source: Sophya/trunk/SophyaLib/Samba/lambdaBuilder.h@ 758

#ifndef LAMBDABUILDER_SEEN
#define LAMBDABUILDER_SEEN

#include <math.h>
#include "ndatablock.h"
#include "tvector.h" 
#include "alm.h"

/*! 
generate Legendre polynomials : use in two steps : 

a) instanciate Legendre(\f$x\f$, \f$lmax\f$) ; \f$x\f$ is the value for wich Legendre polynomials will be required (usually equal to \f$\cos \theta\f$) and \f$lmax\f$ is the MAXIMUM value of the order of polynomials wich will be required in the following code (all polynomials, from \f$l=0 to lmax\f$, are computed once for all by an iterative formula).

b) get the value of Legendre polynomial for a particular value of \f$l\f$ by calling the method getPl.
*/
class Legendre {

 public :
Legendre();
Legendre(r_8 x, int_4 lmax);
 inline r_8 getPl(int_4 l) 
{
  if (l>lmax_) 
    {
      throw (" illegal call of Legendre::getPl with index greater than lmax, which Legendre Class was instanciated with : instanciate Legendre with a greater lmax... ");
    }
return Pl_(l);
}

 private : 
   /*! compute all \f$P_l(x,l_{max})\f$ for \f$l=1,l_{max}\f$ */
  void array_init(int_4 lmax);

  r_8 x_;
  int_4 lmax_;
  TVector<r_8> Pl_;
};



/*! 
This class generate the coefficients :
\f[
            \lambda_l^m=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}
            P_l^m(\cos{\theta})
\f]
where \f$P_l^m\f$ are the associated Legendre polynomials.  The above coefficients contain the theta-dependance of spheric harmonics : 
\f[
            Y_{lm}(\cos{\theta})=\lambda_l^m(\cos{\theta}) e^{im\phi}.
\f]

Each object has a fixed theta (radians), and maximum l and m to be calculated
(lmax and mmax).
 use the class in two steps : 
a) instanciate  LambdaLMBuilder(\f$\theta\f$, \f$lmax\f$, \f$mmax\f$) ;  \f$lmax\f$ and \f$mmax\f$ are  MAXIMUM values for which \f$\lambda_l^m\f$ will be required in the following code (all coefficients, from \f$l=0 to lmax\f$, are computed once for all by an iterative formula).
b) get the values of coefficients for  particular values of \f$l\f$ and \f$m\f$ by calling the method lamlm.
*/
 class LambdaLMBuilder {

 public:

LambdaLMBuilder() {}
LambdaLMBuilder(r_8 theta,int_4 lmax, int_4 mmax);
 ~LambdaLMBuilder() {};

 /*! return the value of the coefficient \f$  \lambda_l^m \f$ */
inline double lamlm(int l, int m) const { return lambda_(l,m); }

 private:
/*! compute a static array of coefficients independant from theta (common to all instances of the LambdaBuilder Class */
 void updateArrayRecurrence(int_4 lmax);

 void array_init(int lmax, int mmax);


 static TriangularMatrix<r_8>* a_recurrence_;
 TriangularMatrix<r_8> lambda_;

 protected :

/*! compute  static arrays of coefficients independant from theta (common to all instances of the derived  classes */
 void updateArrayLamNorm();

 static  TriangularMatrix<r_8>* lam_fact_;
 static  TVector<r_8>*  normal_l_;
 int_4 lmax_;
 int_4 mmax_;
 r_8 cth_;
 r_8 sth_;
 
 };



/*! 

This class generates the coefficients :
\f[
            _{w}\lambda_l^m=-2\sqrt{\frac{2(l-2)!}{(l+2)!}\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}} G^+_{lm}
\f]
\f[
            _{x}\lambda_l^m=-2\sqrt{\frac{2(l-2)!}{(l+2)!}\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}}G^-_{lm}
\f]
where
\f[G^+_{lm}(\cos{\theta})=-\left( \frac{l-m^2}{\sin^2{\theta}}+\frac{1}{2}l\left(l-1\right)\right)P_l^m(\cos{\theta})+\left(l+m\right)\frac{\cos{\theta}}{\sin^2{\theta}}P^m_{l-1}(\cos{\theta})
\f]
and
\f[G^-_{lm}(\cos{\theta})=\frac{m}{\sin^2{\theta}}\left(\left(l-1\right)\cos{\theta}P^m_l(\cos{\theta})-\left(l+m\right)P^m_{l-1}(\cos{\theta})\right)
\f]
 \f$P_l^m\f$ are the associated Legendre polynomials.

The coefficients express the theta-dependance of the \f$W_{lm}(\cos{\theta})\f$ and \f$X_{lm}(\cos{\theta})\f$ functions :
\f[W_{lm}(\cos{\theta}) = \sqrt{\frac{(l+2)!}{2(l-2)!}}_w\lambda_l^m(\cos{\theta})e^{im\phi}
\f]
\f[X_{lm}(\cos{\theta}) = -i\sqrt{\frac{(l+2)!}{2(l-2)!}}_x\lambda_l^m(\cos{\theta})e^{im\phi}
\f]
 where \f$W_{lm}(\cos{\theta})\f$ and \f$X_{lm}(\cos{\theta})\f$ are defined as :

\f[
W_{lm}(\cos{\theta})=-\frac{1}{2}\sqrt{\frac{(l+2)!}{(l-2)!}}\left(
_{+2}Y_l^m(\cos{\theta})+_{-2}Y_l^m(\cos{\theta})\right)
\f]
\f[X_{lm}(\cos{\theta})=-\frac{i}{2}\sqrt{\frac{(l+2)!}{(l-2)!}}\left(
_{+2}Y_l^m(\cos{\theta})-_{-2}Y_l^m(\cos{\theta})\right)
\f]

*/
class LambdaWXBuilder : public LambdaLMBuilder
{
 public:


LambdaWXBuilder() {}

LambdaWXBuilder(r_8 theta, int_4 lmax, int_4 mmax);
 
 /*! return the value of the coefficients \f$ _{w}\lambda_l^m\f$ and \f$_{x}\lambda_l^m\f$ */
inline void lam_wx(int l, int m, r_8& w, r_8& x) const
   {
     w=lamWlm_(l,m);
     x=lamXlm_(l,m);
   }

 private:

 void array_init();


  TriangularMatrix<r_8> lamWlm_;
  TriangularMatrix<r_8> lamXlm_;
  

};

/*! 

This class generates the coefficients
\f[
            _{\pm}\lambda_l^m=2\sqrt{\frac{(l-2)!}{(l+2)!}\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}}\left( G^+_{lm} \mp G^-_{lm}\right)
\f]
where
\f[G^+_{lm}(\cos{\theta})=-\left( \frac{l-m^2}{\sin^2{\theta}}+\frac{1}{2}l\left(l-1\right)\right)P_l^m(\cos{\theta})+\left(l+m\right)\frac{\cos{\theta}}{\sin^2{\theta}}P^m_{l-1}(\cos{\theta})
\f]
and
\f[G^-_{lm}(\cos{\theta})=\frac{m}{\sin^2{\theta}}\left(\left(l-1\right)\cos{\theta}P^m_l(\cos{\theta})-\left(l+m\right)P^m_{l-1}(\cos{\theta})\right)
\f]
and \f$P_l^m\f$ are the associated Legendre polynomials.
The coefficients express the theta-dependance of the  spin-2 spherical harmonics :
\f[_{\pm2}Y_l^m(\cos{\theta})=_\pm\lambda_l^m(\cos{\theta})e^{im\phi}
\f]
*/
class LambdaPMBuilder : public LambdaLMBuilder
{
 public:

LambdaPMBuilder() {}

 LambdaPMBuilder(r_8 theta, int_4 lmax, int_4 mmax);
 /*! return the value of the coefficients \f$ _{+}\lambda_l^m\f$ and \f$_{-}\lambda_l^m\f$ */
inline void lam_pm(int l, int m, r_8& lambda_plus, r_8& lambda_moins) const
   {
      lambda_plus = lamPlm_(l,m);
     lambda_moins = lamMlm_(l,m);
   }

 private:
 void array_init();

  TriangularMatrix<r_8> lamPlm_;
  TriangularMatrix<r_8> lamMlm_;



};

#endif