File content as of revision 15:9c5aaeda36dc:

/*
 * Tiny Vector Matrix Library
 * Dense Vector Matrix Libary of Tiny size using Expression Templates
 *
 * Copyright (C) 2001 - 2007 Olaf Petzold <opetzold@users.sourceforge.net>
 *
 * This library is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public
 * License as published by the Free Software Foundation; either
 * version 2.1 of the License, or (at your option) any later version.
 *
 * This library is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
 * Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public
 * License along with this library; if not, write to the Free Software
 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 *
 * $Id: NumericTraits.h,v 1.16 2007-06-23 15:58:58 opetzold Exp $
 */

#ifndef TVMET_NUMERIC_TRAITS_H
#define TVMET_NUMERIC_TRAITS_H

#if defined(TVMET_HAVE_COMPLEX)
#  include <complex>
#endif
#include <cmath>
#include <limits>

#include <tvmet/CompileTimeError.h>


namespace tvmet {


/**
 * \class NumericTraits NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits for integral types for operations.
 *
 * For each type we have to specialize this traits.
 *
 * \note Keep in mind that the long types long long and long double doesn't
 *       have traits. This is due to the sum_type. We can't give a guarantee
 *       that there is a type of holding the sum. Therefore using this traits
 *       is only safe if you have long long resp. long double types by
 *       working on long ints and doubles. Otherwise you will get not expected
 *       result for some circumstances. Anyway, you can use big integer/float
 *       libraries and specialize the traits by your own.
 *
 * \todo The abs function of complex<non_float_type> can have an
 *       overrun due to numeric computation. Solve it (someone
 *       using value_type=long here?)
 */
template<class T>
struct NumericTraits {
  typedef T                    base_type;
  typedef T                    value_type;
  typedef value_type                sum_type;
  typedef value_type                diff_type;
  typedef value_type                float_type;
  typedef value_type                signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef const value_type&            argument_type;

  static inline
  base_type real(argument_type x);

  static inline
  base_type imag(argument_type x);

  static inline
  value_type conj(argument_type x);

  static inline
  base_type abs(argument_type x);

  static inline
  value_type sqrt(argument_type x);

  static inline
  base_type norm_1(argument_type x) {
    return NumericTraits<base_type>::abs(traits_type::real(x))
         + NumericTraits<base_type>::abs(traits_type::imag(x));
  }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(x)),
            NumericTraits<base_type>::abs(traits_type::imag(x)));
   }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
            traits_type::norm_inf(rhs)),
           std::numeric_limits<base_type>::min());
  }
};


/*
 * numeric traits for standard types
 */


/**
 * \class NumericTraits<char> NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for char.
 */
template<>
struct NumericTraits<char> {
  typedef char                    value_type;
  typedef value_type                base_type;
  typedef long                    sum_type;
  typedef int                    diff_type;
  typedef float                    float_type;
  typedef char                    signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef value_type                argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  /** Complexity on operations. */
  enum {
    ops_plus = 1,    /**< Complexity on plus/minus ops. */
    ops_muls = 1    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits<unsigned char> NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for unsigned char.
 *
 * \note Normally it doesn't make sense to call <tt>conj</tt>
 *       for an unsigned type! An unary minus operator
 *       applied to unsigned type will result unsigned. Therefore
 *       this function is missing here.
 */
template<>
struct NumericTraits<unsigned char> {
  typedef unsigned char             value_type;
  typedef value_type                base_type;
  typedef unsigned long                 sum_type;
  typedef int                     diff_type;
  typedef float                     float_type;
  typedef int                    signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef value_type                argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  /** Complexity on operations. */
  enum {
    ops_plus = 1,    /**< Complexity on plus/minus ops. */
    ops_muls = 1    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits<short int> NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for short int.
 */
template<>
struct NumericTraits<short int> {
  typedef short int                 value_type;
  typedef value_type                base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef long long                 sum_type;
#else
  typedef long                     sum_type;
#endif
  typedef int                     diff_type;
  typedef float                 float_type;
  typedef short int                signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef value_type                argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  /** Complexity on operations. */
  enum {
    ops_plus = 1,    /**< Complexity on plus/minus ops. */
    ops_muls = 1    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits<short unsigned int> NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for short unsigned int.
 *
 * \note Normally it doesn't make sense to call <tt>conj</tt>
 *       for an unsigned type! An unary minus operator
 *       applied to unsigned type will result unsigned. Therefore
 *       this function is missing here.
 */
template<>
struct NumericTraits<short unsigned int> {
  typedef short unsigned int            value_type;
  typedef value_type                base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef unsigned long long            sum_type;
#else
  typedef unsigned long             sum_type;
#endif
  typedef int                     diff_type;
  typedef float                 float_type;
  typedef int                    signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef value_type                argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  /** Complexity on operations. */
  enum {
    ops_plus = 1,    /**< Complexity on plus/minus ops. */
    ops_muls = 1    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits<int> NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for int.
 */
template<>
struct NumericTraits<int> {
  typedef int                     value_type;
  typedef value_type                base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef long long                 sum_type;
#else
  typedef long                     sum_type;
#endif
  typedef int                     diff_type;
  typedef double                 float_type;
  typedef int                    signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef value_type                argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  /** Complexity on operations. */
  enum {
    ops_plus = 1,    /**< Complexity on plus/minus ops. */
    ops_muls = 1    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits<unsigned int> NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for unsigned int.
 *
 * \note Normally it doesn't make sense to call <tt>conj</tt>
 *       for an unsigned type! An unary minus operator
 *       applied to unsigned type will result unsigned. Therefore
 *       this function is missing here.
 */
template<>
struct NumericTraits<unsigned int> {
  typedef unsigned int                  value_type;
  typedef value_type                base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef unsigned long long            sum_type;
#else
  typedef unsigned long             sum_type;
#endif
  typedef int                      diff_type;
  typedef double                 float_type;
  typedef long                     signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef value_type                argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  base_type abs(argument_type x) { return x; }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  /** Complexity on operations. */
  enum {
    ops_plus = 1,    /**< Complexity on plus/minus ops. */
    ops_muls = 1    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits<long> NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for long.
 */
template<>
struct NumericTraits<long> {
  typedef long                      value_type;
  typedef value_type                base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef long long                 sum_type;
#else
  typedef long                      sum_type;
#endif
  typedef long                      diff_type;
  typedef double                 float_type;
  typedef long                      signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef value_type                argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  /** Complexity on operations. */
  enum {
    ops_plus = 1,    /**< Complexity on plus/minus ops. */
    ops_muls = 1    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits<unsigned long> NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for unsigned long.
 *
 * \note Normally it doesn't make sense to call <tt>conj</tt>
 *       for an unsigned type! An unary minus operator
 *       applied to unsigned type will result unsigned. Therefore
 *       this function is missing here.
 */
template<>
struct NumericTraits<unsigned long> {
  typedef unsigned long             value_type;
  typedef value_type                base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef unsigned long long             sum_type;
#else
  typedef unsigned long             sum_type;
#endif
  typedef unsigned long             diff_type;
  typedef double                 float_type;
  typedef long                    signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef value_type                argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  base_type abs(argument_type x) { return x; }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  /** Complexity on operations. */
  enum {
    ops_plus = 1,    /**< Complexity on plus/minus ops. */
    ops_muls = 1    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits<float> NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for float.
 */
template<>
struct NumericTraits<float> {
  typedef float                    value_type;
  typedef value_type                base_type;
  typedef double                 sum_type;
  typedef float                 diff_type;
  typedef float                 float_type;
  typedef float                    signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef value_type                argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) { return std::sqrt(x); }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
            traits_type::norm_inf(rhs)),
           std::numeric_limits<base_type>::min());
  }

  enum { is_complex = false };

  /** Complexity on operations. */
  enum {
    ops_plus = 1,    /**< Complexity on plus/minus ops. */
    ops_muls = 1    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits<double> NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for double.
 */
template<>
struct NumericTraits<double> {
  typedef double                 value_type;
  typedef value_type                base_type;
#if defined(TVMET_HAVE_LONG_DOUBLE)
  typedef long double                 sum_type;
#else
  typedef double                 sum_type;
#endif
  typedef double                diff_type;
  typedef double                 float_type;
  typedef double                signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef value_type                argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) { return std::sqrt(x); }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
            traits_type::norm_inf(rhs)),
           std::numeric_limits<base_type>::min());
  }

  enum { is_complex = false };

  /** Complexity on operations. */
  enum {
    ops_plus = 1,    /**< Complexity on plus/minus ops. */
    ops_muls = 1    /**< Complexity on multiplications. */
  };
};


#if defined(TVMET_HAVE_LONG_DOUBLE)
/**
 * \class NumericTraits<long double> NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for long double.
 */
template<>
struct NumericTraits<long double> {
  typedef long double                 value_type;
  typedef value_type                base_type;
  typedef long double                 sum_type;
  typedef long double                diff_type;
  typedef long double                 float_type;
  typedef long double                signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef value_type                argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) { return std::sqrt(x); }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
            traits_type::norm_inf(rhs)),
           std::numeric_limits<base_type>::min());
  }

  enum { is_complex = false };

  /** Complexity on operations. */
  enum {
    ops_plus = 1,    /**< Complexity on plus/minus ops. */
    ops_muls = 1    /**< Complexity on multiplications. */
  };
};
#endif // TVMET_HAVE_LONG_DOUBLE


/*
 * numeric traits for complex types
 */
#if defined(TVMET_HAVE_COMPLEX)

/**
 * \class NumericTraits< std::complex<int> > NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for std::complex<int>.
 */
template<>
struct NumericTraits< std::complex<int> > {
  typedef int                    base_type;
  typedef std::complex<int>            value_type;
  typedef std::complex<long>             sum_type;
  typedef std::complex<int>            diff_type;
  typedef std::complex<float>            float_type;
  typedef std::complex<int>            signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef const value_type&            argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  value_type conj(argument_type z) { return std::conj(z); }

  static inline
  base_type abs(argument_type z) {
    base_type x = z.real();
    base_type y = z.imag();

    // XXX probably case of overrun; header complex uses scaling
    return static_cast<base_type>(NumericTraits<base_type>::sqrt(x * x + y * y));
  }

  static /* inline */
  value_type sqrt(argument_type z) {
    // borrowed and adapted from header complex
    base_type x = z.real();
    base_type y = z.imag();

    if(x == base_type()) {
    base_type t = NumericTraits<base_type>::sqrt(
                        NumericTraits<base_type>::abs(y) / 2);
    return value_type(t, y < base_type() ? -t : t);
    }
    else {
      base_type t = NumericTraits<base_type>::sqrt(
              2 * (traits_type::abs(z)
                    + NumericTraits<base_type>::abs(x)));
      base_type u = t / 2;
      return x > base_type()
    ? value_type(u, y / t)
    : value_type(NumericTraits<base_type>::abs(y) / t, y < base_type() ? -u : u);
    }
  }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
            NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    return (traits_type::real(lhs) == traits_type::real(rhs))
        && (traits_type::imag(lhs) == traits_type::imag(rhs));
  }

  enum { is_complex = true };

  /** Complexity on operations. */
  enum {
    ops_plus = 2,    /**< Complexity on plus/minus ops. */
    ops_muls = 6    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits< std::complex<unsigned int> > NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for std::complex<unsigned int>.
 *
 * \note Normally it doesn't make sense to call <tt>conj</tt>
 *       for an unsigned type! An unary minus operator
 *       applied to unsigned type will result unsigned. Therefore
 *       this function is missing here.
 */
template<>
struct NumericTraits< std::complex<unsigned int> > {
  typedef unsigned int                base_type;
  typedef std::complex<unsigned int>         value_type;
  typedef std::complex<unsigned long>         sum_type;
  typedef std::complex<int>            diff_type;
  typedef std::complex<float>            float_type;
  typedef std::complex<int>            signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef const value_type&            argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  base_type abs(argument_type z) {
    base_type x = z.real();
    base_type y = z.imag();

    // XXX probably case of overrun; header complex uses scaling
    return static_cast<base_type>(NumericTraits<base_type>::sqrt(x * x + y * y));
  }

  static /* inline */
  value_type sqrt(argument_type z) {
    // borrowed and adapted from header complex
    base_type x = z.real();
    base_type y = z.imag();

    if(x == base_type()) {
    base_type t = NumericTraits<base_type>::sqrt(
                        NumericTraits<base_type>::abs(y) / 2);
    return value_type(t, t);
    }
    else {
      base_type t = NumericTraits<base_type>::sqrt(
              2 * (traits_type::abs(z)
                    + NumericTraits<base_type>::abs(x)));
      return value_type(t / 2, y / t);
    }
  }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
            NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    return (traits_type::real(lhs) == traits_type::real(rhs))
        && (traits_type::imag(lhs) == traits_type::imag(rhs));
  }

  enum { is_complex = true };

  /** Complexity on operations. */
  enum {
    ops_plus = 2,    /**< Complexity on plus/minus ops. */
    ops_muls = 6    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits< std::complex<long> > NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for std::complex<long>.
 */
template<>
struct NumericTraits< std::complex<long> > {
  typedef long                    base_type;
  typedef std::complex<long>            value_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef std::complex<long long>        sum_type;
#else
  typedef std::complex<long>            sum_type;
#endif
  typedef std::complex<int>            diff_type;
  typedef std::complex<float>            float_type;
  typedef std::complex<int>            signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef const value_type&            argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  value_type conj(argument_type z) { return std::conj(z); }

  static inline
  base_type abs(argument_type z) {
    base_type x = z.real();
    base_type y = z.imag();

    // XXX probably case of overrun; header complex uses scaling
    return static_cast<base_type>(NumericTraits<base_type>::sqrt(x * x + y * y));
  }

  static /* inline */
  value_type sqrt(argument_type z) {
    // borrowed and adapted from header complex
    base_type x = z.real();
    base_type y = z.imag();

    if(x == base_type()) {
    base_type t = NumericTraits<base_type>::sqrt(
                        NumericTraits<base_type>::abs(y) / 2);
    return value_type(t, y < base_type() ? -t : t);
    }
    else {
      base_type t = NumericTraits<base_type>::sqrt(
              2 * (traits_type::abs(z)
                    + NumericTraits<base_type>::abs(x)));
      base_type u = t / 2;
      return x > base_type()
    ? value_type(u, y / t)
    : value_type(NumericTraits<base_type>::abs(y) / t, y < base_type() ? -u : u);
    }
  }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
            NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    return (traits_type::real(lhs) == traits_type::real(rhs))
        && (traits_type::imag(lhs) == traits_type::imag(rhs));
  }

  enum { is_complex = true };

  /** Complexity on operations. */
  enum {
    ops_plus = 2,    /**< Complexity on plus/minus ops. */
    ops_muls = 6    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits< std::complex<unsigned long> > NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for std::complex<unsigned long>.
 *
 * \note Normally it doesn't make sense to call <tt>conj</tt>
 *       for an unsigned type! An unary minus operator
 *       applied to unsigned type will result unsigned. Therefore
 *       this function is missing here.
 */
template<>
struct NumericTraits< std::complex<unsigned long> > {
  typedef unsigned long                base_type;
  typedef std::complex<unsigned long>        value_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef std::complex<unsigned long long>    sum_type;
#else
  typedef std::complex<unsigned long>        sum_type;
#endif
  typedef std::complex<long>            diff_type;
  typedef std::complex<float>            float_type;
  typedef std::complex<long>            signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef const value_type&            argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  base_type abs(argument_type z) {
    base_type x = z.real();
    base_type y = z.imag();

    // XXX probably case of overrun; header complex uses scaling
    return static_cast<base_type>(NumericTraits<base_type>::sqrt(x * x + y * y));
  }

  static /* inline */
  value_type sqrt(argument_type z) {
    // borrowed and adapted from header complex
    base_type x = z.real();
    base_type y = z.imag();

    if(x == base_type()) {
    base_type t = NumericTraits<base_type>::sqrt(
                        NumericTraits<base_type>::abs(y) / 2);
    return value_type(t, t);
    }
    else {
      base_type t = NumericTraits<base_type>::sqrt(
              2 * (traits_type::abs(z)
                    + NumericTraits<base_type>::abs(x)));
      return value_type(t / 2, y / t);
    }
  }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
            NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    return (traits_type::real(lhs) == traits_type::real(rhs))
        && (traits_type::imag(lhs) == traits_type::imag(rhs));
  }

  enum { is_complex = true };

  /** Complexity  on operations.*/
  enum {
    ops_plus = 2,    /**< Complexity on plus/minus ops. */
    ops_muls = 6    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits< std::complex<float> > NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for std::complex<float>.
 */
template<>
struct NumericTraits< std::complex<float> > {
  typedef float                    base_type;
  typedef std::complex<float>            value_type;
  typedef std::complex<double>            sum_type;
  typedef std::complex<float>            diff_type;
  typedef std::complex<float>            float_type;
  typedef std::complex<float>            signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef const value_type&            argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  value_type conj(argument_type z) { return std::conj(z); }

  static inline
  base_type abs(argument_type z) { return std::abs(z); }

  static inline
  value_type sqrt(argument_type z) { return std::sqrt(z); }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
                NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

 static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
            traits_type::norm_inf(rhs)),
           std::numeric_limits<base_type>::min());
  }

  enum { is_complex = true };

  /** Complexity on operations. */
  enum {
    ops_plus = 2,    /**< Complexity on plus/minus ops. */
    ops_muls = 6    /**< Complexity on multiplications. */
  };
};


/**
 * \class NumericTraits< std::complex<double> > NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for std::complex<double>.
 */
template<>
struct NumericTraits< std::complex<double> > {
  typedef double                base_type;
  typedef std::complex<double>            value_type;
#if defined(TVMET_HAVE_LONG_DOUBLE)
  typedef std::complex<long double>         sum_type;
#else
  typedef std::complex<double>            sum_type;
#endif
  typedef std::complex<double>            diff_type;
  typedef std::complex<double>            float_type;
  typedef std::complex<double>            signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef const value_type&            argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  value_type conj(argument_type z) { return std::conj(z); }

  static inline
  base_type abs(argument_type z) { return std::abs(z); }

  static inline
  value_type sqrt(argument_type z) { return std::sqrt(z); }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
                NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

 static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
            traits_type::norm_inf(rhs)),
           std::numeric_limits<base_type>::min());
  }

  enum { is_complex = true };

  /** Complexity on operations. */
  enum {
    ops_plus = 2,    /**< Complexity on plus/minus ops. */
    ops_muls = 6    /**< Complexity on multiplications. */
  };
};


#if defined(TVMET_HAVE_LONG_DOUBLE)
/**
 * \class NumericTraits< std::complex<long double> > NumericTraits.h "tvmet/NumericTraits.h"
 * \brief Traits specialized for std::complex<double>.
 */
template<>
struct NumericTraits< std::complex<long double> > {
  typedef long double                base_type;
  typedef std::complex<long double>        value_type;
  typedef std::complex<long double>         sum_type;
  typedef std::complex<long double>        diff_type;
  typedef std::complex<long double>        float_type;
  typedef std::complex<long double>        signed_type;

  typedef NumericTraits<value_type>        traits_type;
  typedef const value_type&            argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  value_type conj(argument_type z) { return std::conj(z); }

  static inline
  base_type abs(argument_type z) { return std::abs(z); }

  static inline
  value_type sqrt(argument_type z) { return std::sqrt(z); }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
                NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
            traits_type::norm_inf(rhs)),
           std::numeric_limits<base_type>::min());
  }

  enum { is_complex = true };

  /** Complexity on operations. */
  enum {
    ops_plus = 2,    /**< Complexity on plus/minus ops. */
    ops_muls = 6    /**< Complexity on multiplications. */
  };
};
#endif // defined(TVMET_HAVE_LONG_DOUBLE)


#endif // defined(TVMET_HAVE_COMPLEX)


} // namespace tvmet


#endif //  TVMET_NUMERIC_TRAITS_H


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