polynomial.h

// ---------------------------------------------------------------------
//
// Copyright (C) 2000 - 2017 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, 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.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------

#ifndef dealii__polynomial_h
#define dealii__polynomial_h



#include <deal.II/base/config.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/subscriptor.h>
#include <deal.II/base/point.h>
#include <deal.II/base/std_cxx11/shared_ptr.h>

#include <vector>

DEAL_II_NAMESPACE_OPEN

namespace Polynomials
{

  template <typename number>
  class Polynomial : public Subscriptor
  {
  public:
    Polynomial (const std::vector<number> &coefficients);

    Polynomial (const unsigned int n);

    Polynomial (const std::vector<Point<1> > &lagrange_support_points,
                const unsigned int            evaluation_point);

    Polynomial ();

    number value (const number x) const;

    void value (const number         x,
                std::vector<number> &values) const;

    void value (const number         x,
                const unsigned int n_derivatives,
                number *values) const;

    unsigned int degree () const;

    void scale (const number factor);

    template <typename number2>
    void shift (const number2 offset);

    Polynomial<number> derivative () const;

    Polynomial<number> primitive () const;

    Polynomial<number> &operator *= (const double s);

    Polynomial<number> &operator *= (const Polynomial<number> &p);

    Polynomial<number> &operator += (const Polynomial<number> &p);

    Polynomial<number> &operator -= (const Polynomial<number> &p);

    bool operator == (const Polynomial<number> &p)  const;

    void print(std::ostream &out) const;

    template <class Archive>
    void serialize (Archive &ar, const unsigned int version);

  protected:

    static void scale (std::vector<number> &coefficients,
                       const number         factor);

    template <typename number2>
    static void shift (std::vector<number> &coefficients,
                       const number2        shift);

    static void multiply (std::vector<number> &coefficients,
                          const number factor);

    void transform_into_standard_form ();

    std::vector<number> coefficients;

    bool in_lagrange_product_form;

    std::vector<number> lagrange_support_points;

    number lagrange_weight;
  };


  template <typename number>
  class Monomial : public Polynomial<number>
  {
  public:
    Monomial(const unsigned int n,
             const double coefficient = 1.);

    static
    std::vector<Polynomial<number> >
    generate_complete_basis (const unsigned int degree);

  private:
    static std::vector<number> make_vector(unsigned int n,
                                           const double coefficient);
  };


  class LagrangeEquidistant: public Polynomial<double>
  {
  public:
    LagrangeEquidistant (const unsigned int n,
                         const unsigned int support_point);

    static
    std::vector<Polynomial<double> >
    generate_complete_basis (const unsigned int degree);

  private:

    static
    void
    compute_coefficients (const unsigned int n,
                          const unsigned int support_point,
                          std::vector<double> &a);
  };



  std::vector<Polynomial<double> >
  generate_complete_Lagrange_basis (const std::vector<Point<1> > &points);



  class Legendre : public Polynomial<double>
  {
  public:
    Legendre (const unsigned int p);

    static
    std::vector<Polynomial<double> >
    generate_complete_basis (const unsigned int degree);
  };

  class Lobatto : public Polynomial<double>
  {
  public:
    Lobatto (const unsigned int p = 0);

    static std::vector<Polynomial<double> >
    generate_complete_basis (const unsigned int p);

  private:
    std::vector<double> compute_coefficients (const unsigned int p);
  };



  class Hierarchical : public Polynomial<double>
  {
  public:
    Hierarchical (const unsigned int p);

    static
    std::vector<Polynomial<double> >
    generate_complete_basis (const unsigned int degree);

  private:
    static void compute_coefficients (const unsigned int p);

    static const std::vector<double> &
    get_coefficients (const unsigned int p);

    static std::vector<std_cxx11::shared_ptr<const std::vector<double> > > recursive_coefficients;
  };


  class HermiteInterpolation : public Polynomial<double>
  {
  public:
    HermiteInterpolation (const unsigned int p);

    static std::vector<Polynomial<double> >
    generate_complete_basis (const unsigned int p);
  };
}


/* -------------------------- inline functions --------------------- */

namespace Polynomials
{
  template <typename number>
  inline
  Polynomial<number>::Polynomial ()
    :
    in_lagrange_product_form (false),
    lagrange_weight          (1.)
  {}



  template <typename number>
  inline
  unsigned int
  Polynomial<number>::degree () const
  {
    if (in_lagrange_product_form == true)
      {
        return lagrange_support_points.size();
      }
    else
      {
        Assert (coefficients.size()>0, ExcEmptyObject());
        return coefficients.size() - 1;
      }
  }



  template <typename number>
  inline
  number
  Polynomial<number>::value (const number x) const
  {
    if (in_lagrange_product_form == false)
      {
        Assert (coefficients.size() > 0, ExcEmptyObject());

        // Horner scheme
        const unsigned int m=coefficients.size();
        number value = coefficients.back();
        for (int k=m-2; k>=0; --k)
          value = value*x + coefficients[k];
        return value;
      }
    else
      {
        // direct evaluation of Lagrange polynomial
        const unsigned int m = lagrange_support_points.size();
        number value = 1.;
        for (unsigned int j=0; j<m; ++j)
          value *= x-lagrange_support_points[j];
        value *= lagrange_weight;
        return value;
      }
  }



  template <typename number>
  template <class Archive>
  inline
  void
  Polynomial<number>::serialize (Archive &ar, const unsigned int)
  {
    // forward to serialization function in the base class.
    ar &static_cast<Subscriptor &>(*this);
    ar &coefficients;
    ar &in_lagrange_product_form;
    ar &lagrange_support_points;
    ar &lagrange_weight;
  }

}
DEAL_II_NAMESPACE_CLOSE

#endif