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$$$$ LEVM     NOTICE  CHAT      11/09/12    21:17:01     7124           
                                             DATE     11/09/12

    Operateur LEVM                            Voir aussi : EXCE MOCA
    --------------                                         AJUSTE     
    LREE5 CHI2  = LEVM  'ABSC' LREE1 'ORDO' LREE2 'SIGM' LREE3 
                      'PARA' LREE4 'PROC' PRO1 ;




    Description :
    ------------
 
    The operator LEVM yields the best proposal for a sequence of 
    parameters of a function which aims to close a specified set 
    of points (abscissa, ordinate). The criteria is the weigthed sum
    of the squares of the ordonates differences. The algorithm is  
    the so-called Levenberg-Marquard's method. The operator is not
    reset when interrupted by the user.

    Remarks :
    ________

    LREE5 : list of proposed parameters (LISTREEL type)

    CHI2  : final criteria value (FLOTTANT type)

    LREE1 : list of abscissas (LISTREEL type)

    LREE2 : list of ordinates (LISTREEL type)

    LREE3 : list of each point weigths (LISTREEL type)

    LREE4 :  list of initial parameters (LISTREEL type)
            (it is hinted to provide non zero reals in the same scale
            than the expected values)

    PRO1 : procedure gibiane to compute the ordonates and partial 
           derivatives for each value LREE1. One should check the 
           convenient precision to compute the partial derivatives.
           Example of data :

           DEBPROC PRO2 LREEX*LISTREEL LREEA*LISTREEL ;

           * calcul de la fonction parametree 
           * compute the parameters dependant function
           * LREEX : liste des abscisses / abscissas
           * LREEA : liste des parametres / parameters
           * LREEY : liste des ordonnees / ordinates

           FINPROC LREEY ;

           DEBPROC PRO1 LREEX*LISTREEL LREEA*LISTREEL ; 
             
           * fonction argument
           * input procedure for LEVM
           * LREEX : liste des abscisses / abscissas
           * LREEA : liste des parametres / parameters
           * LREEY : liste des ordonnees / ordinates
           * calcul de LREEY
           LREEY = PRO2 LREEX LREEA ;

           * derivees partielles / partial derivatives
           TLRE = TABLE ;
           REPETER BPAR (DIME LREEA) ;
             ai = EXTR LREEA &BPAR ;
             LREEB = COPIE LREEA ;
             REMP LREEB &BPAR (ai * (1. + 1.e-2)) ;
             TLRE . &BPAR = PRO2 LREEX LREEB ;
           FIN BPAR ;

           * LREDY : derivees partielles / partial derivatives
           LREDY = PROG ;
           REPETER BX (DIME LREEX) ;
             REPETER BPAR (DIME LREEA) ;
               dyi = ((EXTR TLRE . &BPAR &BX) - (EXTR LREEY &BX) ) 
                        / 1.e-2 / (EXTR LREEA &BPAR) ;
             FIN BPAR ;
           FIN BX ;
               
           FINPROC LREEY LREDY ;