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file: //  Copyright John Maddock 2006, 2007.
  //  Copyright Paul A. Bristow 2006, 2007.
  //  Use, modification and distribution are subject to the
  //  Boost Software License, Version 1.0. (See accompanying file
  //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
  
  #ifndef BOOST_STATS_TRIANGULAR_HPP
  #define BOOST_STATS_TRIANGULAR_HPP
  
  // http://mathworld.wolfram.com/TriangularDistribution.html
  // Note that the 'constructors' defined by Wolfram are difference from those here,
  // for example
  // N[variance[triangulardistribution{1, +2}, 1.5], 50] computes 
  // 0.041666666666666666666666666666666666666666666666667
  // TriangularDistribution{1, +2}, 1.5 is the analog of triangular_distribution(1, 1.5, 2)
  
  // http://en.wikipedia.org/wiki/Triangular_distribution
  
  #include <boost/math/distributions/fwd.hpp>
  #include <boost/math/special_functions/expm1.hpp>
  #include <boost/math/distributions/detail/common_error_handling.hpp>
  #include <boost/math/distributions/complement.hpp>
  #include <boost/math/constants/constants.hpp>
  
  #include <utility>
  
  namespace boost{ namespace math
  {
    namespace detail
    {
      template <class RealType, class Policy>
      inline bool check_triangular_lower(
        const char* function,
        RealType lower,
        RealType* result, const Policy& pol)
      {
        if((boost::math::isfinite)(lower))
        { // Any finite value is OK.
          return true;
        }
        else
        { // Not finite: infinity or NaN.
          *result = policies::raise_domain_error<RealType>(
            function,
            "Lower parameter is %1%, but must be finite!", lower, pol);
          return false;
        }
      } // bool check_triangular_lower(
  
      template <class RealType, class Policy>
      inline bool check_triangular_mode(
        const char* function,
        RealType mode,
        RealType* result, const Policy& pol)
      {
        if((boost::math::isfinite)(mode))
        { // any finite value is OK.
          return true;
        }
        else
        { // Not finite: infinity or NaN.
          *result = policies::raise_domain_error<RealType>(
            function,
            "Mode parameter is %1%, but must be finite!", mode, pol);
          return false;
        }
      } // bool check_triangular_mode(
  
      template <class RealType, class Policy>
      inline bool check_triangular_upper(
        const char* function,
        RealType upper,
        RealType* result, const Policy& pol)
      {
        if((boost::math::isfinite)(upper))
        { // any finite value is OK.
          return true;
        }
        else
        { // Not finite: infinity or NaN.
          *result = policies::raise_domain_error<RealType>(
            function,
            "Upper parameter is %1%, but must be finite!", upper, pol);
          return false;
        }
      } // bool check_triangular_upper(
  
      template <class RealType, class Policy>
      inline bool check_triangular_x(
        const char* function,
        RealType const& x,
        RealType* result, const Policy& pol)
      {
        if((boost::math::isfinite)(x))
        { // Any finite value is OK
          return true;
        }
        else
        { // Not finite: infinity or NaN.
          *result = policies::raise_domain_error<RealType>(
            function,
            "x parameter is %1%, but must be finite!", x, pol);
          return false;
        }
      } // bool check_triangular_x
  
      template <class RealType, class Policy>
      inline bool check_triangular(
        const char* function,
        RealType lower,
        RealType mode,
        RealType upper,
        RealType* result, const Policy& pol)
      {
        if ((check_triangular_lower(function, lower, result, pol) == false)
          || (check_triangular_mode(function, mode, result, pol) == false)
          || (check_triangular_upper(function, upper, result, pol) == false))
        { // Some parameter not finite.
          return false;
        }
        else if (lower >= upper) // lower == upper NOT useful.
        { // lower >= upper.
          *result = policies::raise_domain_error<RealType>(
            function,
            "lower parameter is %1%, but must be less than upper!", lower, pol);
          return false;
        }
        else
        { // Check lower <= mode <= upper.
          if (mode < lower)
          {
            *result = policies::raise_domain_error<RealType>(
              function,
              "mode parameter is %1%, but must be >= than lower!", lower, pol);
            return false;
          }
          if (mode > upper)
          {
            *result = policies::raise_domain_error<RealType>(
              function,
              "mode parameter is %1%, but must be <= than upper!", upper, pol);
            return false;
          }
          return true; // All OK.
        }
      } // bool check_triangular
    } // namespace detail
  
    template <class RealType = double, class Policy = policies::policy<> >
    class triangular_distribution
    {
    public:
      typedef RealType value_type;
      typedef Policy policy_type;
  
      triangular_distribution(RealType l_lower = -1, RealType l_mode = 0, RealType l_upper = 1)
        : m_lower(l_lower), m_mode(l_mode), m_upper(l_upper) // Constructor.
      { // Evans says 'standard triangular' is lower 0, mode 1/2, upper 1,
        // has median sqrt(c/2) for c <=1/2 and 1 - sqrt(1-c)/2 for c >= 1/2
        // But this -1, 0, 1 is more useful in most applications to approximate normal distribution,
        // where the central value is the most likely and deviations either side equally likely.
        RealType result;
        detail::check_triangular("boost::math::triangular_distribution<%1%>::triangular_distribution",l_lower, l_mode, l_upper, &result, Policy());
      }
      // Accessor functions.
      RealType lower()const
      {
        return m_lower;
      }
      RealType mode()const
      {
        return m_mode;
      }
      RealType upper()const
      {
        return m_upper;
      }
    private:
      // Data members:
      RealType m_lower;  // distribution lower aka a
      RealType m_mode;  // distribution mode aka c
      RealType m_upper;  // distribution upper aka b
    }; // class triangular_distribution
  
    typedef triangular_distribution<double> triangular;
  
    template <class RealType, class Policy>
    inline const std::pair<RealType, RealType> range(const triangular_distribution<RealType, Policy>& /* dist */)
    { // Range of permissible values for random variable x.
      using boost::math::tools::max_value;
      return std::pair<RealType, RealType>(-max_value<RealType>(), max_value<RealType>());
    }
  
    template <class RealType, class Policy>
    inline const std::pair<RealType, RealType> support(const triangular_distribution<RealType, Policy>& dist)
    { // Range of supported values for random variable x.
      // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
      return std::pair<RealType, RealType>(dist.lower(), dist.upper());
    }
  
    template <class RealType, class Policy>
    RealType pdf(const triangular_distribution<RealType, Policy>& dist, const RealType& x)
    {
      static const char* function = "boost::math::pdf(const triangular_distribution<%1%>&, %1%)";
      RealType lower = dist.lower();
      RealType mode = dist.mode();
      RealType upper = dist.upper();
      RealType result = 0; // of checks.
      if(false == detail::check_triangular(function, lower, mode, upper, &result, Policy()))
      {
        return result;
      }
      if(false == detail::check_triangular_x(function, x, &result, Policy()))
      {
        return result;
      }
      if((x < lower) || (x > upper))
      {
        return 0;
      }
      if (x == lower)
      { // (mode - lower) == 0 which would lead to divide by zero!
        return (mode == lower) ? 2 / (upper - lower) : RealType(0);
      }
      else if (x == upper)
      {
        return (mode == upper) ? 2 / (upper - lower) : RealType(0);
      }
      else if (x <= mode)
      {
        return 2 * (x - lower) / ((upper - lower) * (mode - lower));
      }
      else
      {  // (x > mode)
        return 2 * (upper - x) / ((upper - lower) * (upper - mode));
      }
    } // RealType pdf(const triangular_distribution<RealType, Policy>& dist, const RealType& x)
  
    template <class RealType, class Policy>
    inline RealType cdf(const triangular_distribution<RealType, Policy>& dist, const RealType& x)
    {
      static const char* function = "boost::math::cdf(const triangular_distribution<%1%>&, %1%)";
      RealType lower = dist.lower();
      RealType mode = dist.mode();
      RealType upper = dist.upper();
      RealType result = 0; // of checks.
      if(false == detail::check_triangular(function, lower, mode, upper, &result, Policy()))
      {
        return result;
      }
      if(false == detail::check_triangular_x(function, x, &result, Policy()))
      {
        return result;
      }
      if((x <= lower))
      {
        return 0;
      }
      if (x >= upper)
      {
        return 1;
      }
      // else lower < x < upper
      if (x <= mode)
      {
        return ((x - lower) * (x - lower)) / ((upper - lower) * (mode - lower));
      }
      else
      {
        return 1 - (upper - x) *  (upper - x) / ((upper - lower) * (upper - mode));
      }
    } // RealType cdf(const triangular_distribution<RealType, Policy>& dist, const RealType& x)
  
    template <class RealType, class Policy>
    RealType quantile(const triangular_distribution<RealType, Policy>& dist, const RealType& p)
    {
      BOOST_MATH_STD_USING  // for ADL of std functions (sqrt).
      static const char* function = "boost::math::quantile(const triangular_distribution<%1%>&, %1%)";
      RealType lower = dist.lower();
      RealType mode = dist.mode();
      RealType upper = dist.upper();
      RealType result = 0; // of checks
      if(false == detail::check_triangular(function,lower, mode, upper, &result, Policy()))
      {
        return result;
      }
      if(false == detail::check_probability(function, p, &result, Policy()))
      {
        return result;
      }
      if(p == 0)
      {
        return lower;
      }
      if(p == 1)
      {
        return upper;
      }
      RealType p0 = (mode - lower) / (upper - lower);
      RealType q = 1 - p;
      if (p < p0)
      {
        result = sqrt((upper - lower) * (mode - lower) * p) + lower;
      }
      else if (p == p0)
      {
        result = mode;
      }
      else // p > p0
      {
        result = upper - sqrt((upper - lower) * (upper - mode) * q);
      }
      return result;
  
    } // RealType quantile(const triangular_distribution<RealType, Policy>& dist, const RealType& q)
  
    template <class RealType, class Policy>
    RealType cdf(const complemented2_type<triangular_distribution<RealType, Policy>, RealType>& c)
    {
      static const char* function = "boost::math::cdf(const triangular_distribution<%1%>&, %1%)";
      RealType lower = c.dist.lower();
      RealType mode = c.dist.mode();
      RealType upper = c.dist.upper();
      RealType x = c.param;
      RealType result = 0; // of checks.
      if(false == detail::check_triangular(function, lower, mode, upper, &result, Policy()))
      {
        return result;
      }
      if(false == detail::check_triangular_x(function, x, &result, Policy()))
      {
        return result;
      }
      if (x <= lower)
      {
        return 1;
      }
      if (x >= upper)
      {
        return 0;
      }
      if (x <= mode)
      {
        return 1 - ((x - lower) * (x - lower)) / ((upper - lower) * (mode - lower));
      }
      else
      {
        return (upper - x) *  (upper - x) / ((upper - lower) * (upper - mode));
      }
    } // RealType cdf(const complemented2_type<triangular_distribution<RealType, Policy>, RealType>& c)
  
    template <class RealType, class Policy>
    RealType quantile(const complemented2_type<triangular_distribution<RealType, Policy>, RealType>& c)
    {
      BOOST_MATH_STD_USING  // Aid ADL for sqrt.
      static const char* function = "boost::math::quantile(const triangular_distribution<%1%>&, %1%)";
      RealType l = c.dist.lower();
      RealType m = c.dist.mode();
      RealType u = c.dist.upper();
      RealType q = c.param; // probability 0 to 1.
      RealType result = 0; // of checks.
      if(false == detail::check_triangular(function, l, m, u, &result, Policy()))
      {
        return result;
      }
      if(false == detail::check_probability(function, q, &result, Policy()))
      {
        return result;
      }
      if(q == 0)
      {
        return u;
      }
      if(q == 1)
      {
        return l;
      }
      RealType lower = c.dist.lower();
      RealType mode = c.dist.mode();
      RealType upper = c.dist.upper();
  
      RealType p = 1 - q;
      RealType p0 = (mode - lower) / (upper - lower);
      if(p < p0)
      {
        RealType s = (upper - lower) * (mode - lower);
        s *= p;
        result = sqrt((upper - lower) * (mode - lower) * p) + lower;
      }
      else if (p == p0)
      {
        result = mode;
      }
      else // p > p0
      {
        result = upper - sqrt((upper - lower) * (upper - mode) * q);
      }
      return result;
    } // RealType quantile(const complemented2_type<triangular_distribution<RealType, Policy>, RealType>& c)
  
    template <class RealType, class Policy>
    inline RealType mean(const triangular_distribution<RealType, Policy>& dist)
    {
      static const char* function = "boost::math::mean(const triangular_distribution<%1%>&)";
      RealType lower = dist.lower();
      RealType mode = dist.mode();
      RealType upper = dist.upper();
      RealType result = 0;  // of checks.
      if(false == detail::check_triangular(function, lower, mode, upper, &result, Policy()))
      {
        return result;
      }
      return (lower + upper + mode) / 3;
    } // RealType mean(const triangular_distribution<RealType, Policy>& dist)
  
  
    template <class RealType, class Policy>
    inline RealType variance(const triangular_distribution<RealType, Policy>& dist)
    {
      static const char* function = "boost::math::mean(const triangular_distribution<%1%>&)";
      RealType lower = dist.lower();
      RealType mode = dist.mode();
      RealType upper = dist.upper();
      RealType result = 0; // of checks.
      if(false == detail::check_triangular(function, lower, mode, upper, &result, Policy()))
      {
        return result;
      }
      return (lower * lower + upper * upper + mode * mode - lower * upper - lower * mode - upper * mode) / 18;
    } // RealType variance(const triangular_distribution<RealType, Policy>& dist)
  
    template <class RealType, class Policy>
    inline RealType mode(const triangular_distribution<RealType, Policy>& dist)
    {
      static const char* function = "boost::math::mode(const triangular_distribution<%1%>&)";
      RealType mode = dist.mode();
      RealType result = 0; // of checks.
      if(false == detail::check_triangular_mode(function, mode, &result, Policy()))
      { // This should never happen!
        return result;
      }
      return mode;
    } // RealType mode
  
    template <class RealType, class Policy>
    inline RealType median(const triangular_distribution<RealType, Policy>& dist)
    {
      BOOST_MATH_STD_USING // ADL of std functions.
      static const char* function = "boost::math::median(const triangular_distribution<%1%>&)";
      RealType mode = dist.mode();
      RealType result = 0; // of checks.
      if(false == detail::check_triangular_mode(function, mode, &result, Policy()))
      { // This should never happen!
        return result;
      }
      RealType lower = dist.lower();
      RealType upper = dist.upper();
      if (mode >= (upper + lower) / 2)
      {
        return lower + sqrt((upper - lower) * (mode - lower)) / constants::root_two<RealType>();
      }
      else
      {
        return upper - sqrt((upper - lower) * (upper - mode)) / constants::root_two<RealType>();
      }
    } // RealType mode
  
    template <class RealType, class Policy>
    inline RealType skewness(const triangular_distribution<RealType, Policy>& dist)
    {
      BOOST_MATH_STD_USING  // for ADL of std functions
      using namespace boost::math::constants; // for root_two
      static const char* function = "boost::math::skewness(const triangular_distribution<%1%>&)";
  
      RealType lower = dist.lower();
      RealType mode = dist.mode();
      RealType upper = dist.upper();
      RealType result = 0; // of checks.
      if(false == boost::math::detail::check_triangular(function,lower, mode, upper, &result, Policy()))
      {
        return result;
      }
      return root_two<RealType>() * (lower + upper - 2 * mode) * (2 * lower - upper - mode) * (lower - 2 * upper + mode) /
        (5 * pow((lower * lower + upper * upper + mode * mode 
          - lower * upper - lower * mode - upper * mode), RealType(3)/RealType(2)));
      // #11768: Skewness formula for triangular distribution is incorrect -  corrected 29 Oct 2015 for release 1.61.
    } // RealType skewness(const triangular_distribution<RealType, Policy>& dist)
  
    template <class RealType, class Policy>
    inline RealType kurtosis(const triangular_distribution<RealType, Policy>& dist)
    { // These checks may be belt and braces as should have been checked on construction?
      static const char* function = "boost::math::kurtosis(const triangular_distribution<%1%>&)";
      RealType lower = dist.lower();
      RealType upper = dist.upper();
      RealType mode = dist.mode();
      RealType result = 0;  // of checks.
      if(false == detail::check_triangular(function,lower, mode, upper, &result, Policy()))
      {
        return result;
      }
      return static_cast<RealType>(12)/5; //  12/5 = 2.4;
    } // RealType kurtosis_excess(const triangular_distribution<RealType, Policy>& dist)
  
    template <class RealType, class Policy>
    inline RealType kurtosis_excess(const triangular_distribution<RealType, Policy>& dist)
    { // These checks may be belt and braces as should have been checked on construction?
      static const char* function = "boost::math::kurtosis_excess(const triangular_distribution<%1%>&)";
      RealType lower = dist.lower();
      RealType upper = dist.upper();
      RealType mode = dist.mode();
      RealType result = 0;  // of checks.
      if(false == detail::check_triangular(function,lower, mode, upper, &result, Policy()))
      {
        return result;
      }
      return static_cast<RealType>(-3)/5; // - 3/5 = -0.6
      // Assuming mathworld really means kurtosis excess?  Wikipedia now corrected to match this.
    }
  
  } // namespace math
  } // namespace boost
  
  // This include must be at the end, *after* the accessors
  // for this distribution have been defined, in order to
  // keep compilers that support two-phase lookup happy.
  #include <boost/math/distributions/detail/derived_accessors.hpp>
  
  #endif // BOOST_STATS_TRIANGULAR_HPP
  
  
  
  