oomph-lib

00001 //LIC// ====================================================================
00002 //LIC// This file forms part of oomph-lib, the object-oriented, 
00003 //LIC// multi-physics finite-element library, available 
00004 //LIC// at http://www.oomph-lib.org.
00005 //LIC// 
00006 //LIC//           Version 0.90. August 3, 2009.
00007 //LIC// 
00008 //LIC// Copyright (C) 2006-2009 Matthias Heil and Andrew Hazel
00009 //LIC// 
00010 //LIC// This library is free software; you can redistribute it and/or
00011 //LIC// modify it under the terms of the GNU Lesser General Public
00012 //LIC// License as published by the Free Software Foundation; either
00013 //LIC// version 2.1 of the License, or (at your option) any later version.
00014 //LIC// 
00015 //LIC// This library is distributed in the hope that it will be useful,
00016 //LIC// but WITHOUT ANY WARRANTY; without even the implied warranty of
00017 //LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
00018 //LIC// Lesser General Public License for more details.
00019 //LIC// 
00020 //LIC// You should have received a copy of the GNU Lesser General Public
00021 //LIC// License along with this library; if not, write to the Free Software
00022 //LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
00023 //LIC// 02110-1301  USA.
00024 //LIC// 
00025 //LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
00026 //LIC// 
00027 //LIC//====================================================================
00028 //Header functions for functions used to generate orthogonal polynomials
00029 //Include guards to prevent multiple inclusion of the header
00030 
00031 #ifndef OOMPH_ORTHPOLY_HEADER
00032 #define OOMPH_ORTHPOLY_HEADER
00033 
00034 // Config header generated by autoconfig
00035 #ifdef HAVE_CONFIG_H
00036   #include <oomph-lib-config.h>
00037 #endif
00038 
00039 //Oomph-lib include
00040 #include "Vector.h"
00041 #include "oomph_utilities.h"
00042 
00043 #include <cmath>
00044 
00045 namespace oomph
00046 {
00047 
00048 //Let's put these things in a namespace
00049 namespace Orthpoly
00050 {
00051  const double pi = 4.0*std::atan(1.0);
00052  const double eps = 1.0e-15;
00053  
00054  ///\short Calculates Legendre polynomial of degree p at x
00055  /// using the three term recurrence relation 
00056  /// \f$ (n+1) P_{n+1} = (2n+1)xP_{n} - nP_{n-1} \f$ 
00057  inline double legendre(const unsigned &p, const double &x)
00058   {
00059    //Return the constant value
00060    if(p==0) return 1.0;
00061    //Return the linear polynomial
00062    else if(p==1) return x;
00063    //Otherwise use the recurrence relation
00064    else{
00065     //Initialise the terms in the recurrence relation, we're going
00066     //to shift down before using the relation.
00067     double L0 = 1.0, L1 = 1.0, L2 = x;
00068     //Loop over the remaining polynomials
00069     for(unsigned n=1;n<p;n++)
00070      {
00071       //Shift down the values
00072       L0 = L1; L1 = L2;
00073       //Use the three term recurrence
00074       L2 = ((2*n + 1)*x*L1 - n*L0)/(n+1);
00075      }
00076    //Once we've finished return the final value
00077     return L2;
00078    }
00079   }
00080 
00081 
00082  /// \short Calculates Legendre polynomial of degree p at x
00083  /// using three term recursive formula. Returns all polynomials up to
00084  /// order p in the vector
00085  inline void legendre_vector(const unsigned &p, const double &x, 
00086                              Vector<double> &polys)
00087   {
00088    //Set the constant term
00089    polys[0] = 1.0; 
00090    //If we're only asked for the constant term return
00091    if(p==0) {return;}
00092    //Set the linear polynomial
00093    polys[1] = x;
00094    //Initialise terms for the recurrence relation
00095    double L0 = 1.0, L1 = 1.0, L2 = x;
00096    //Loop over the remaining terms
00097    for(unsigned n=1;n<p;n++){
00098     //Shift down the values
00099     L0=L1;
00100     L1=L2;
00101     //Use the recurrence relation
00102     L2 = ((2*n+1)*x*L1 - n*L0)/(n+1);
00103     //Set the newly calculated polynomial
00104     polys[n+1] = L2;
00105    }
00106   }
00107 
00108 
00109  /// \short  Calculates first derivative of Legendre 
00110  /// polynomial of degree p at x
00111  /// using three term recursive formula. 
00112  /// \f$ nP_{n+1}^{'} = (2n+1)xP_{n}^{'} - (n+1)P_{n-1}^{'} \f$
00113  inline double dlegendre(const unsigned &p, const double &x)
00114   {
00115    double dL1 = 1.0, dL2 = 3*x, dL3=0.0;
00116    if(p==0) return 0.0;
00117    else if(p==1) return dL1;
00118    else if(p==2) return dL2;
00119    else{
00120     for(unsigned n=2;n<p;n++){
00121      dL3 = 1.0/n*((2.0*n+1.0)*x*dL2-(n+1.0)*dL1); 
00122      dL1=dL2;
00123      dL2=dL3;
00124     }
00125     return dL3;
00126    }
00127   }
00128  
00129  /// \short Calculates second derivative of Legendre 
00130  /// polynomial of degree p at x
00131  /// using three term recursive formula. 
00132  inline double ddlegendre(const unsigned &p, const double &x)
00133   {
00134    double ddL2 = 3.0, ddL3 = 15*x, ddL4=0.0;
00135    if(p==0) return 0.0;
00136    else if(p==1) return 0.0;
00137    else if(p==2) return ddL2;
00138    else if(p==3) return ddL3;
00139    else{
00140     for(unsigned i=3;i<p;i++){
00141      ddL4 = 1.0/(i-1.0)*((2.0*i+1.0)*x*ddL3-(i+2.0)*ddL2); 
00142      ddL2=ddL3;
00143      ddL3=ddL4;
00144     }
00145     return ddL4;
00146    }
00147   }
00148  
00149  /// \short Calculate the Jacobi polnomials
00150  inline double jacobi(const int &alpha, const int &beta, const unsigned &p, 
00151                       const double &x)
00152   {
00153    double P0 = 1.0;
00154    double P1 = 0.5*(alpha-beta+(alpha+beta+2.0)*x);
00155    double P2;
00156    if(p==0) return P0;
00157    else if(p==1) return P1;
00158    else{
00159     for(unsigned n=1;n<p;n++){
00160      double a1n = 2*(n+1)*(n+alpha+beta+1)*(2*n+alpha+beta);
00161      double a2n = (2*n+alpha+beta+1)*(alpha*alpha-beta*beta);
00162      double a3n = (2*n+alpha+beta)*(2*n+alpha+beta+1)*(2*n+alpha+beta+2);
00163      double a4n = 2*(n+alpha)*(n+beta)*(2*n+alpha+beta+2);
00164      P2 = ( (a2n+a3n*x)*P1 - a4n*P0 ) / a1n;
00165      P0 = P1;
00166      P1 = P2;
00167     }
00168     return P2;
00169    }
00170   }
00171 
00172  /// \short Calculate the Jacobi polnomials all in one goe
00173  inline void jacobi(const int &alpha, const int &beta, const unsigned &p, 
00174                     const double &x,
00175                     Vector<double> &polys)
00176   {
00177    //Set the constant term
00178    polys[0] = 1.0;
00179    //If we've only been asked for the constant term, bin out
00180    if(p==0) {return;}
00181    //Set the linear polynomial
00182    polys[1] = 0.5*(alpha-beta+(alpha+beta+2.0)*x);   
00183    //Initialise the terms for the recurrence relation
00184    double P0 = 1.0;
00185    double P1 = 1.0;
00186    double P2 = 0.5*(alpha-beta+(alpha+beta+2.0)*x);
00187    //Loop over the remaining terms
00188    for(unsigned n=1;n<p;n++)
00189     {
00190      //Shift down the terms
00191      P0 = P1;
00192      P1 = P2;
00193      //Set the constants
00194      double a1n = 2*(n+1)*(n+alpha+beta+1)*(2*n+alpha+beta);
00195      double a2n = (2*n+alpha+beta+1)*(alpha*alpha-beta*beta);
00196      double a3n = (2*n+alpha+beta)*(2*n+alpha+beta+1)*(2*n+alpha+beta+2);
00197      double a4n = 2*(n+alpha)*(n+beta)*(2*n+alpha+beta+2);
00198      //Set the latest term
00199      P2 = ( (a2n+a3n*x)*P1 - a4n*P0 ) / a1n;
00200      //Set the newly calculate polynomial
00201      polys[n+1] = P2;
00202     }
00203   }
00204 
00205 
00206 /// Calculates the Gauss Lobatto Legendre abscissas for degree p = NNode-1
00207  void gll_nodes(const unsigned &Nnode, Vector<double> &x);
00208  
00209 // This version of gll_nodes calculates the abscissas AND weights
00210  void gll_nodes(const unsigned &Nnode, Vector<double> &x, Vector<double> &w);
00211 
00212 // Calculates the Gauss Legendre abscissas of degree p=Nnode-1
00213  void gl_nodes(const unsigned &Nnode, Vector<double> &x);
00214 
00215 // This version of gl_nodes calculates the abscissas AND weights
00216  void gl_nodes(const unsigned &Nnode, Vector<double> &x, Vector<double> &w);
00217 
00218 } //End of the namespace
00219 
00220 }
00221 
00222 #endif
orthpoly.h
