In tracking codes there is the need to obtain, at run time, various machine quantities which depend parametically on things such as momentum or quadrupole strength. To this end we have overloaded (in FORTRAN 90) Berz' DA package [1] as well as the analysis library LieLib [2,3] which is based on this package and we have created polymorphic types. Runtime polymorphism is not interpretation as in COSY-INFINITY [4] and is more appropriate to large ring tracking codes. Consequently we have applied these tools to the code SixTrack [6]

Forest, E

Schmidt, F

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN - SL DIVISIONIn tracking codes there is the need to obtain, at run time, various machine quantities which dependparametically on things such as momentum or quadrupole strength.  To this end we have overloaded (inFORTRAN 90) Berz' DA package [1] as well as the analysis library LieLib [2,3] which is based on thispackage and we have created polymorphic types. Runtime polymorphism is not interpretation as inCOSY-INFINITY [4] and is more appropriate to large ring tracking codes.  Consequently we haveapplied these tools to the code SixTrack [6].* KEKCERN-SL-2000-048 paper presented  at  EPAC2000, 26-30 June 2000, Vienna, AustriaGeneva, Switzerland 03.08.00SL/Div RepsMap creation and analysis via overloaded tools in FORTRAN 90AP  E. Forest*, F. SchmidtMap Creation and Analysis via overloaded Tools in FORTRAN 90E. Forest, KEK; F. Schmidt, CERNAbstractIn tracking codes there is the need to obtain, at run time,various machine quantities which depend parametricallyon things such as momentum or quadrupole strength. Tothis end we have overloaded (in FORTRAN90) Berz’s DA-package [1] as well as the analysis library LieLib [2, 3]which is based on this package and we have created poly-morphic types. Runtime polymorphism is not interpreta-tion as in COSY-INFINITY [4] and is more appropriate tolarge ring tracking codes. Consequently we have appliedthese tools to the code SixTrack [6].1 INTRODUCTIONA substantial amount of accelerator theory reduces to thecalculation of a normal form. The normal form is a for-mal statement concerning the stability properties of the ringunder consideration. For example, if one states that a lin-ear system can be normalised into a rotation, and then thisstatement is mathematically equivalent to the so-called lin-ear Courant-Snyder theory. Normal form concepts can beapplied to the equations of motion or finite “s” maps. Be-cause they are applicable to finite “s” maps the entire ma-chinery of automatic differentiation, introduced by Berz toour field, can be used. Libraries such as our Lielib can bewritten to normalise these maps. Finally the normalisingtransformations can be propagated using the same trackingcode that produced the maps in the first place.There are two impediments to the usage of these tech-niques in accelerators. First there is a cultural barrier: un-less one learns how to rephrase the standard theory in termsof finite maps, then the machinery of Berz is not usable.Map methods in rings, in conjunction with Lie methods,give us an elegant and powerful practical way to do linearlattice calculations which are easily extendible to nonlinearproblems. Unless one is willing to sit down and (re)learnlinear theory, then these finite “s” map methods will be dif-ficult if not impossible to swallow.Secondly, if one swims against the cultural current andlearns the map methods, there is still a formidable taskin dealing with FORTRAN77 libraries such as the “DA-package” and Lielib. Berz, first to notice this problem,started the COSY-INFINITY program in which he devisedan interpreted language. The COSY language permits usersto access Taylor series and manipulate them using a nicersyntax. For example, the addition of two Taylors A andB could become C=A+B rather than using a call to a sub-routine such as CALL DAADD(A,B,C). COSY-INFINITYreads a file, which it then interprets. This solution is cer-tainly acceptable to single pass systems but fits poorly intotracking codes for large rings.Modern languages such as C++ and also FORTRAN90permit operator overloading and thus allows one to use thesyntax C=A+B in a compiled code. In addition, one wouldlike to do “parameter dependence at run time.” The au-thors had included this feature into their code SixTrackand DESPOT respectively in a very cumbersome way us-ing the old FORTRAN77 “DA-package.” Bengtsson, to ourknowledge, was first to realize that in languages like C++,one could develop a delegated (run time) polymorphic typethat permit precisely this kind of run time conversion. Re-markably, and somewhat contrary to public belief, suchpolymorphic types can be constructed in FORTRAN90 aswell (see Szymanski [5]).We decided to go back to Bengtsson’s idea which con-sisted in linking directly Berz’s package and Lielib to theappropriate polymorphic classes. This work could cer-tainly have been done in C++ [8] ; we are not advocatingany language over another.2 THE POLYMORPHIC PACKAGEThe library is a hierarchical set of packages (modules inFORTRAN90) and they are displayed in Figure 1. The firstmodules definition.f90 contains definitions of types as thename indicates. For example, a type called taylor:TYPE TAYLORINTEGER I ! integer is a pointer in old da-packageTYPE (TAYLORLOW) J ! taylorlow is the newda taylor seriesEND TYPE TAYLORTaylor is the type for a polynomial. The integer I pointsto the integer used by Berz in his package to locate a poly-nomial. The type Taylorlow illustrates the possibility ofputting several TPSA libraries under the “hood.”The next 3 modules overload the old “TPSA-Package”of Berz and the physics library LIELIB. The basic TPSAoperations are defined in TPSA.F90, basic maps and theirconcatenation are defined in TPSALIE.F90 and finallymore advanced options such as normal forms and map fac-torisations are defined in TPSALIE ANALYSIS.F90. Upto this point we have overloaded the basic functionality ofthe “DA-package” and LIELIB. A code using these toolsonly could resemble very much the COSY-INFINITY inputfile except that it requires compilation: it is not interpreted.The first addition to this edifice was COMPLEX TAY-LOR.F90. Here we defined a complex polynomial as twopolynomials of type taylor: the real and imaginary parts.TYPE(COMPLEXTAYLOR)TYPE (TAYLOR) RTYPE (TAYLOR) IEND TYPE COMPLEXTAYLORThe two final packages are REAL POLYMORH.F90and COMPLEX POLYMORPH.F90. In these packageswe define a real and a complex polymorphic type. We willtpsalie_analysis.f90newlielib.f90Lielib.ftpsalie.90Tpsa.f90definition.f90dabnew.fnewda.f90Berz’s TPSA Package:polynomials are representedby integer pointers.New TPSA PackageAnalysis RoutinesBoth TPSA Taylor series are merged here.A new type called taylor combines taylorlowand the integer pointer of Berz’s TPSA. Thecomplex taylor type  and the polymorphictypes are also defined there.define_newda.f90 Defines in newtype called taylorlowBasic Operations of the TPSA packages involving Taylor series are overloaded here,for example +,-,/, etc... as well as other things suchas derivatives. Here we really overload dab.f andnewda.f90Here operations on maps are overloaded. A new typedamap is introduced (as well as other useful types).Map operations are overloaded, for example,concatenation and inverse.Finally, at the top, this package overloads varioususeful parameterization of a map: Dragt-Finn,inverse Dragt-Finn,   vector fields, vector fieldsin resonance basis, and, of course, normal forms.Code that overloads the Taylor packages(including that of Berz) as well as theanalysis routines of LielibFree standing TPSA packageswhich could be used on theirown. Of course that defeatsthe purpose of overloading!real_polymorph.f90complex_taylor.90Allows complex TPSA Polymorphic types: Can change a runtimeComplex_polymorph.f90Complex type is overloaded. The complex typeis made of 2 taylors: T YPE complextaylortype (taylor) rtype (taylor) iEND T YPE complextaylorFigure 1: Schematic Overview of the Full Polymorphic Package (FPP), down-load from web [7].Figure 1:not discuss the complex polymorph type but instead con-centrate on the real polymorphic type denoted by Real 8 inour package by analogy to REAL*8 in FORTRAN77.The Real 8 type is defined as follows:TYPE REAL_8TYPE (TAYLOR) TDOUBLE PRECISION RLOGICAL ALLOCINTEGER KINDINTEGER I ! assignment integerDOUBLE PRECISION S ! special scaling factorEND TYPE REAL_8A real polymorph is either double precision or taylor. Ifit is double precision then the integer kind is set to one andthe value of the polymorph is contained in the field “r.” Ifthe polymorph is a polynomial then the kind is set to twoand the logical alloc is true. This logical is necessary forgarbage collection of the polynomials.If a code is written entirely with types Real 8 rather thantrue double precisions, and if nothing else is done, then thecode will always use the field “r” and the TPSA packagewill never be invoked. Such a calculation is typically 4 to5 times slower than a true real floating point calculation asestimated long time ago by Bengtsson. The real power ofpolymorphism is the ability for a real number to turn Taylorat run time. This is achieved by overloading the equal sign(=). Suppose that the variable A is type real 8: A=5.d0.If the kind of A is set to 1, then this line simply assigns5.d0 to A%r. However if the user has set the kind of A to0, then this assignment will set polynomial part of A, A%t,to 5.d0 + s ∗ XI. This means that the variable A is now apolynomial in XI. The quantity A%s is normally defaultedto one unless the user desires otherwise. Of course uponcompletion of this assignment, the kind of A is set to 2. Itis now a polynomial.To illustrate how these things are done, we will showhere the functions that overload addition of two type taylorin TPSA.F90 (polynomials) and the equivalent routine inreal polymorph.f90.FUNCTION ADD( S1, S2 )TYPE (TAYLOR) ADDTYPE (TAYLOR), INTENT (IN) :: S1, S2CALL ASS(ADD) !assigns add to a global scratch variableIF(OLD) THEN ! old=.true. refers to berz’s da-packageCALL DAADD(S1%I,S2%I,ADD%I)ELSECALL NEWDAADD(S1%J,S2%J,ADD%J)ENDIFEND FUNCTION ADDThis function ADD, in TPSA.F90, overloads the addi-tion of two polynomials by calling the appropriate subrou-tine of the DA-package. There are routines for all possi-ble operations (adding scalar to polynomials, multiplica-tion etc. . . ) Now let us look at the equivalent function forthe polymorph.FUNCTION ADD( S1, S2 )TYPE (REAL_8) ADDTYPE (REAL_8), INTENT (IN) :: S1, S2INTEGER LOCALMASTERSELECT CASE(S1%KIND+3*S2%KIND)CASE(4)ADD%R=S1%R+S2%RADD%KIND=1CASE(5,7,8)LOCALMASTER=MASTERCALL ASS(ADD)SELECT CASE(S1%KIND+3*S2%KIND)CASE(5)ADD%T= S1%T+S2%RCASE(7) ;ADD%T= S1%R+S2%T ;CASE(8); ADD%T= S1%T+S2%TEND SELECTMASTER=LOCALMASTERCASE DEFAULTWRITE(6,*) " TROUBLE IN ADD "PAUSEEND SELECTEND FUNCTION ADDLooking at the above function, if S1%kind and S2%kindare both equal to one, then CASE(4) will be selected. The“double precision” field of the polymorphs S1 and S2 willbe added and assigned to the double precision field ofADD. If both kinds are 2, then CASE(8) will be selectedand the taylor parts will be added. This will invoke theother function ADD contained in TPSA.F90. Of coursethere are also the cases corresponding to a real added to ataylor series. These are CASE(7) and CASE(8). We men-tioned that a polymorph could have kind=0; this shouldnot happen during an operation and therefore the code willreach the DEFAULT case and the program will pause.The techniques used here could be written more com-pactly in C++. Also the assignment of global vari-ables is rather involved here: MASTER, LOCALMAS-TER,etc. . . are all related to the global scratch variablesgymnastics. We will not describe these things here.This completes our short description of polymorphism.Now we will show some examples of its use in a trackingcode.3 EXAMPLE OF USAGE IN SIXTRACKPROGRAM TESTUSE TRACKINGTYPE(BEAM_LINE) ALSREAL*8 CLOSED_ORBIT(6)TYPE(REAL_8) Y(6)TYPE(DAMAP) ATYPE(TAYLOR) BETA(2)TYPE(NORMALFORM) NORMAL_FORMINTEGER I,ik, ND2DO I=1,6 ;CLOSED_ORBIT(I)=0.D0 ;ENDDOCALL read_BEAM_LINE(ALS,’ALS.TXT’)write(6,*) " Number of quadrupoles 0 or 2 "read(5,*) ik ! If k=0 the quadrupoles are not parametersIF(IK==0) CALL KILL(ALS) ! MAKES ALL POLYMORPHIC INDEX p%I TO 0CALL FIND_orbit(ALS,CLOSED_ORBIT,1,nocavity)if(ik==2) then ! Two families of quadrupolesPARA_REMAIN=.TRUE.do i=1,als%nif(als%mag(i)%name(1:2)==’Q1’) als%magp(i)%bn(2)%i=6if(als%mag(i)%name(1:2)==’Q2’) als%magp(i)%bn(2)%i=7enddoendifND2=4call init(3,ND2/2,ik+1,0,BERZ)call alloc(y,6);call alloc(NORMAL_FORM)call alloc(BETA,2);call alloc(A)y=5do i=1,6; y(i)=CLOSED_ORBIT(i) ;enddoCALL TRACK(ALS,Y,1,1,-nocavity)NORMAL_FORM=yY=5do i=1,ND2;y(i)=NORMAL_FORM%A_t%V(i)+CLOSED_ORBIT(I);enddoDO I=ND2+1,6;Y(I)=CLOSED_ORBIT(I);ENDDOdo i=1,als%n-1CALL TRACK(ALS,Y,I,I+1,-NOCAVITY) !Tracking of NORMAL_FORM%A_tA=Y !POLYMORPHS MADE INTO A DAMAPWRITE(16,*) ALS%MAG(1)%NAMEBETA(1)=(A%V(1).PAR.’1000’)**2+(A%V(1).PAR.’0100’)**2CALL DAPRINT(BETA(1),16)BETA(2)=(A%V(3).PAR.’0010’)**2+(A%V(3).PAR.’0001’)**2CALL DAPRINT(BETA(2),16)BETA(1)=BETA(1).D.5 ; CALL DAPRINT(BETA(1),16)BETA(2)=BETA(2).D.5 ; CALL DAPRINT(BETA(2),16)enddoEND PROGRAM TESTIn this short program, the one-turn fully coupled map iscomputed around its fixed point and then normalised. Thecanonical transformation is contained in the field NOR-MAL FORM%A T. From the theory of normal forms,it is expected that this transformation evolves like themap itself. Therefore it is propagated using the trackingcommand TRACK(ALS,Y,I,I+1,-NOCAVITY) through el-ement number “I.” In this particular example, we trackedthe beta functions and their derivatives with respect to delta(fifth variable). The reader notices that in this example thequadrupole strengths are potential runtime parameters.4 CONCLUSIONThe full package is operational and its usefulness has beendemonstrated in conjunction with the tracking tools for theLHC, HERA, and ALS lattices. Besides runtime param-eter dependence, it also includes stochastic envelopes (forradiation). All calculations use arbitrary vector fields.5 ACKNOWLEDGEMENTSOne of the authors (E.F.) would like to thank F. Ruggierofor his invitation to join the SL–AP group at CERN fora period of 3 months. He also acknowledges the help ofA. Xiao at DESY and the hospitality of F. Willeke.6 REFERENCES[1] M. Berz, Part. Accel., 1989, Vol. 24, pp. 109–124.[2] M. Berz, ´E. Forest and J. Irwin, Part. Accel., 1989, Vol. 24,pp. 91–107.[3] ´E. Forest, LBL differential algebra package and LieLib, un-published.[4] M. Berz, “COSY INFINITY 8”, NSCL Technical ReportMSUCL-1088, Michigan State University, 1998.[5] V.K. Decyk, C.D. Norton, and B.K. Szymanski, “How toSupport Inheritance and Run-Time Polymorphism in For-tran 90”, In Computer Physics Communications, Vol. 115,pp. 9-17, 1998.[6] F. Schmidt, CERN SL/94–56(AP) (1994).[7] F. Schmidt, “SixTrack web page”,http://wwwslap.cern.ch/˜frs.[8] Visit the following web sites: http://www-ap.fnal.gov/ michelot/, http://wwwslap.cern.ch/classic/.

Map creation and analysis via overloaded tools in FORTRAN 90

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