polynomials_integrated_legendre_sz.cc

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// -----------------------------------------------------------------------------
//
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception OR LGPL-2.1-or-later
// Copyright (C) 2017 - 2026 by the deal.II authors
//
// This file is part of the deal.II library.
//
// Detailed license information governing the source code and contributions
// can be found in LICENSE.md and CONTRIBUTING.md at the top level directory.
//
// -----------------------------------------------------------------------------
 
 
#include <deal.II/base/config.h>
 
#include <deal.II/base/polynomial.h>
#include <deal.II/base/polynomials_integrated_legendre_sz.h>
 
#include <vector>
 
DEAL_II_NAMESPACE_OPEN
 
IntegratedLegendreSZ::IntegratedLegendreSZ(const unsigned int k)
  : Polynomials::Polynomial<double>(get_coefficients(k))
{}
 
 
 
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