
// 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>
// Disable ICC's std::complex operator specializations so we can use our own.
#define _OVERRIDE_COMPLEX_SPECIALIZATION_ 1
#include <complex>
#include <deque>
#include <queue>
#include <cassert>
#include <list>
#if __cplusplus >= 201103L || (defined(_MSVC_LANG) && _MSVC_LANG >= 201103L)
#include <random>
#include <chrono>
#ifdef EIGEN_USE_THREADS
#include <future>
#endif
#endif

// Same for cuda_fp16.h
#if defined(__CUDACC__) && !defined(EIGEN_NO_CUDA)
  // Means the compiler is either nvcc or clang with CUDA enabled
  #define EIGEN_CUDACC __CUDACC__
#endif
#if defined(EIGEN_CUDACC)
#include <cuda.h>
  #define EIGEN_CUDA_SDK_VER (CUDA_VERSION * 10)
#else
  #define EIGEN_CUDA_SDK_VER 0
#endif
#if EIGEN_CUDA_SDK_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__) && !defined(EIGEN_USE_SYCL)
  //
  // 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


// test possible conflicts
struct real {};
struct imag {};

#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
// `I` may be defined by complex.h:
#define I  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)
    {
      get_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 get_registered_tests(); }
  protected:
    static std::vector<EigenTest*>& get_registered_tests()
    {
      static std::vector<EigenTest*>* ms_registered_tests = new std::vector<EigenTest*>();
      return *ms_registered_tests;
    }
    std::string m_name;
    void (*m_func)(void);
  };

  // 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 and generateRandomMatrixSvs

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_CWISE_EQUAL(a, b) VERIFY(test_isCwiseApprox(a, b, true))
#define VERIFY_IS_CWISE_APPROX(a, b) VERIFY(test_isCwiseApprox(a, b, false))

#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 {

// Forward declarations to avoid ICC warnings
template<typename T, typename U>
bool test_is_equal(const T& actual, const U& expected, bool expect_equal=true);

template<typename MatrixType>
void createRandomPIMatrixOfRank(Index desired_rank, Index rows, Index cols, MatrixType& m);

template<typename PermutationVectorType>
void randomPermutationVector(PermutationVectorType& v, Index size);

template<typename MatrixType>
MatrixType generateRandomUnitaryMatrix(const Index dim);

template<typename MatrixType, typename RealScalarVectorType>
void generateRandomMatrixSvs(const RealScalarVectorType &svs, const Index rows, const Index cols, MatrixType& M);

template<typename VectorType, typename RealScalar>
VectorType setupRandomSvs(const Index dim, const RealScalar max);

template<typename VectorType, typename RealScalar>
VectorType setupRangeSvs(const Index dim, const RealScalar min, const RealScalar max);

} // end namespace Eigen

// Forward declaration to avoid ICC warnings
template<typename T> std::string type_name();


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>(); }

#define EIGEN_TEST_SCALAR_TEST_OVERLOAD(TYPE)                             \
  inline bool test_isApprox(TYPE a, TYPE b)                               \
  { return internal::isApprox(a, b, test_precision<TYPE>()); }            \
  inline bool test_isMuchSmallerThan(TYPE a, TYPE b)                      \
  { return internal::isMuchSmallerThan(a, b, test_precision<TYPE>()); }   \
  inline bool test_isApproxOrLessThan(TYPE a, TYPE b)                     \
  { return internal::isApproxOrLessThan(a, b, test_precision<TYPE>()); }

EIGEN_TEST_SCALAR_TEST_OVERLOAD(short)
EIGEN_TEST_SCALAR_TEST_OVERLOAD(unsigned short)
EIGEN_TEST_SCALAR_TEST_OVERLOAD(int)
EIGEN_TEST_SCALAR_TEST_OVERLOAD(unsigned int)
EIGEN_TEST_SCALAR_TEST_OVERLOAD(long)
EIGEN_TEST_SCALAR_TEST_OVERLOAD(unsigned long)
#if EIGEN_HAS_CXX11
EIGEN_TEST_SCALAR_TEST_OVERLOAD(long long)
EIGEN_TEST_SCALAR_TEST_OVERLOAD(unsigned long long)
#endif
EIGEN_TEST_SCALAR_TEST_OVERLOAD(float)
EIGEN_TEST_SCALAR_TEST_OVERLOAD(double)
EIGEN_TEST_SCALAR_TEST_OVERLOAD(half)
EIGEN_TEST_SCALAR_TEST_OVERLOAD(bfloat16)

#undef EIGEN_TEST_SCALAR_TEST_OVERLOAD

#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

// 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))/(numext::mini)(RealScalar(numext::abs2(a)),RealScalar(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>());
}

// Checks component-wise, works with infs and nans.
template<typename Derived1, typename Derived2>
bool test_isCwiseApprox(const DenseBase<Derived1>& m1,
                        const DenseBase<Derived2>& m2,
                        bool exact) {
  if (m1.rows() != m2.rows()) {
    return false;
  }
  if (m1.cols() != m2.cols()) {
    return false;
  }
  for (Index r = 0; r < m1.rows(); ++r) {
    for (Index c = 0; c < m1.cols(); ++c) {
      if (m1(r, c) != m2(r, c)
          && !((numext::isnan)(m1(r, c)) && (numext::isnan)(m2(r, c))) 
          && (exact || !test_isApprox(m1(r, c), m2(r, c)))) {
        return false;
      }
    }
  }
  return 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;
}

// Forward declaration to avoid ICC warning
template<typename MatrixType>
void createRandomPIMatrixOfRank(Index desired_rank, Index rows, Index cols, MatrixType& m);
/**
 * 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.
 *
 * @tparam MatrixType type of random partial isometry matrix
 * @param desired_rank rank requested for the random partial isometry matrix
 * @param rows row dimension of requested random partial isometry matrix
 * @param cols column dimension of requested random partial isometry matrix
 * @param m random partial isometry matrix
 */
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 remain
  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);
/**
 * Generate random permutation vector.
 *
 * @tparam PermutationVectorType type of vector used to store permutation
 * @param v permutation vector
 * @param size length of permutation vector
 */
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));
  }
}

/**
 * Generate a random unitary matrix of prescribed dimension.
 *
 * The algorithm is using a random Householder sequence to produce
 * a random unitary matrix.
 *
 * @tparam MatrixType type of matrix to generate
 * @param dim row and column dimension of the requested square matrix
 * @return random unitary matrix
 */
template<typename MatrixType>
MatrixType generateRandomUnitaryMatrix(const Index dim)
{
  typedef typename internal::traits<MatrixType>::Scalar Scalar;
  typedef Matrix<Scalar, Dynamic, 1> VectorType;

  MatrixType v = MatrixType::Identity(dim, dim);
  VectorType h = VectorType::Zero(dim);
  for (Index i = 0; i < dim; ++i)
  {
    v.col(i).tail(dim - i - 1) = VectorType::Random(dim - i - 1);
    h(i) = 2 / v.col(i).tail(dim - i).squaredNorm();
  }

  const Eigen::HouseholderSequence<MatrixType, VectorType> HSeq(v, h);
  return MatrixType(HSeq);
}

/**
 * Generation of random matrix with prescribed singular values.
 *
 * We generate random matrices with given singular values by setting up
 * a singular value decomposition. By choosing the number of zeros as
 * singular values we can specify the rank of the matrix.
 * Moreover, we also control its spectral norm, which is the largest
 * singular value, as well as its condition number with respect to the
 * l2-norm, which is the quotient of the largest and smallest singular
 * value.
 *
 * Reference: For details on the method see e.g. Section 8.1 (pp. 62 f) in
 *
 *   C. C. Paige, M. A. Saunders,
 *   LSQR: An algorithm for sparse linear equations and sparse least squares.
 *   ACM Transactions on Mathematical Software 8(1), pp. 43-71, 1982.
 *   https://web.stanford.edu/group/SOL/software/lsqr/lsqr-toms82a.pdf
 *
 * and also the LSQR webpage https://web.stanford.edu/group/SOL/software/lsqr/.
 *
 * @tparam MatrixType matrix type to generate
 * @tparam RealScalarVectorType vector type with real entries used for singular values
 * @param svs vector of desired singular values
 * @param rows row dimension of requested random matrix
 * @param cols column dimension of requested random matrix
 * @param M generated matrix with prescribed singular values
 */
template<typename MatrixType, typename RealScalarVectorType>
void generateRandomMatrixSvs(const RealScalarVectorType &svs, const Index rows, const Index cols, MatrixType& M)
{
  enum { Rows = MatrixType::RowsAtCompileTime, Cols = MatrixType::ColsAtCompileTime };
  typedef typename internal::traits<MatrixType>::Scalar Scalar;
  typedef Matrix<Scalar, Rows, Rows> MatrixAType;
  typedef Matrix<Scalar, Cols, Cols> MatrixBType;

  const Index min_dim = (std::min)(rows, cols);

  const MatrixAType U = generateRandomUnitaryMatrix<MatrixAType>(rows);
  const MatrixBType V = generateRandomUnitaryMatrix<MatrixBType>(cols);

  M = U.block(0, 0, rows, min_dim) * svs.asDiagonal() * V.block(0, 0, cols, min_dim).transpose();
}

/**
 * Setup a vector of random singular values with prescribed upper limit.
 * For use with generateRandomMatrixSvs().
 *
 * Singular values are non-negative real values. By convention (to be consistent with
 * singular value decomposition) we sort them in decreasing order.
 *
 * This strategy produces random singular values in the range [0, max], in particular
 * the singular values can be zero or arbitrarily close to zero.
 *
 * @tparam VectorType vector type with real entries used for singular values
 * @tparam RealScalar data type used for real entry
 * @param dim number of singular values to generate
 * @param max upper bound for singular values
 * @return vector of singular values
 */
template<typename VectorType, typename RealScalar>
VectorType setupRandomSvs(const Index dim, const RealScalar max)
{
  VectorType svs = max / RealScalar(2) * (VectorType::Random(dim) + VectorType::Ones(dim));
  std::sort(svs.begin(), svs.end(), std::greater<RealScalar>());
  return svs;
}

/**
 * Setup a vector of random singular values with prescribed range.
 * For use with generateRandomMatrixSvs().
 *
 * Singular values are non-negative real values. By convention (to be consistent with
 * singular value decomposition) we sort them in decreasing order.
 *
 * For dim > 1 this strategy generates a vector with largest entry max, smallest entry
 * min, and remaining entries in the range [min, max]. For dim == 1 the only entry is
 * min.
 *
 * @tparam VectorType vector type with real entries used for singular values
 * @tparam RealScalar data type used for real entry
 * @param dim number of singular values to generate
 * @param min smallest singular value to use
 * @param max largest singular value to use
 * @return vector of singular values
 */
template<typename VectorType, typename RealScalar>
VectorType setupRangeSvs(const Index dim, const RealScalar min, const RealScalar max)
{
  VectorType svs = VectorType::Random(dim);
  if(dim == 0)
    return svs;
  if(dim == 1)
  {
    svs(0) = min;
    return svs;
  }
  std::sort(svs.begin(), svs.end(), std::greater<RealScalar>());

  // scale to range [min, max]
  const RealScalar c_min = svs(dim - 1), c_max = svs(0);
  svs = (svs - VectorType::Constant(dim, c_min)) / (c_max - c_min);
  return min * (VectorType::Ones(dim) - svs) + max * svs;
}

/**
 * Check if number is "not a number" (NaN).
 *
 * @tparam T input type
 * @param x input value
 * @return true, if input value is "not a number" (NaN)
 */
template<typename T> bool isNotNaN(const T& x)
{
  return x==x;
}

/**
 * Check if number is plus infinity.
 *
 * @tparam T input type
 * @param x input value
 * @return true, if input value is plus infinity
 */
template<typename T> bool isPlusInf(const T& x)
{
  return x > NumTraits<T>::highest();
}

/**
 * Check if number is minus infinity.
 *
 * @tparam T input type
 * @param x input value
 * @return true, if input value is minus infinity
 */
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; };

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;

/**
 * Set number of repetitions for unit test from input string.
 *
 * @param str input string
 */
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;
}

/**
 * Set seed for randomized unit tests from input string.
 *
 * @param str input string
 */
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