EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

Matching Methods

MADX currently supports four different matching algorithms:

• LMDIF: Fast Gradient Minimisation

  The LMDIF command minimises the sum of squares of the constraint functions using their numerical derivatives:

  LMDIF,CALLS=integer,TOLERANCE=real;
  

  It is the fastest minimisation method available in MAD. The command has two attributes:
  • CALLS: The maximum number of calls to the penalty function (default: 1000).
  • TOLERANCE: The desired tolerance for the minimum (default: 10**(-6)).
  Example:

  LMDIF,CALLS=2000,TOLERANCE=1.0E-8;
  

• MIGRAD: Gradient Minimisation

  The MIGRAD command minimises the penalty function using the numerical derivatives of the sum of squares:

  MIGRAD,CALLS=integer,TOLERANCE=real,STRATEGY=1;
  

  The command has three attributes:
  • CALLS: The maximum number of calls to the penalty function (default: 1000).
  • TOLERANCE: The desired tolerance for the minimum (default: 10**(-6)).
  • STRATEGY: A code for the strategy to be used (default: 1). Details are given in [James].
  Example:

  MIGRAD,CALLS=2000,TOLERANCE=1.0E-8;
  

• SIMPLEX: Simplex Minimisation

  The SIMPLEX command minimises the penalty function by the simplex method:

  SIMPLEX,CALLS=integer,TOLERANCE=real;
  

  Details are given in [James]. The command has two attributes:
  • CALLS: The maximum number of calls to the penalty function (default: 1000).
  • TOLERANCE: The desired tolerance for the minimum (default: 10**(-6)).
  Example:

  SIMPLEX,CALLS=2000,TOLERANCE=1.0E-8;
  

• JACOBIAN: Newton Minimisation

  The JACOBIAN command minimises the penalty function calculating the Jacobian and solving the linear problem. A QR or LQ decomposition is performed when the system is over or under-determined. Before starting the matching routine two optional transformations (COOL and RANDOM) are performed.

  JACOBIAN,CALLS=integer,TOLERANCE=real,REPEAT=integer,STRATEGY=integer,COOL=real,BALANCE=real, random=real;
  

  The command has the attributes:
  • CALLS: The maximum number of calls to the penalty function (default: 30).
  • TOLERANCE: The desired tolerance for the minimum (default: 10**(-6)).
  • REPEAT: The number of call of the JACOBIAN routine (default: 1).
  • BISEC: Selects the maximum number of iteratation used to determin the step length which reduces the penalty function during the main iteration. A large number (i.e. 6) reduce the probability to diverge from the solution, but increase the one for being trapped in a local minum.
  • STRATEGY: A code for the strategy to be used (default: 3). If STRATEGY=1 the routine resets the values of the variables which exceeds the limits. If STRATEGY=2 the routine print the Jacobian and exit without matching. If STRATEGY=3 the routine disables the variables which exceeds the limits keeping however the number of variables greater or equal to the number of the constraints.
  • COOL, BALANCE: The factors which specify the following transformation:

      if "balance" >=0
         newval=(1-cool)*oldval+cool*( (1-balance)*maxval+balance*minval )
      else
         newval=(1-cool)*oldval+cool* optval
    

    where newval is the new value after the transformation, oldval is the previous value, maxval, minval, optval are the maximum value, minimum value, optimal value of the variable specified in the VARY command.
  • RANDOM: The factors which specify the following transformation:

       newval= (1+ random * rand() ) * oldval
    

    where newval is the new value after the transformation, oldval is the previous value, rand() is a stochastic variable with a uniform (-0.5,0.5) distribution.
  Example:

  JACOBIAN,CALLS=20,TOLERANCE=1.0E-8,STRATEGY=3,COOL=0.1,BALANCE=0.5,RANDOM=0.01;