polynomial.h

// ---------------------------------------------------------------------
//
// Copyright (C) 2000 - 2018 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.md at
// the top level directory of deal.II.
//
// ---------------------------------------------------------------------

#ifndef dealii_polynomial_h
#define dealii_polynomial_h



#include <deal.II/base/config.h>

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

#include <memory>
#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::unique_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);
  };



  class HermiteLikeInterpolation : public Polynomial<double>
  {
  public:
    HermiteLikeInterpolation(const unsigned int degree,
                             const unsigned int index);

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



  /*
   * Evaluate a Jacobi polynomial @f$ P_n^{\alpha, \beta}(x) @f$ specified by the
   * parameters @p alpha, @p beta, @p n, where @p n is the degree of the
   * Jacobi polynomial.
   *
   * @note The Jacobi polynomials are not orthonormal and are defined on the
   * unit interval @f$[0, 1]@f$ as usual for deal.II, rather than @f$[-1, +1]@f$ often
   * used in literature. @p x is the point of evaluation.
   */
  template <typename Number>
  Number
  jacobi_polynomial_value(const unsigned int degree,
                          const int          alpha,
                          const int          beta,
                          const Number       x);


  template <typename Number>
  std::vector<Number>
  jacobi_polynomial_roots(const unsigned int degree,
                          const int          alpha,
                          const int          beta);
} // namespace Polynomials


/* -------------------------- 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;
  }



  template <typename Number>
  Number
  jacobi_polynomial_value(const unsigned int degree,
                          const int          alpha,
                          const int          beta,
                          const Number       x)
  {
    Assert(alpha >= 0 && beta >= 0,
           ExcNotImplemented("Negative alpha/beta coefficients not supported"));
    // the Jacobi polynomial is evaluated using a recursion formula.
    Number p0, p1;

    // The recursion formula is defined for the interval [-1, 1], so rescale
    // to that interval here
    const Number xeval = Number(-1) + 2. * x;

    // initial values P_0(x), P_1(x):
    p0 = 1.0;
    if (degree == 0)
      return p0;
    p1 = ((alpha + beta + 2) * xeval + (alpha - beta)) / 2;
    if (degree == 1)
      return p1;

    for (unsigned int i = 1; i < degree; ++i)
      {
        const Number v  = 2 * i + (alpha + beta);
        const Number a1 = 2 * (i + 1) * (i + (alpha + beta + 1)) * v;
        const Number a2 = (v + 1) * (alpha * alpha - beta * beta);
        const Number a3 = v * (v + 1) * (v + 2);
        const Number a4 = 2 * (i + alpha) * (i + beta) * (v + 2);

        const Number pn = ((a2 + a3 * xeval) * p1 - a4 * p0) / a1;
        p0              = p1;
        p1              = pn;
      }
    return p1;
  }



  template <typename Number>
  std::vector<Number>
  jacobi_polynomial_roots(const unsigned int degree,
                          const int          alpha,
                          const int          beta)
  {
    std::vector<Number> x(degree, 0.5);

    // compute zeros with a Newton algorithm.

    // Set tolerance. For long double we might not always get the additional
    // precision in a run time environment (e.g. with valgrind), so we must
    // limit the tolerance to double. Since we do a Newton iteration, doing
    // one more iteration after the residual has indicated convergence will be
    // enough for all number types due to the quadratic convergence of
    // Newton's method

    const Number tolerance =
      4 * std::max(static_cast<Number>(std::numeric_limits<double>::epsilon()),
                   std::numeric_limits<Number>::epsilon());

    // The following implementation follows closely the one given in the
    // appendix of the book by Karniadakis and Sherwin: Spectral/hp element
    // methods for computational fluid dynamics (Oxford University Press,
    // 2005)

    // If symmetric, we only need to compute the half of points
    const unsigned int n_points = (alpha == beta ? degree / 2 : degree);
    for (unsigned int k = 0; k < n_points; ++k)
      {
        // we take the zeros of the Chebyshev polynomial (alpha=beta=-0.5) as
        // initial values, corrected by the initial value
        Number r = 0.5 - 0.5 * std::cos(static_cast<Number>(2 * k + 1) /
                                        (2 * degree) * numbers::PI);
        if (k > 0)
          r = (r + x[k - 1]) / 2;

        unsigned int converged = numbers::invalid_unsigned_int;
        for (unsigned int it = 1; it < 1000; ++it)
          {
            Number s = 0.;
            for (unsigned int i = 0; i < k; ++i)
              s += 1. / (r - x[i]);

            // derivative of P_n^{alpha,beta}, rescaled to [0, 1]
            const Number J_x =
              (alpha + beta + degree + 1) *
              jacobi_polynomial_value(degree - 1, alpha + 1, beta + 1, r);

            // value of P_n^{alpha,beta}
            const Number f = jacobi_polynomial_value(degree, alpha, beta, r);
            const Number delta = f / (f * s - J_x);
            r += delta;
            if (converged == numbers::invalid_unsigned_int &&
                std::abs(delta) < tolerance)
              converged = it;

            // do one more iteration to ensure accuracy also for tighter
            // types than double (e.g. long double)
            if (it == converged + 1)
              break;
          }

        Assert(converged != numbers::invalid_unsigned_int,
               ExcMessage("Newton iteration for zero of Jacobi polynomial "
                          "did not converge."));

        x[k] = r;
      }

    // in case we assumed symmetry, fill up the missing values
    for (unsigned int k = n_points; k < degree; ++k)
      x[k] = 1.0 - x[degree - k - 1];

    return x;
  }

} // namespace Polynomials
DEAL_II_NAMESPACE_CLOSE

#endif