Changeset 1218 in Sophya for trunk/SophyaLib/Samba/lambdaBuilder.cc

Timestamp:

Oct 3, 2000, 2:14:14 PM (25 years ago)

Author:

ansari

Message:

doc dans .cc

File:

• trunk/SophyaLib/Samba/lambdaBuilder.cc (modified) (7 diffs)

trunk/SophyaLib/Samba/lambdaBuilder.cc

@@ -1 +1 @@
 #include "lambdaBuilder.h"
 #include "nbconst.h"
+
+
+/*!
+  \class SOPHYA::Legendre
+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.
+
+*/
 
 
@@ -12 +23 @@
   array_init(lmax);
 }
+   
+/*! \fn void SOPHYA::Legendre::array_init(int_4 lmax)
+
+compute all \f$P_l(x,l_{max})\f$ for \f$l=1,l_{max}\f$ 
+*/
 void Legendre::array_init(int_4 lmax)
 {
@@ -28 +44 @@
 TVector<r_8>* LambdaLMBuilder::normal_l_     = NULL;
 
+
+
+/*! \class SOPHYA::LambdaLMBuilder
+
+
+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.
+*/
+
+
 LambdaLMBuilder::LambdaLMBuilder(r_8 theta,int_4 lmax, int_4 mmax)
     {
@@ -84 +123 @@
 
 
+/*! \fn void  SOPHYA::LambdaLMBuilder::updateArrayRecurrence(int_4 lmax)
+
+ compute a static array of coefficients independant from theta (common to all instances of the LambdaBuilder Class 
+*/
 void  LambdaLMBuilder::updateArrayRecurrence(int_4 lmax)
    {
@@ -98 +141 @@
    }
 
-
+/*! \fn void  SOPHYA::LambdaLMBuilder::updateArrayLamNorm()
+
+ compute  static arrays of coefficients independant from theta (common to all instances of the derived  classes 
+*/
 void  LambdaLMBuilder::updateArrayLamNorm()
      {
@@ -119 +165 @@
 
 
+/*! \class SOPHYA::LambdaWXBuilder
+
+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]
+
+*/
 
 
@@ -173 +252 @@
    }
 
+/*!   \class SOPHYA::LambdaPMBuilder
+
+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]
+*/
 
 LambdaPMBuilder::LambdaPMBuilder(r_8 theta, int_4 lmax, int_4 mmax) : LambdaLMBuilder(theta, lmax, mmax)