ZGOUBI USERS' GUIDE - HEP

4.4 Optical Elements and related numerical procedures 87´Ú¾’¾ÝÚqY’ÝY‰l ¡ jl j^Ú¾4Å’¾ Ý>ÚY’ÝYÚ¾’¾ ÝÚY’ÝYÚ¾’ÝYintervene in the derivatives of the compound functions ÚœÛ&Ü£ÝÝ d ¾ "5ÚœÛ&ÜÝ d ¾ "ÞÚÁÛ&Ü.d ¾ "ÞÚœÛ&Ü£ÝThese ingredients allow calculating the derivatives ÚœÛ&Ü£Ýd ¾d ¾d ¾d ¾" which, in turn,V µV µÉ V µÉ V µ"ÚÁÛ&Ü ÝÛ Ú¨ V µÛ ÚÛ Ú˜ V µÛ ÚInterpolation method :"RÚœÛ&Ü Ýß V µÛ ÚÛ ÚÛ ÚThe expression ¡ 1interpolation grid in the median plane centered on the projectionFig. 20. A polynomial interpolation is involved, of the form"/Å in Eq. 4.4.12 is, in this case, computed at the ž žnodes U( or in practice) of a “flying”of the actual particle position as schemed inN ž2 44k47ˆ44‡77k74 > >¡ 1> 4that yields the requested derivatives, using ´ ´ ´"/Å ´Å ´ÅZ ´j[l Note that, the source code contains the explicit analytical expressions of the ´ coefficientsequations, so that the operation is not so CPU time consuming.^ Åj[lsolutions of the normalœ A U A ^interpolationgridδsB2particletrajectoryBm m 1B1 0 3Figure 20: Interpolation method. à‹á and àãâ are the projections in the median plane of particle positions ä:á and äØâ separated byone integration step åHæ .Extrapolation Off Median PlaneThe vector field ¡ and its derivatives in the median plane are calculated by means of a second or fourth order polynomialinterpolation, depending on the value of the parameter IORDRE (IORDRE=2, 25 or 4, see section 1.4.2). The transformationfrom polar to Cartesian coordinates is performed following eqs (1.4.9 or 1.4.10). Extrapolation off median planeis then performed by means of Taylor expansions, following the procedure described in section 1.3.2.