polynomials_integrated_legendre_sz.cc

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// ---------------------------------------------------------------------
//
// Copyright (C) 2015 - 2019 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.
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#include <deal.II/base/polynomials_integrated_legendre_sz.h>

DEAL_II_NAMESPACE_OPEN

IntegratedLegendreSZ::IntegratedLegendreSZ(const unsigned int k)
  : Polynomials::Polynomial<double>(get_coefficients(k))
{}



const std::vector<double>
IntegratedLegendreSZ::get_coefficients(const unsigned int k)
{
  std::vector<double> coefficients(k + 1);

  // first two polynomials are hard-coded:
  if (k == 0)
    {
      coefficients[0] = -1.;
      return coefficients;
    }
  else if (k == 1)
    {
      coefficients[0] = 0.;
      coefficients[1] = 1.;
      return coefficients;
    }

  // General formula is:
  // k*L_{k}(x) = (2*k-3)*x*L_{k-1} - (k-3)*L_{k-2}.
  std::vector<double> coefficients_km2 = get_coefficients(k - 2);
  std::vector<double> coefficients_km1 = get_coefficients(k - 1);

  const double a = 1.0 / k;
  const double b = 2.0 * k - 3.0;
  const double c = k - 3.0;

  // To maintain stability, delay the division (multiplication by a) until the
  // end.
  for (unsigned int i = 1; i <= k - 2; i++)
    {
      coefficients[i] = b * coefficients_km1[i - 1] - c * coefficients_km2[i];
    }

  coefficients[0]     = -c * coefficients_km2[0];
  coefficients[k]     = b * coefficients_km1[k - 1];
  coefficients[k - 1] = b * coefficients_km1[k - 2];

  for (double &coefficient : coefficients)
    {
      coefficient *= a;
    }

  return coefficients;
}



std::vector<Polynomials::Polynomial<double>>
IntegratedLegendreSZ::generate_complete_basis(const unsigned int degree)
{
  std::vector<Polynomials::Polynomial<double>> v;
  v.reserve(degree + 1);
  for (unsigned int i = 0; i <= degree; ++i)
    {
      v.push_back(IntegratedLegendreSZ(i));
    }
  return v;
}



DEAL_II_NAMESPACE_CLOSE