orthpoly.h

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//LIC// ====================================================================
//LIC// This file forms part of oomph-lib, the object-oriented, 
//LIC// multi-physics finite-element library, available 
//LIC// at http://www.oomph-lib.org.
//LIC// 
//LIC//    Version 1.0; svn revision $LastChangedRevision$
//LIC//
//LIC// $LastChangedDate$
//LIC// 
//LIC// Copyright (C) 2006-2016 Matthias Heil and Andrew Hazel
//LIC// 
//LIC// This library is free software; you can redistribute it and/or
//LIC// modify it under the terms of the GNU Lesser General Public
//LIC// License as published by the Free Software Foundation; either
//LIC// version 2.1 of the License, or (at your option) any later version.
//LIC// 
//LIC// This library is distributed in the hope that it will be useful,
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//LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
//LIC// Lesser General Public License for more details.
//LIC// 
//LIC// You should have received a copy of the GNU Lesser General Public
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//LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
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//LIC// 
//LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
//LIC// 
//LIC//====================================================================
//Header functions for functions used to generate orthogonal polynomials
//Include guards to prevent multiple inclusion of the header

#ifndef OOMPH_ORTHPOLY_HEADER
#define OOMPH_ORTHPOLY_HEADER

// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
  #include <oomph-lib-config.h>
#endif

#ifdef OOMPH_HAS_MPI
#include "mpi.h"
#endif

//Oomph-lib include
#include "Vector.h"
#include "oomph_utilities.h"

#include <cmath>

namespace oomph
{

//Let's put these things in a namespace
namespace Orthpoly
{
 const double eps = 1.0e-15;
 
 ///\short Calculates Legendre polynomial of degree p at x
 /// using the three term recurrence relation 
 /// \f$ (n+1) P_{n+1} = (2n+1)xP_{n} - nP_{n-1} \f$ 
 inline double legendre(const unsigned &p, const double &x)
  {
   //Return the constant value
   if(p==0) return 1.0;
   //Return the linear polynomial
   else if(p==1) return x;
   //Otherwise use the recurrence relation
   else{
    //Initialise the terms in the recurrence relation, we're going
    //to shift down before using the relation.
    double L0 = 1.0, L1 = 1.0, L2 = x;
    //Loop over the remaining polynomials
    for(unsigned n=1;n<p;n++)
     {
      //Shift down the values
      L0 = L1; L1 = L2;
      //Use the three term recurrence
      L2 = ((2*n + 1)*x*L1 - n*L0)/(n+1);
     }
   //Once we've finished return the final value
    return L2;
   }
  }


 /// \short Calculates Legendre polynomial of degree p at x
 /// using three term recursive formula. Returns all polynomials up to
 /// order p in the vector
 inline void legendre_vector(const unsigned &p, const double &x, 
                             Vector<double> &polys)
  {
   //Set the constant term
   polys[0] = 1.0; 
   //If we're only asked for the constant term return
   if(p==0) {return;}
   //Set the linear polynomial
   polys[1] = x;
   //Initialise terms for the recurrence relation
   double L0 = 1.0, L1 = 1.0, L2 = x;
   //Loop over the remaining terms
   for(unsigned n=1;n<p;n++){
    //Shift down the values
    L0=L1;
    L1=L2;
    //Use the recurrence relation
    L2 = ((2*n+1)*x*L1 - n*L0)/(n+1);
    //Set the newly calculated polynomial
    polys[n+1] = L2;
   }
  }


 /// \short  Calculates first derivative of Legendre 
 /// polynomial of degree p at x
 /// using three term recursive formula. 
 /// \f$ nP_{n+1}^{'} = (2n+1)xP_{n}^{'} - (n+1)P_{n-1}^{'} \f$
 inline double dlegendre(const unsigned &p, const double &x)
  {
   double dL1 = 1.0, dL2 = 3*x, dL3=0.0;
   if(p==0) return 0.0;
   else if(p==1) return dL1;
   else if(p==2) return dL2;
   else{
    for(unsigned n=2;n<p;n++){
     dL3 = 1.0/n*((2.0*n+1.0)*x*dL2-(n+1.0)*dL1); 
     dL1=dL2;
     dL2=dL3;
    }
    return dL3;
   }
  }
 
 /// \short Calculates second derivative of Legendre 
 /// polynomial of degree p at x
 /// using three term recursive formula. 
 inline double ddlegendre(const unsigned &p, const double &x)
  {
   double ddL2 = 3.0, ddL3 = 15*x, ddL4=0.0;
   if(p==0) return 0.0;
   else if(p==1) return 0.0;
   else if(p==2) return ddL2;
   else if(p==3) return ddL3;
   else{
    for(unsigned i=3;i<p;i++){
     ddL4 = 1.0/(i-1.0)*((2.0*i+1.0)*x*ddL3-(i+2.0)*ddL2); 
     ddL2=ddL3;
     ddL3=ddL4;
    }
    return ddL4;
   }
  }
 
 /// \short Calculate the Jacobi polnomials
 inline double jacobi(const int &alpha, const int &beta, const unsigned &p, 
                      const double &x)
  {
   double P0 = 1.0;
   double P1 = 0.5*(alpha-beta+(alpha+beta+2.0)*x);
   double P2;
   if(p==0) return P0;
   else if(p==1) return P1;
   else{
    for(unsigned n=1;n<p;n++){
     double a1n = 2*(n+1)*(n+alpha+beta+1)*(2*n+alpha+beta);
     double a2n = (2*n+alpha+beta+1)*(alpha*alpha-beta*beta);
     double a3n = (2*n+alpha+beta)*(2*n+alpha+beta+1)*(2*n+alpha+beta+2);
     double a4n = 2*(n+alpha)*(n+beta)*(2*n+alpha+beta+2);
     P2 = ( (a2n+a3n*x)*P1 - a4n*P0 ) / a1n;
     P0 = P1;
     P1 = P2;
    }
    return P2;
   }
  }

 /// \short Calculate the Jacobi polnomials all in one goe
 inline void jacobi(const int &alpha, const int &beta, const unsigned &p, 
                    const double &x,
                    Vector<double> &polys)
  {
   //Set the constant term
   polys[0] = 1.0;
   //If we've only been asked for the constant term, bin out
   if(p==0) {return;}
   //Set the linear polynomial
   polys[1] = 0.5*(alpha-beta+(alpha+beta+2.0)*x);   
   //Initialise the terms for the recurrence relation
   double P0 = 1.0;
   double P1 = 1.0;
   double P2 = 0.5*(alpha-beta+(alpha+beta+2.0)*x);
   //Loop over the remaining terms
   for(unsigned n=1;n<p;n++)
    {
     //Shift down the terms
     P0 = P1;
     P1 = P2;
     //Set the constants
     double a1n = 2*(n+1)*(n+alpha+beta+1)*(2*n+alpha+beta);
     double a2n = (2*n+alpha+beta+1)*(alpha*alpha-beta*beta);
     double a3n = (2*n+alpha+beta)*(2*n+alpha+beta+1)*(2*n+alpha+beta+2);
     double a4n = 2*(n+alpha)*(n+beta)*(2*n+alpha+beta+2);
     //Set the latest term
     P2 = ( (a2n+a3n*x)*P1 - a4n*P0 ) / a1n;
     //Set the newly calculate polynomial
     polys[n+1] = P2;
    }
  }


/// Calculates the Gauss Lobatto Legendre abscissas for degree p = NNode-1
 void gll_nodes(const unsigned &Nnode, Vector<double> &x);
 
// This version of gll_nodes calculates the abscissas AND weights
 void gll_nodes(const unsigned &Nnode, Vector<double> &x, Vector<double> &w);

// Calculates the Gauss Legendre abscissas of degree p=Nnode-1
 void gl_nodes(const unsigned &Nnode, Vector<double> &x);

// This version of gl_nodes calculates the abscissas AND weights
 void gl_nodes(const unsigned &Nnode, Vector<double> &x, Vector<double> &w);

} //End of the namespace

}

#endif