text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2006-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
  //
  // This Source Code Form is subject to the terms of the Mozilla
  // Public License v. 2.0. If a copy of the MPL was not distributed
  // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
  
  #ifndef EIGEN_NUMTRAITS_H
  #define EIGEN_NUMTRAITS_H
  
  namespace Eigen {
  
      namespace internal {
  
  // default implementation of digits10(), based on numeric_limits if specialized,
  // 0 for integer types, and log10(epsilon()) otherwise.
          template<typename T,
                  bool use_numeric_limits = std::numeric_limits<T>::is_specialized,
                  bool is_integer = NumTraits<T>::IsInteger>
          struct default_digits10_impl {
              static int run() { return std::numeric_limits<T>::digits10; }
          };
  
          template<typename T>
          struct default_digits10_impl<T, false, false> // Floating point
          {
              static int run() {
                  using std::log10;
                  using std::ceil;
                  typedef typename NumTraits<T>::Real Real;
                  return int(ceil(-log10(NumTraits<Real>::epsilon())));
              }
          };
  
          template<typename T>
          struct default_digits10_impl<T, false, true> // Integer
          {
              static int run() { return 0; }
          };
  
      } // end namespace internal
  
  /** \class NumTraits
    * \ingroup Core_Module
    *
    * \brief Holds information about the various numeric (i.e. scalar) types allowed by Eigen.
    *
    * \tparam T the numeric type at hand
    *
    * This class stores enums, typedefs and static methods giving information about a numeric type.
    *
    * The provided data consists of:
    * \li A typedef \c Real, giving the "real part" type of \a T. If \a T is already real,
    *     then \c Real is just a typedef to \a T. If \a T is \c std::complex<U> then \c Real
    *     is a typedef to \a U.
    * \li A typedef \c NonInteger, giving the type that should be used for operations producing non-integral values,
    *     such as quotients, square roots, etc. If \a T is a floating-point type, then this typedef just gives
    *     \a T again. Note however that many Eigen functions such as internal::sqrt simply refuse to
    *     take integers. Outside of a few cases, Eigen doesn't do automatic type promotion. Thus, this typedef is
    *     only intended as a helper for code that needs to explicitly promote types.
    * \li A typedef \c Literal giving the type to use for numeric literals such as "2" or "0.5". For instance, for \c std::complex<U>, Literal is defined as \c U.
    *     Of course, this type must be fully compatible with \a T. In doubt, just use \a T here.
    * \li A typedef \a Nested giving the type to use to nest a value inside of the expression tree. If you don't know what
    *     this means, just use \a T here.
    * \li An enum value \a IsComplex. It is equal to 1 if \a T is a \c std::complex
    *     type, and to 0 otherwise.
    * \li An enum value \a IsInteger. It is equal to \c 1 if \a T is an integer type such as \c int,
    *     and to \c 0 otherwise.
    * \li Enum values ReadCost, AddCost and MulCost representing a rough estimate of the number of CPU cycles needed
    *     to by move / add / mul instructions respectively, assuming the data is already stored in CPU registers.
    *     Stay vague here. No need to do architecture-specific stuff.
    * \li An enum value \a IsSigned. It is equal to \c 1 if \a T is a signed type and to 0 if \a T is unsigned.
    * \li An enum value \a RequireInitialization. It is equal to \c 1 if the constructor of the numeric type \a T must
    *     be called, and to 0 if it is safe not to call it. Default is 0 if \a T is an arithmetic type, and 1 otherwise.
    * \li An epsilon() function which, unlike <a href="http://en.cppreference.com/w/cpp/types/numeric_limits/epsilon">std::numeric_limits::epsilon()</a>,
    *     it returns a \a Real instead of a \a T.
    * \li A dummy_precision() function returning a weak epsilon value. It is mainly used as a default
    *     value by the fuzzy comparison operators.
    * \li highest() and lowest() functions returning the highest and lowest possible values respectively.
    * \li digits10() function returning the number of decimal digits that can be represented without change. This is
    *     the analogue of <a href="http://en.cppreference.com/w/cpp/types/numeric_limits/digits10">std::numeric_limits<T>::digits10</a>
    *     which is used as the default implementation if specialized.
    */
  
      template<typename T>
      struct GenericNumTraits {
          enum {
              IsInteger = std::numeric_limits<T>::is_integer,
              IsSigned = std::numeric_limits<T>::is_signed,
              IsComplex = 0,
              RequireInitialization = internal::is_arithmetic<T>::value ? 0 : 1,
              ReadCost = 1,
              AddCost = 1,
              MulCost = 1
          };
  
          typedef T Real;
          typedef typename internal::conditional<
                  IsInteger,
                  typename internal::conditional<sizeof(T) <= 2, float, double>::type,
                  T
          >::type NonInteger;
          typedef T Nested;
          typedef T Literal;
  
          EIGEN_DEVICE_FUNC
          static inline Real epsilon() {
              return numext::numeric_limits<T>::epsilon();
          }
  
          EIGEN_DEVICE_FUNC
          static inline int digits10() {
              return internal::default_digits10_impl<T>::run();
          }
  
          EIGEN_DEVICE_FUNC
          static inline Real dummy_precision() {
              // make sure to override this for floating-point types
              return Real(0);
          }
  
  
          EIGEN_DEVICE_FUNC
          static inline T highest() {
              return (numext::numeric_limits<T>::max)();
          }
  
          EIGEN_DEVICE_FUNC
          static inline T lowest() {
              return IsInteger ? (numext::numeric_limits<T>::min)() : (-(numext::numeric_limits<T>::max)());
          }
  
          EIGEN_DEVICE_FUNC
          static inline T infinity() {
              return numext::numeric_limits<T>::infinity();
          }
  
          EIGEN_DEVICE_FUNC
          static inline T quiet_NaN() {
              return numext::numeric_limits<T>::quiet_NaN();
          }
      };
  
      template<typename T>
      struct NumTraits : GenericNumTraits<T> {
      };
  
      template<>
      struct NumTraits<float>
              : GenericNumTraits<float> {
          EIGEN_DEVICE_FUNC
          static inline float dummy_precision() { return 1e-5f; }
      };
  
      template<>
      struct NumTraits<double> : GenericNumTraits<double> {
          EIGEN_DEVICE_FUNC
          static inline double dummy_precision() { return 1e-12; }
      };
  
      template<>
      struct NumTraits<long double>
              : GenericNumTraits<long double> {
          static inline long double dummy_precision() { return 1e-15l; }
      };
  
      template<typename _Real>
      struct NumTraits<std::complex<_Real> >
              : GenericNumTraits<std::complex<_Real> > {
          typedef _Real Real;
          typedef typename NumTraits<_Real>::Literal Literal;
          enum {
              IsComplex = 1,
              RequireInitialization = NumTraits<_Real>::RequireInitialization,
              ReadCost = 2 * NumTraits<_Real>::ReadCost,
              AddCost = 2 * NumTraits<Real>::AddCost,
              MulCost = 4 * NumTraits<Real>::MulCost + 2 * NumTraits<Real>::AddCost
          };
  
          EIGEN_DEVICE_FUNC
          static inline Real epsilon() { return NumTraits<Real>::epsilon(); }
  
          EIGEN_DEVICE_FUNC
          static inline Real dummy_precision() { return NumTraits<Real>::dummy_precision(); }
  
          EIGEN_DEVICE_FUNC
          static inline int digits10() { return NumTraits<Real>::digits10(); }
      };
  
      template<typename Scalar, int Rows, int Cols, int Options, int MaxRows, int MaxCols>
      struct NumTraits<Array < Scalar, Rows, Cols, Options, MaxRows, MaxCols> > {
      typedef Array <Scalar, Rows, Cols, Options, MaxRows, MaxCols> ArrayType;
      typedef typename NumTraits<Scalar>::Real RealScalar;
      typedef Array <RealScalar, Rows, Cols, Options, MaxRows, MaxCols> Real;
      typedef typename NumTraits<Scalar>::NonInteger NonIntegerScalar;
      typedef Array <NonIntegerScalar, Rows, Cols, Options, MaxRows, MaxCols> NonInteger;
      typedef ArrayType &Nested;
      typedef typename NumTraits<Scalar>::Literal Literal;
  
      enum {
          IsComplex = NumTraits<Scalar>::IsComplex,
          IsInteger = NumTraits<Scalar>::IsInteger,
          IsSigned = NumTraits<Scalar>::IsSigned,
          RequireInitialization = 1,
          ReadCost =
          ArrayType::SizeAtCompileTime == Dynamic ? HugeCost : ArrayType::SizeAtCompileTime * NumTraits<Scalar>::ReadCost,
          AddCost =
          ArrayType::SizeAtCompileTime == Dynamic ? HugeCost : ArrayType::SizeAtCompileTime * NumTraits<Scalar>::AddCost,
          MulCost =
          ArrayType::SizeAtCompileTime == Dynamic ? HugeCost : ArrayType::SizeAtCompileTime * NumTraits<Scalar>::MulCost
      };
  
      EIGEN_DEVICE_FUNC
      static inline RealScalar epsilon() { return NumTraits<RealScalar>::epsilon(); }
  
      EIGEN_DEVICE_FUNC
      static inline RealScalar dummy_precision() { return NumTraits<RealScalar>::dummy_precision(); }
  
      static inline int digits10() { return NumTraits<Scalar>::digits10(); }
  };
  
  template<>
  struct NumTraits<std::string>
          : GenericNumTraits<std::string> {
      enum {
          RequireInitialization = 1,
          ReadCost = HugeCost,
          AddCost = HugeCost,
          MulCost = HugeCost
      };
  
      static inline int digits10() { return 0; }
  
  private:
      static inline std::string epsilon();
  
      static inline std::string dummy_precision();
  
      static inline std::string lowest();
  
      static inline std::string highest();
  
      static inline std::string infinity();
  
      static inline std::string quiet_NaN();
  };
  
  // Empty specialization for void to allow template specialization based on NumTraits<T>::Real with T==void and SFINAE.
  template<>
  struct NumTraits<void> {
  };
  
  } // end namespace Eigen
  
  #endif // EIGEN_NUMTRAITS_H
  