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// Copyright (C) 2016 EDF
// All Rights Reserved
// This code is published under the GNU Lesser General Public License (GNU LGPL)
#ifndef _POLYNOMIALS1D_H
#define _POLYNOMIALS1D_H
/**
* \file Polynomials1D.h
* \brief Implementation of one variate polynomial
* \author Jérôme Lelong
*/
namespace StOpt
{
/**
* \defgroup Polynomial Single variate polynomials
* \brief It Implements single variate polynomials. each class must provide
* three methods f, Df, D2f to compute the function value, its first
* derivatives and its second derivative.
*/
/**@{*/
/**
* \class Canonical
* \brief Implementation of the canonical polynomials with one variate
* P(X, n) = X^n
*/
class Canonical
{
public:
/**
* \brief Evaluate the canonical Polynomial
* \param p_x point at which to evaluate the polynomial
* \param p_n degree of the polynomial
* \return P(p_x, p_n)
*/
double F(double p_x, int p_n) const;
/**
* \brief Evaluate the first derivative of the canonical Polynomial
* \param p_x point at which to evaluate the polynomial
* \param p_n degree of the polynomial
* \return P'(p_x, p_n)
*/
double DF(double p_x, int p_n) const;
/**
* \brief Evaluate the second derivative of the canonical Polynomial
* \param p_x point at which to evaluate the polynomial
* \param p_n degree of the polynomial
* \return P''(p_x, p_n)
*/
double D2F(double p_x, int p_n) const;
};
/**
* \class Tchebychev
* \brief Implementation of the Tchebychev polynomials with one variate
* P(X, n + 2) = 2 X P(X, n + 1) - P(X, n)
*/
class Tchebychev
{
public:
/**
* \brief Evaluate the Tchebychev Polynomial
*
* \param p_x point at which to evaluate the polynomial
* \param p_n degree of the polynomial
* \return P(p_x, p_n)
*/
double F(double p_x, int p_n) const;
/**
* \brief Evaluate the first derivative of the Tchebychev Polynomial
*
* \param p_x point at which to evaluate the polynomial
* \param p_n degree of the polynomial
* \return P'(p_x, p_n)
*/
double DF(double p_x, int p_n) const;
/**
* \brief Evaluate the second derivative of the Tchebychev Polynomial
*
* \param p_x point at which to evaluate the polynomial
* \param p_n degree of the polynomial
* \return P''(p_x, p_n)
*/
double D2F(double p_x, int p_n) const;
private:
/**
* \brief The terminal recursive function to compute Tchebychev polynomials of any
* order. This function is only used for order > 7
* \param p_x the evaluation point
* \param p_n the order of the polynomial to be evaluated
* \param p_n0 initialization of the recurrence
* \param p_fn0 used to store the polynomial of order p_n0.
* \param p_fn0_1 used to store the polynomial of order p_n0 - 1.
*/
double FRec(double p_x, int p_n, int p_n0, double &p_fn0, double &p_fn0_1) const;
/**
* \brief The terminal recursive function to compute the first derivative of the
* Tchebychev polynomials of any order. This function is only used for order > 7
*
* \param p_x the evaluation point
* \param p_n the order of the polynomial to be evaluated
* \param p_n0 initialization of the recurrence
* \param Dfn0 used to store the derivative of the polynomial of order p_n0.
* \param Dfn0_1 used to store the derivative of the polynomial of order p_n0 - 1.
*/
double DFRec(double p_x, int p_n, int p_n0, double &p_dfn0, double &p_dfn0_1) const;
/**
* \brief The terminal recursive function to compute the second derivative of the
* Tchebychev polynomials of any order. This function is only used for order > 7
*
* \param p_x point to evaluate at
* \param p_n the order of the polynomial
* \param p_n0 initialization of the recurrence
* \param D2fn0 used to store the second derivative of the polynomial of order p_n0.
* \param D2fn0_1 used to store the second derivative of the polynomial of order p_n0 - 1.
*/
double D2FRec(double p_x, int p_n, int p_n0, double &p_d2fn0, double &p_d2fn0_1) const;
};
/**
* \class Hermite
* \brief Implementation of the Hermite polynomials with one variate.
* P(X, n + 1) = X P(X, n) - 2 n P(X, n - 1)
* These polynomials are orthogonal for the normal distribution and morover if
* G is a standard normal random variable E[P(G, n)] = n!
*/
class Hermite
{
public:
/**
* \brief Evaluate the Hermite Polynomial
*
* \param p_x point at which to evaluate the polynomial
* \param p_n degree of the polynomial
* \return P(p_x, p_n)
*/
double F(double p_x, int p_n) const;
/**
* \brief Evaluate the first derivative of the Hermite Polynomial
*
* \param p_x point at which to evaluate the polynomial
* \param p_n degree of the polynomial
* \return P'(p_x, p_n)
*/
double DF(double p_x, int p_n) const;
/**
* \brief Evaluate the second derivative of the Hermite Polynomial
*
* \param p_x point at which to evaluate the polynomial
* \param p_n degree of the polynomial
* \return P''(p_x, p_n)
*/
double D2F(double p_x, int p_n) const;
private:
/**
* \brief The terminal recursive function to compute Hermite polynomials of any
* order. This function is only used for order > 7
*
* \param p_x point at which to evaluate the polynomial
* \param p_n the order of the polynomial to be evaluated
* \param p_n0 rank of initialization
* \param p_fn0 used to store the polynomial of order p_n0.
* \param p_fn0_1 used to store the polynomial of order p_n0 - 1.
*/
double FRec(double p_x, int p_n, int p_n0, double &p_fn0, double &p_fn0_1) const;
};
/**@}*/
}
#endif /* _POLYNOMIALS1D_H */
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