef.h as part of the regular
  // Windows system interfaces and many other Windows APIs depend on these
  // macros being available.  To prevent the macro expansion of min/max and to
  // make Eigen compatible with the Windows environment all function calls of
  // std::min() and std::max() have to be written with parenthesis around the
  // function name.
  //
  // All STL headers used by Eigen should be included here.  Because main.h is
  // included before any Eigen header and because the STL headers are guarded
  // against multiple inclusions, no STL header will see our own min/max macro
  // definitions.
  #include <limits>
  #include <algorithm>
  #include <complex>
  #include <deque>
  #include <queue>
  #include <cassert>
  #include <list>
  #if __cplusplus >= 201103L
  #include <random>
  #ifdef EIGEN_USE_THREADS
  #include <future>
  #endif
  #endif
  
  // Same for cuda_fp16.h
  #if defined(__CUDACC_VER_MAJOR__) && (__CUDACC_VER_MAJOR__ >= 9)
  #define EIGEN_TEST_CUDACC_VER  ((__CUDACC_VER_MAJOR__ * 10000) + (__CUDACC_VER_MINOR__ * 100))
  #elif defined(__CUDACC_VER__)
  #define EIGEN_TEST_CUDACC_VER __CUDACC_VER__
  #else
  #define EIGEN_TEST_CUDACC_VER 0
  #endif
  
  #if EIGEN_TEST_CUDACC_VER >= 70500
  #include <cuda_fp16.h>
  #endif
  
  // To test that all calls from Eigen code to std::min() and std::max() are
  // protected by parenthesis against macro expansion, the min()/max() macros
  // are defined here and any not-parenthesized min/max call will cause a
  // compiler error.
  #if !defined(__HIPCC__)
    //
    // HIP header files include the following files
    //  <thread>
    //  <regex>
    //  <unordered_map>
    // which seem to contain not-parenthesized calls to "max"/"min", triggering the following check and causing the compile to fail
    //
    // Including those header files before the following macro definition for "min" / "max", only partially resolves the issue
    // This is because other HIP header files also define "isnan" / "isinf" / "isfinite" functions, which are needed in other
    // headers.
    //
    // So instead choosing to simply disable this check for HIP
    //
    #define min(A,B) please_protect_your_min_with_parentheses
    #define max(A,B) please_protect_your_max_with_parentheses
    #define isnan(X) please_protect_your_isnan_with_parentheses
    #define isinf(X) please_protect_your_isinf_with_parentheses
    #define isfinite(X) please_protect_your_isfinite_with_parentheses
  #endif
  
  #ifdef M_PI
  #undef M_PI
  #endif
  #define M_PI please_use_EIGEN_PI_instead_of_M_PI
  
  #define FORBIDDEN_IDENTIFIER (this_identifier_is_forbidden_to_avoid_clashes) this_identifier_is_forbidden_to_avoid_clashes
  // B0 is defined in POSIX header termios.h
  #define B0 FORBIDDEN_IDENTIFIER
  
  // Unit tests calling Eigen's blas library must preserve the default blocking size
  // to avoid troubles.
  #ifndef EIGEN_NO_DEBUG_SMALL_PRODUCT_BLOCKS
  #define EIGEN_DEBUG_SMALL_PRODUCT_BLOCKS
  #endif
  
  // shuts down ICC's remark #593: variable "XXX" was set but never used
  #define TEST_SET_BUT_UNUSED_VARIABLE(X) EIGEN_UNUSED_VARIABLE(X)
  
  #ifdef TEST_ENABLE_TEMPORARY_TRACKING
  
  static long int nb_temporaries;
  static long int nb_temporaries_on_assert = -1;
  
  inline void on_temporary_creation(long int size) {
    // here's a great place to set a breakpoint when debugging failures in this test!
    if(size!=0) nb_temporaries++;
    if(nb_temporaries_on_assert>0) assert(nb_temporaries<nb_temporaries_on_assert);
  }
  
  #define EIGEN_DENSE_STORAGE_CTOR_PLUGIN { on_temporary_creation(size); }
  
  #define VERIFY_EVALUATION_COUNT(XPR,N) {\
      nb_temporaries = 0; \
      XPR; \
      if(nb_temporaries!=(N)) { std::cerr << "nb_temporaries == " << nb_temporaries << "\n"; }\
      VERIFY( (#XPR) && nb_temporaries==(N) ); \
    }
  
  #endif
  
  #include "split_test_helper.h"
  
  #ifdef NDEBUG
  #undef NDEBUG
  #endif
  
  // On windows CE, NDEBUG is automatically defined <assert.h> if NDEBUG is not defined.
  #ifndef DEBUG
  #define DEBUG
  #endif
  
  // bounds integer values for AltiVec
  #if defined(__ALTIVEC__) || defined(__VSX__)
  #define EIGEN_MAKING_DOCS
  #endif
  
  #define DEFAULT_REPEAT 10
  
  namespace Eigen
  {
    static std::vector<std::string> g_test_stack;
    // level == 0 <=> abort if test fail
    // level >= 1 <=> warning message to std::cerr if test fail
    static int g_test_level = 0;
    static int g_repeat = 1;
    static unsigned int g_seed = 0;
    static bool g_has_set_repeat = false, g_has_set_seed = false;
  
    class EigenTest
    {
    public:
      EigenTest() : m_func(0) {}
      EigenTest(const char* a_name, void (*func)(void))
        : m_name(a_name), m_func(func)
      {
        ms_registered_tests.push_back(this);
      }
      const std::string& name() const { return m_name; }
      void operator()() const { m_func(); }
  
      static const std::vector<EigenTest*>& all() { return ms_registered_tests; }
    protected:
      std::string m_name;
      void (*m_func)(void);
      static std::vector<EigenTest*> ms_registered_tests;
    };
  
    std::vector<EigenTest*> EigenTest::ms_registered_tests;
  
    // Declare and register a test, e.g.:
    //    EIGEN_DECLARE_TEST(mytest) { ... }
    // will create a function:
    //    void test_mytest() { ... }
    // that will be automatically called.
    #define EIGEN_DECLARE_TEST(X) \
      void EIGEN_CAT(test_,X) (); \
      static EigenTest EIGEN_CAT(test_handler_,X) (EIGEN_MAKESTRING(X), & EIGEN_CAT(test_,X)); \
      void EIGEN_CAT(test_,X) ()
  }
  
  #define TRACK std::cerr << __FILE__ << " " << __LINE__ << std::endl
  // #define TRACK while()
  
  #define EIGEN_DEFAULT_IO_FORMAT IOFormat(4, 0, "  ", "\n", "", "", "", "")
  
  #if (defined(_CPPUNWIND) || defined(__EXCEPTIONS)) && !defined(__CUDA_ARCH__) && !defined(__HIP_DEVICE_COMPILE__) && !defined(__SYCL_DEVICE_ONLY__)
    #define EIGEN_EXCEPTIONS
  #endif
  
  #ifndef EIGEN_NO_ASSERTION_CHECKING
  
    namespace Eigen
    {
      static const bool should_raise_an_assert = false;
  
      // Used to avoid to raise two exceptions at a time in which
      // case the exception is not properly caught.
      // This may happen when a second exceptions is triggered in a destructor.
      static bool no_more_assert = false;
      static bool report_on_cerr_on_assert_failure = true;
  
      struct eigen_assert_exception
      {
        eigen_assert_exception(void) {}
        ~eigen_assert_exception() { Eigen::no_more_assert = false; }
      };
  
      struct eigen_static_assert_exception
      {
        eigen_static_assert_exception(void) {}
        ~eigen_static_assert_exception() { Eigen::no_more_assert = false; }
      };
    }
    // If EIGEN_DEBUG_ASSERTS is defined and if no assertion is triggered while
    // one should have been, then the list of executed assertions is printed out.
    //
    // EIGEN_DEBUG_ASSERTS is not enabled by default as it
    // significantly increases the compilation time
    // and might even introduce side effects that would hide
    // some memory errors.
    #ifdef EIGEN_DEBUG_ASSERTS
  
      namespace Eigen
      {
        namespace internal
        {
          static bool push_assert = false;
        }
        static std::vector<std::string> eigen_assert_list;
      }
      #define eigen_assert(a)                       \
        if( (!(a)) && (!no_more_assert) )     \
        { \
          if(report_on_cerr_on_assert_failure) \
            std::cerr <<  #a << " " __FILE__ << "(" << __LINE__ << ")\n"; \
          Eigen::no_more_assert = true;       \
          EIGEN_THROW_X(Eigen::eigen_assert_exception()); \
        }                                     \
        else if (Eigen::internal::push_assert)       \
        {                                     \
          eigen_assert_list.push_back(std::string(EIGEN_MAKESTRING(__FILE__) " (" EIGEN_MAKESTRING(__LINE__) ") : " #a) ); \
        }
  
      #ifdef EIGEN_EXCEPTIONS
      #define VERIFY_RAISES_ASSERT(a)                                                   \
        {                                                                               \
          Eigen::no_more_assert = false;                                                \
          Eigen::eigen_assert_list.clear();                                             \
          Eigen::internal::push_assert = true;                                          \
          Eigen::report_on_cerr_on_assert_failure = false;                              \
          try {                                                                         \
            a;                                                                          \
            std::cerr << "One of the following asserts should have been triggered:\n";  \
            for (uint ai=0 ; ai<eigen_assert_list.size() ; ++ai)                        \
              std::cerr << "  " << eigen_assert_list[ai] << "\n";                       \
            VERIFY(Eigen::should_raise_an_assert && # a);                               \
          } catch (Eigen::eigen_assert_exception) {                                     \
            Eigen::internal::push_assert = false; VERIFY(true);                         \
          }                                                                             \
          Eigen::report_on_cerr_on_assert_failure = true;                               \
          Eigen::internal::push_assert = false;                                         \
        }
      #endif //EIGEN_EXCEPTIONS
  
    #elif !defined(__CUDACC__) && !defined(__HIPCC__) && !defined(__SYCL_DEVICE_ONLY__) // EIGEN_DEBUG_ASSERTS
      // see bug 89. The copy_bool here is working around a bug in gcc <= 4.3
      #define eigen_assert(a) \
        if( (!Eigen::internal::copy_bool(a)) && (!no_more_assert) )\
        {                                       \
          Eigen::no_more_assert = true;         \
          if(report_on_cerr_on_assert_failure)  \
            eigen_plain_assert(a);              \
          else                                  \
            EIGEN_THROW_X(Eigen::eigen_assert_exception()); \
        }
  
      #ifdef EIGEN_EXCEPTIONS
        #define VERIFY_RAISES_ASSERT(a) {                           \
          Eigen::no_more_assert = false;                            \
          Eigen::report_on_cerr_on_assert_failure = false;          \
          try {                                                     \
            a;                                                      \
            VERIFY(Eigen::should_raise_an_assert && # a);           \
          }                                                         \
          catch (Eigen::eigen_assert_exception&) { VERIFY(true); }  \
          Eigen::report_on_cerr_on_assert_failure = true;           \
        }
      #endif // EIGEN_EXCEPTIONS
    #endif // EIGEN_DEBUG_ASSERTS
  
    #if defined(TEST_CHECK_STATIC_ASSERTIONS) && defined(EIGEN_EXCEPTIONS)
      #define EIGEN_STATIC_ASSERT(a,MSG) \
        if( (!Eigen::internal::copy_bool(a)) && (!no_more_assert) )\
        {                                       \
          Eigen::no_more_assert = true;         \
          if(report_on_cerr_on_assert_failure)  \
            eigen_plain_assert((a) && #MSG);      \
          else                                  \
            EIGEN_THROW_X(Eigen::eigen_static_assert_exception()); \
        }
      #define VERIFY_RAISES_STATIC_ASSERT(a) {                    \
        Eigen::no_more_assert = false;                            \
        Eigen::report_on_cerr_on_assert_failure = false;          \
        try {                                                     \
          a;                                                      \
          VERIFY(Eigen::should_raise_an_assert && # a);           \
        }                                                         \
        catch (Eigen::eigen_static_assert_exception&) { VERIFY(true); }  \
        Eigen::report_on_cerr_on_assert_failure = true;           \
      }
    #endif // TEST_CHECK_STATIC_ASSERTIONS
  
  #ifndef VERIFY_RAISES_ASSERT
    #define VERIFY_RAISES_ASSERT(a) \
      std::cout << "Can't VERIFY_RAISES_ASSERT( " #a " ) with exceptions disabled\n";
  #endif
  #ifndef VERIFY_RAISES_STATIC_ASSERT
    #define VERIFY_RAISES_STATIC_ASSERT(a) \
      std::cout << "Can't VERIFY_RAISES_STATIC_ASSERT( " #a " ) with exceptions disabled\n";
  #endif
  
    #if !defined(__CUDACC__) && !defined(__HIPCC__) && !defined(__SYCL_DEVICE_ONLY__)
    #define EIGEN_USE_CUSTOM_ASSERT
    #endif
  
  #else // EIGEN_NO_ASSERTION_CHECKING
  
    #define VERIFY_RAISES_ASSERT(a) {}
    #define VERIFY_RAISES_STATIC_ASSERT(a) {}
  
  #endif // EIGEN_NO_ASSERTION_CHECKING
  
  #define EIGEN_INTERNAL_DEBUGGING
  #include <Eigen/QR> // required for createRandomPIMatrixOfRank
  
  inline void verify_impl(bool condition, const char *testname, const char *file, int line, const char *condition_as_string)
  {
    if (!condition)
    {
      if(Eigen::g_test_level>0)
        std::cerr << "WARNING: ";
      std::cerr << "Test " << testname << " failed in " << file << " (" << line << ")"
        << std::endl << "    " << condition_as_string << std::endl;
      std::cerr << "Stack:\n";
      const int test_stack_size = static_cast<int>(Eigen::g_test_stack.size());
      for(int i=test_stack_size-1; i>=0; --i)
        std::cerr << "  - " << Eigen::g_test_stack[i] << "\n";
      std::cerr << "\n";
      if(Eigen::g_test_level==0)
        abort();
    }
  }
  
  #define VERIFY(a) ::verify_impl(a, g_test_stack.back().c_str(), __FILE__, __LINE__, EIGEN_MAKESTRING(a))
  
  #define VERIFY_GE(a, b) ::verify_impl(a >= b, g_test_stack.back().c_str(), __FILE__, __LINE__, EIGEN_MAKESTRING(a >= b))
  #define VERIFY_LE(a, b) ::verify_impl(a <= b, g_test_stack.back().c_str(), __FILE__, __LINE__, EIGEN_MAKESTRING(a <= b))
  
  
  #define VERIFY_IS_EQUAL(a, b) VERIFY(test_is_equal(a, b, true))
  #define VERIFY_IS_NOT_EQUAL(a, b) VERIFY(test_is_equal(a, b, false))
  #define VERIFY_IS_APPROX(a, b) VERIFY(verifyIsApprox(a, b))
  #define VERIFY_IS_NOT_APPROX(a, b) VERIFY(!test_isApprox(a, b))
  #define VERIFY_IS_MUCH_SMALLER_THAN(a, b) VERIFY(test_isMuchSmallerThan(a, b))
  #define VERIFY_IS_NOT_MUCH_SMALLER_THAN(a, b) VERIFY(!test_isMuchSmallerThan(a, b))
  #define VERIFY_IS_APPROX_OR_LESS_THAN(a, b) VERIFY(test_isApproxOrLessThan(a, b))
  #define VERIFY_IS_NOT_APPROX_OR_LESS_THAN(a, b) VERIFY(!test_isApproxOrLessThan(a, b))
  
  #define VERIFY_IS_UNITARY(a) VERIFY(test_isUnitary(a))
  
  #define STATIC_CHECK(COND) EIGEN_STATIC_ASSERT( (COND) , EIGEN_INTERNAL_ERROR_PLEASE_FILE_A_BUG_REPORT )
  
  #define CALL_SUBTEST(FUNC) do { \
      g_test_stack.push_back(EIGEN_MAKESTRING(FUNC)); \
      FUNC; \
      g_test_stack.pop_back(); \
    } while (0)
  
  
  namespace Eigen {
  
  template<typename T1,typename T2>
  typename internal::enable_if<internal::is_same<T1,T2>::value,bool>::type
  is_same_type(const T1&, const T2&)
  {
    return true;
  }
  
  template<typename T> inline typename NumTraits<T>::Real test_precision() { return NumTraits<T>::dummy_precision(); }
  template<> inline float test_precision<float>() { return 1e-3f; }
  template<> inline double test_precision<double>() { return 1e-6; }
  template<> inline long double test_precision<long double>() { return 1e-6l; }
  template<> inline float test_precision<std::complex<float> >() { return test_precision<float>(); }
  template<> inline double test_precision<std::complex<double> >() { return test_precision<double>(); }
  template<> inline long double test_precision<std::complex<long double> >() { return test_precision<long double>(); }
  
  inline bool test_isApprox(const short& a, const short& b)
  { return internal::isApprox(a, b, test_precision<short>()); }
  inline bool test_isApprox(const unsigned short& a, const unsigned short& b)
  { return internal::isApprox(a, b, test_precision<unsigned short>()); }
  inline bool test_isApprox(const unsigned int& a, const unsigned int& b)
  { return internal::isApprox(a, b, test_precision<unsigned int>()); }
  inline bool test_isApprox(const long& a, const long& b)
  { return internal::isApprox(a, b, test_precision<long>()); }
  inline bool test_isApprox(const unsigned long& a, const unsigned long& b)
  { return internal::isApprox(a, b, test_precision<unsigned long>()); }
  
  inline bool test_isApprox(const int& a, const int& b)
  { return internal::isApprox(a, b, test_precision<int>()); }
  inline bool test_isMuchSmallerThan(const int& a, const int& b)
  { return internal::isMuchSmallerThan(a, b, test_precision<int>()); }
  inline bool test_isApproxOrLessThan(const int& a, const int& b)
  { return internal::isApproxOrLessThan(a, b, test_precision<int>()); }
  
  inline bool test_isApprox(const float& a, const float& b)
  { return internal::isApprox(a, b, test_precision<float>()); }
  inline bool test_isMuchSmallerThan(const float& a, const float& b)
  { return internal::isMuchSmallerThan(a, b, test_precision<float>()); }
  inline bool test_isApproxOrLessThan(const float& a, const float& b)
  { return internal::isApproxOrLessThan(a, b, test_precision<float>()); }
  
  inline bool test_isApprox(const double& a, const double& b)
  { return internal::isApprox(a, b, test_precision<double>()); }
  inline bool test_isMuchSmallerThan(const double& a, const double& b)
  { return internal::isMuchSmallerThan(a, b, test_precision<double>()); }
  inline bool test_isApproxOrLessThan(const double& a, const double& b)
  { return internal::isApproxOrLessThan(a, b, test_precision<double>()); }
  
  #ifndef EIGEN_TEST_NO_COMPLEX
  inline bool test_isApprox(const std::complex<float>& a, const std::complex<float>& b)
  { return internal::isApprox(a, b, test_precision<std::complex<float> >()); }
  inline bool test_isMuchSmallerThan(const std::complex<float>& a, const std::complex<float>& b)
  { return internal::isMuchSmallerThan(a, b, test_precision<std::complex<float> >()); }
  
  inline bool test_isApprox(const std::complex<double>& a, const std::complex<double>& b)
  { return internal::isApprox(a, b, test_precision<std::complex<double> >()); }
  inline bool test_isMuchSmallerThan(const std::complex<double>& a, const std::complex<double>& b)
  { return internal::isMuchSmallerThan(a, b, test_precision<std::complex<double> >()); }
  
  #ifndef EIGEN_TEST_NO_LONGDOUBLE
  inline bool test_isApprox(const std::complex<long double>& a, const std::complex<long double>& b)
  { return internal::isApprox(a, b, test_precision<std::complex<long double> >()); }
  inline bool test_isMuchSmallerThan(const std::complex<long double>& a, const std::complex<long double>& b)
  { return internal::isMuchSmallerThan(a, b, test_precision<std::complex<long double> >()); }
  #endif
  #endif
  
  #ifndef EIGEN_TEST_NO_LONGDOUBLE
  inline bool test_isApprox(const long double& a, const long double& b)
  {
      bool ret = internal::isApprox(a, b, test_precision<long double>());
      if (!ret) std::cerr
          << std::endl << "    actual   = " << a
          << std::endl << "    expected = " << b << std::endl << std::endl;
      return ret;
  }
  
  inline bool test_isMuchSmallerThan(const long double& a, const long double& b)
  { return internal::isMuchSmallerThan(a, b, test_precision<long double>()); }
  inline bool test_isApproxOrLessThan(const long double& a, const long double& b)
  { return internal::isApproxOrLessThan(a, b, test_precision<long double>()); }
  #endif // EIGEN_TEST_NO_LONGDOUBLE
  
  inline bool test_isApprox(const half& a, const half& b)
  { return internal::isApprox(a, b, test_precision<half>()); }
  inline bool test_isMuchSmallerThan(const half& a, const half& b)
  { return internal::isMuchSmallerThan(a, b, test_precision<half>()); }
  inline bool test_isApproxOrLessThan(const half& a, const half& b)
  { return internal::isApproxOrLessThan(a, b, test_precision<half>()); }
  
  // test_relative_error returns the relative difference between a and b as a real scalar as used in isApprox.
  template<typename T1,typename T2>
  typename NumTraits<typename T1::RealScalar>::NonInteger test_relative_error(const EigenBase<T1> &a, const EigenBase<T2> &b)
  {
    using std::sqrt;
    typedef typename NumTraits<typename T1::RealScalar>::NonInteger RealScalar;
    typename internal::nested_eval<T1,2>::type ea(a.derived());
    typename internal::nested_eval<T2,2>::type eb(b.derived());
    return sqrt(RealScalar((ea-eb).cwiseAbs2().sum()) / RealScalar((std::min)(eb.cwiseAbs2().sum(),ea.cwiseAbs2().sum())));
  }
  
  template<typename T1,typename T2>
  typename T1::RealScalar test_relative_error(const T1 &a, const T2 &b, const typename T1::Coefficients* = 0)
  {
    return test_relative_error(a.coeffs(), b.coeffs());
  }
  
  template<typename T1,typename T2>
  typename T1::Scalar test_relative_error(const T1 &a, const T2 &b, const typename T1::MatrixType* = 0)
  {
    return test_relative_error(a.matrix(), b.matrix());
  }
  
  template<typename S, int D>
  S test_relative_error(const Translation<S,D> &a, const Translation<S,D> &b)
  {
    return test_relative_error(a.vector(), b.vector());
  }
  
  template <typename S, int D, int O>
  S test_relative_error(const ParametrizedLine<S,D,O> &a, const ParametrizedLine<S,D,O> &b)
  {
    return (std::max)(test_relative_error(a.origin(), b.origin()), test_relative_error(a.origin(), b.origin()));
  }
  
  template <typename S, int D>
  S test_relative_error(const AlignedBox<S,D> &a, const AlignedBox<S,D> &b)
  {
    return (std::max)(test_relative_error((a.min)(), (b.min)()), test_relative_error((a.max)(), (b.max)()));
  }
  
  template<typename Derived> class SparseMatrixBase;
  template<typename T1,typename T2>
  typename T1::RealScalar test_relative_error(const MatrixBase<T1> &a, const SparseMatrixBase<T2> &b)
  {
    return test_relative_error(a,b.toDense());
  }
  
  template<typename Derived> class SparseMatrixBase;
  template<typename T1,typename T2>
  typename T1::RealScalar test_relative_error(const SparseMatrixBase<T1> &a, const MatrixBase<T2> &b)
  {
    return test_relative_error(a.toDense(),b);
  }
  
  template<typename Derived> class SparseMatrixBase;
  template<typename T1,typename T2>
  typename T1::RealScalar test_relative_error(const SparseMatrixBase<T1> &a, const SparseMatrixBase<T2> &b)
  {
    return test_relative_error(a.toDense(),b.toDense());
  }
  
  template<typename T1,typename T2>
  typename NumTraits<typename NumTraits<T1>::Real>::NonInteger test_relative_error(const T1 &a, const T2 &b, typename internal::enable_if<internal::is_arithmetic<typename NumTraits<T1>::Real>::value, T1>::type* = 0)
  {
    typedef typename NumTraits<typename NumTraits<T1>::Real>::NonInteger RealScalar;
    return numext::sqrt(RealScalar(numext::abs2(a-b))/RealScalar((numext::mini)(numext::abs2(a),numext::abs2(b))));
  }
  
  template<typename T>
  T test_relative_error(const Rotation2D<T> &a, const Rotation2D<T> &b)
  {
    return test_relative_error(a.angle(), b.angle());
  }
  
  template<typename T>
  T test_relative_error(const AngleAxis<T> &a, const AngleAxis<T> &b)
  {
    return (std::max)(test_relative_error(a.angle(), b.angle()), test_relative_error(a.axis(), b.axis()));
  }
  
  template<typename Type1, typename Type2>
  inline bool test_isApprox(const Type1& a, const Type2& b, typename Type1::Scalar* = 0) // Enabled for Eigen's type only
  {
    return a.isApprox(b, test_precision<typename Type1::Scalar>());
  }
  
  // get_test_precision is a small wrapper to test_precision allowing to return the scalar precision for either scalars or expressions
  template<typename T>
  typename NumTraits<typename T::Scalar>::Real get_test_precision(const T&, const typename T::Scalar* = 0)
  {
    return test_precision<typename NumTraits<typename T::Scalar>::Real>();
  }
  
  template<typename T>
  typename NumTraits<T>::Real get_test_precision(const T&,typename internal::enable_if<internal::is_arithmetic<typename NumTraits<T>::Real>::value, T>::type* = 0)
  {
    return test_precision<typename NumTraits<T>::Real>();
  }
  
  // verifyIsApprox is a wrapper to test_isApprox that outputs the relative difference magnitude if the test fails.
  template<typename Type1, typename Type2>
  inline bool verifyIsApprox(const Type1& a, const Type2& b)
  {
    bool ret = test_isApprox(a,b);
    if(!ret)
    {
      std::cerr << "Difference too large wrt tolerance " << get_test_precision(a)  << ", relative error is: " << test_relative_error(a,b) << std::endl;
    }
    return ret;
  }
  
  // The idea behind this function is to compare the two scalars a and b where
  // the scalar ref is a hint about the expected order of magnitude of a and b.
  // WARNING: the scalar a and b must be positive
  // Therefore, if for some reason a and b are very small compared to ref,
  // we won't issue a false negative.
  // This test could be: abs(a-b) <= eps * ref
  // However, it seems that simply comparing a+ref and b+ref is more sensitive to true error.
  template<typename Scalar,typename ScalarRef>
  inline bool test_isApproxWithRef(const Scalar& a, const Scalar& b, const ScalarRef& ref)
  {
    return test_isApprox(a+ref, b+ref);
  }
  
  template<typename Derived1, typename Derived2>
  inline bool test_isMuchSmallerThan(const MatrixBase<Derived1>& m1,
                                     const MatrixBase<Derived2>& m2)
  {
    return m1.isMuchSmallerThan(m2, test_precision<typename internal::traits<Derived1>::Scalar>());
  }
  
  template<typename Derived>
  inline bool test_isMuchSmallerThan(const MatrixBase<Derived>& m,
                                     const typename NumTraits<typename internal::traits<Derived>::Scalar>::Real& s)
  {
    return m.isMuchSmallerThan(s, test_precision<typename internal::traits<Derived>::Scalar>());
  }
  
  template<typename Derived>
  inline bool test_isUnitary(const MatrixBase<Derived>& m)
  {
    return m.isUnitary(test_precision<typename internal::traits<Derived>::Scalar>());
  }
  
  // Forward declaration to avoid ICC warning
  template<typename T, typename U>
  bool test_is_equal(const T& actual, const U& expected, bool expect_equal=true);
  
  template<typename T, typename U>
  bool test_is_equal(const T& actual, const U& expected, bool expect_equal)
  {
      if ((actual==expected) == expect_equal)
          return true;
      // false:
      std::cerr
          << "\n    actual   = " << actual
          << "\n    expected " << (expect_equal ? "= " : "!=") << expected << "\n\n";
      return false;
  }
  
  /** Creates a random Partial Isometry matrix of given rank.
    *
    * A partial isometry is a matrix all of whose singular values are either 0 or 1.
    * This is very useful to test rank-revealing algorithms.
    */
  // Forward declaration to avoid ICC warning
  template<typename MatrixType>
  void createRandomPIMatrixOfRank(Index desired_rank, Index rows, Index cols, MatrixType& m);
  template<typename MatrixType>
  void createRandomPIMatrixOfRank(Index desired_rank, Index rows, Index cols, MatrixType& m)
  {
    typedef typename internal::traits<MatrixType>::Scalar Scalar;
    enum { Rows = MatrixType::RowsAtCompileTime, Cols = MatrixType::ColsAtCompileTime };
  
    typedef Matrix<Scalar, Dynamic, 1> VectorType;
    typedef Matrix<Scalar, Rows, Rows> MatrixAType;
    typedef Matrix<Scalar, Cols, Cols> MatrixBType;
  
    if(desired_rank == 0)
    {
      m.setZero(rows,cols);
      return;
    }
  
    if(desired_rank == 1)
    {
      // here we normalize the vectors to get a partial isometry
      m = VectorType::Random(rows).normalized() * VectorType::Random(cols).normalized().transpose();
      return;
    }
  
    MatrixAType a = MatrixAType::Random(rows,rows);
    MatrixType d = MatrixType::Identity(rows,cols);
    MatrixBType  b = MatrixBType::Random(cols,cols);
  
    // set the diagonal such that only desired_rank non-zero entries reamain
    const Index diag_size = (std::min)(d.rows(),d.cols());
    if(diag_size != desired_rank)
      d.diagonal().segment(desired_rank, diag_size-desired_rank) = VectorType::Zero(diag_size-desired_rank);
  
    HouseholderQR<MatrixAType> qra(a);
    HouseholderQR<MatrixBType> qrb(b);
    m = qra.householderQ() * d * qrb.householderQ();
  }
  
  // Forward declaration to avoid ICC warning
  template<typename PermutationVectorType>
  void randomPermutationVector(PermutationVectorType& v, Index size);
  template<typename PermutationVectorType>
  void randomPermutationVector(PermutationVectorType& v, Index size)
  {
    typedef typename PermutationVectorType::Scalar Scalar;
    v.resize(size);
    for(Index i = 0; i < size; ++i) v(i) = Scalar(i);
    if(size == 1) return;
    for(Index n = 0; n < 3 * size; ++n)
    {
      Index i = internal::random<Index>(0, size-1);
      Index j;
      do j = internal::random<Index>(0, size-1); while(j==i);
      std::swap(v(i), v(j));
    }
  }
  
  template<typename T> bool isNotNaN(const T& x)
  {
    return x==x;
  }
  
  template<typename T> bool isPlusInf(const T& x)
  {
    return x > NumTraits<T>::highest();
  }
  
  template<typename T> bool isMinusInf(const T& x)
  {
    return x < NumTraits<T>::lowest();
  }
  
  } // end namespace Eigen
  
  template<typename T> struct GetDifferentType;
  
  template<> struct GetDifferentType<float> { typedef double type; };
  template<> struct GetDifferentType<double> { typedef float type; };
  template<typename T> struct GetDifferentType<std::complex<T> >
  { typedef std::complex<typename GetDifferentType<T>::type> type; };
  
  // Forward declaration to avoid ICC warning
  template<typename T> std::string type_name();
  template<typename T> std::string type_name()                    { return "other"; }
  template<> std::string type_name<float>()                       { return "float"; }
  template<> std::string type_name<double>()                      { return "double"; }
  template<> std::string type_name<long double>()                 { return "long double"; }
  template<> std::string type_name<int>()                         { return "int"; }
  template<> std::string type_name<std::complex<float> >()        { return "complex<float>"; }
  template<> std::string type_name<std::complex<double> >()       { return "complex<double>"; }
  template<> std::string type_name<std::complex<long double> >()  { return "complex<long double>"; }
  template<> std::string type_name<std::complex<int> >()          { return "complex<int>"; }
  
  using namespace Eigen;
  
  inline void set_repeat_from_string(const char *str)
  {
    errno = 0;
    g_repeat = int(strtoul(str, 0, 10));
    if(errno || g_repeat <= 0)
    {
      std::cout << "Invalid repeat value " << str << std::endl;
      exit(EXIT_FAILURE);
    }
    g_has_set_repeat = true;
  }
  
  inline void set_seed_from_string(const char *str)
  {
    errno = 0;
    g_seed = int(strtoul(str, 0, 10));
    if(errno || g_seed == 0)
    {
      std::cout << "Invalid seed value " << str << std::endl;
      exit(EXIT_FAILURE);
    }
    g_has_set_seed = true;
  }
  
  int main(int argc, char *argv[])
  {
      g_has_set_repeat = false;
      g_has_set_seed = false;
      bool need_help = false;
  
      for(int i = 1; i < argc; i++)
      {
        if(argv[i][0] == 'r')
        {
          if(g_has_set_repeat)
          {
            std::cout << "Argument " << argv[i] << " conflicting with a former argument" << std::endl;
            return 1;
          }
          set_repeat_from_string(argv[i]+1);
        }
        else if(argv[i][0] == 's')
        {
          if(g_has_set_seed)
          {
            std::cout << "Argument " << argv[i] << " conflicting with a former argument" << std::endl;
            return 1;
          }
           set_seed_from_string(argv[i]+1);
        }
        else
        {
          need_help = true;
        }
      }
  
      if(need_help)
      {
        std::cout << "This test application takes the following optional arguments:" << std::endl;
        std::cout << "  rN     Repeat each test N times (default: " << DEFAULT_REPEAT << ")" << std::endl;
        std::cout << "  sN     Use N as seed for random numbers (default: based on current time)" << std::endl;
        std::cout << std::endl;
        std::cout << "If defined, the environment variables EIGEN_REPEAT and EIGEN_SEED" << std::endl;
        std::cout << "will be used as default values for these parameters." << std::endl;
        return 1;
      }
  
      char *env_EIGEN_REPEAT = getenv("EIGEN_REPEAT");
      if(!g_has_set_repeat && env_EIGEN_REPEAT)
        set_repeat_from_string(env_EIGEN_REPEAT);
      char *env_EIGEN_SEED = getenv("EIGEN_SEED");
      if(!g_has_set_seed && env_EIGEN_SEED)
        set_seed_from_string(env_EIGEN_SEED);
  
      if(!g_has_set_seed) g_seed = (unsigned int) time(NULL);
      if(!g_has_set_repeat) g_repeat = DEFAULT_REPEAT;
  
      std::cout << "Initializing random number generator with seed " << g_seed << std::endl;
      std::stringstream ss;
      ss << "Seed: " << g_seed;
      g_test_stack.push_back(ss.str());
      srand(g_seed);
      std::cout << "Repeating each test " << g_repeat << " times" << std::endl;
  
      VERIFY(EigenTest::all().size()>0);
  
      for(std::size_t i=0; i<EigenTest::all().size(); ++i)
      {
        const EigenTest& current_test = *EigenTest::all()[i];
        Eigen::g_test_stack.push_back(current_test.name());
        current_test();
        Eigen::g_test_stack.pop_back();
      }
  
      return 0;
  }
  
  // These warning are disabled here such that they are still ON when parsing Eigen's header files.
  #if defined __INTEL_COMPILER
    // remark #383: value copied to temporary, reference to temporary used
    //  -> this warning is raised even for legal usage as: g_test_stack.push_back("foo"); where g_test_stack is a std::vector<std::string>
    // remark #1418: external function definition with no prior declaration
    //  -> this warning is raised for all our test functions. Declaring them static would fix the issue.
    // warning #279: controlling expression is constant
    // remark #1572: floating-point equality and inequality comparisons are unreliable
    #pragma warning disable 279 383 1418 1572
  #endif
  
  #ifdef _MSC_VER
    // 4503 - decorated name length exceeded, name was truncated
    #pragma warning( disable : 4503)
  #endif
  