  
  [1X6 Random elements[0X
  
  In this chapter we describe some fundamental mechanisms to produce (pseudo-)
  random  elements  that are used later in Chapter [14X7[0m about searching in groups
  and orbits.
  
  
  [1X6.1 Randomizing mutable objects[0X
  
  For  certain  types of mutable objects one can get a "random one" by calling
  the following operation:
  
  [1X6.1-1 Randomize[0m
  
  [2X> Randomize( [0X[3Xob[, rs][0X[2X ) ___________________________________________[0Xoperation
  [6XReturns:[0X  nothing
  
  The  mutable  object [3Xob[0m is changed in place. The value afterwards is random.
  The  optional  second  argument  [3Xrs[0m  must  be a random source and the random
  numbers  used  to  randomize  [3Xob[0m are created using the random source [3Xrs[0m (see
  [14X'Reference: Random Sources'[0m). If [3Xrs[0m is not given, then the global [5XGAP[0m random
  number generator is used.
  
  Currently, there are [2XRandomize[0m methods for compressed vectors and compressed
  matrices over finite fields. See also the [5Xcvec[0m package for methods for [10Xcvec[0ms
  and [10Xcmat[0ms.
  
  For vectors and one-dimensional subspaces there are two special functions to
  create a list of random objects:
  
  [1X6.1-2 MakeRandomVectors[0m
  
  [2X> MakeRandomVectors( [0X[3Xsample, number[, rs][0X[2X ) ________________________[0Xfunction
  [6XReturns:[0X  a list of random vectors
  
  [3Xsample[0m must be a vector for the mutable copies of which [2XRandomize[0m ([14X6.1-1[0m) is
  applicable  and  [3Xnumber[0m  must  be a positive integer. If given, [3Xrs[0m must be a
  random  source.  This  function creates a list of [3Xnumber[0m random vectors with
  the  same type as [3Xsample[0m using [2XRandomize[0m ([14X6.1-1[0m). For the creation of random
  numbers the random source [3Xrs[0m is used or, if not given, the global [5XGAP[0m random
  number generator.
  
  [1X6.1-3 MakeRandomLines[0m
  
  [2X> MakeRandomLines( [0X[3Xsample, number[, rs][0X[2X ) __________________________[0Xfunction
  [6XReturns:[0X  a list of normalised random vectors
  
  [3Xsample[0m must be a vector for the mutable copies of which [2XRandomize[0m ([14X6.1-1[0m) is
  applicable  and  [3Xnumber[0m  must  be a positive integer. If given, [3Xrs[0m must be a
  random  source.  This  function  creates  a list of [3Xnumber[0m normalised random
  vectors  with  the same type as [3Xsample[0m using [2XRandomize[0m ([14X6.1-1[0m). "Normalised"
  here  means  that  the first non-zero entry in the vector is equal to 1. For
  the  creation  of  random  numbers  the  random source [3Xrs[0m is used or, if not
  given, the global [5XGAP[0m random number generator.
  
  
  [1X6.2 Product replacement[0X
  
  For  computations  in  finite  groups  product  replacement algorithms are a
  viable method of generating pseudo-random elements. This section describes a
  framework and an object type to provide these algorithms. Roughly speaking a
  "product  replacer object" is something that is created with a list of group
  generators  and  produces  a  sequence of pseudo random group elements using
  some random source for random numbers.
  
  [1X6.2-1 ProductReplacer[0m
  
  [2X> ProductReplacer( [0X[3Xgens[, opt][0X[2X ) __________________________________[0Xoperation
  [6XReturns:[0X  a new product replacer object
  
  [3Xgens[0m  must be a list of group generators. If given, [3Xopt[0m is a [5XGAP[0m record with
  options.  The  operation  creates  a  new  product replacer object producing
  pseudo random elements in the group generated by the generators [3Xgens[0m.
  
  The  exact  algorithm  used  is explained below after the description of the
  options.
  
  The following components in the options record have a defined meaning:
  
  [8X[10Xrandomsource[0m[8X[0m
        A  random  source  object  that is used to generate the random numbers
        used.  If  none is specified the global [5XGAP[0m random number generator is
        used.
  
  [8X[10Xscramble[0m[8X[0m
        The  [10Xscramble[0m  value in the algorithm described below can be set using
        this option. The default value is 100.
  
  [8X[10Xscramblefactor[0m[8X[0m
        The  [10Xscramblefactor[0m  value in the algorithm described below can be set
        using this option. The default value is 10.
  
  [8X[10Xaddslots[0m[8X[0m
        The  [10Xaddslots[0m  value in the algorithm described below can be set using
        this option. The default value is 10.
  
  [8X[10Xmaxdepth[0m[8X[0m
        If  [10Xmaxdepth[0m  is  set,  then  the production of pseudo random elements
        starts  all  over  whenever  [10Xmaxdepth[0m  product  replacements have been
        performed.  The  rationale  behind  this  is that the elements created
        should   be  evenly  distributed  but  that  the  expressions  in  the
        generators should not be too long. A good compromise is usually to set
        [10Xmaxdepth[0m to 200 or 300.
  
  [8X[10Xnoaccu[0m[8X[0m
        Without  this  option  set  to  [9Xtrue[0m  the  "rattle" version of product
        replacement  is  used  which  involves  an  accumulator  and  uses two
        products  per random element. To use the "shake" version with only one
        product replacement per random element set this component to [9Xtrue[0m.
  
  [8X[10Xnormalin[0m[8X[0m
        There  is a variant of the product replacement algorithm that produces
        elements  in  the  normal  closure of the group generated by a list of
        elements.  It  needs random elements in the ambient group in which the
        normal  closure  is  defined.  This is implemented here by setting the
        [10Xnormalin[0m component to a product replacer object working in the ambient
        group.  It is recommended to switch off the accumulator in the product
        replacer  object  for  the  ambient  group. Then to produce one random
        element in the normal closure needs three multiplications.
  
  The  algorithm  used  does  the  following:  A list of [10XLength([0m[3Xgens[0m[10X)+addslots[0m
  elements is created that starts with the elements [3Xgens[0m and is filled up with
  random  generators  from  [3Xgens[0m.  A  product replacement randomly chooses two
  elements in the list and replaces one of them by the product of the two. One
  step  in  the  algorithm  is  to  do  one  product  replacement  followed by
  post-multiplying  the  result  to  the  accumulator  if  one  is used. First
  [10XMaximum(Length([0m[3Xgens[0m[10X)*scramblefactor,scramble)[0m  steps  are  performed.  After
  this initialisation for every random element requested one step is done done
  and the resulting element returned.
  
  [1X6.2-2 Next[0m
  
  [2X> Next( [0X[3Xpr[0X[2X ) ______________________________________________________[0Xoperation
  [6XReturns:[0X  a (pseudo-) random group element g
  
  [3Xpr[0m  must  be  a  product  replacer  object.  This operation makes the object
  generate the next random element and return it.
  
  [1X6.2-3 Reset[0m
  
  [2X> Reset( [0X[3Xpr[0X[2X ) _____________________________________________________[0Xoperation
  [6XReturns:[0X  nothing
  
  [3Xpr[0m  must  be  a product replacer object. This operation resets the object in
  the  sense  that  it resets its random source (see [2XReset[0m ([14XReference: Reset[0m))
  and reinitialises the random element generation as described above.
  
