chap2.txt

  
  2 Usage
  
  If  you  are  just interested in using the Gauss package with homalg, you do
  not  need  to  know  much  about  GaussForHomalg,  as  it  will  work in the
  background, telling homalg which functions to call.
  
  However,  you might be interested in writing your own XyzForHomalg, enabling
  homalg  to  assist  you  with  your  computations.  For this purpose, I will
  provide  an overview of the GaussForHomalg code. Please note that Gauss is a
  GAP  package,  therefore  this  is  not  a  reference  guide for the package
  RingsForHomalg,  which  utilizes the IO-stream functionality of IO_ForHomalg
  to send commands to external computer algebra systems. If you wish to tie an
  external system to homalg, RingsForHomalg is the better reference package.
  
  The  file  for  all  low-level operations is GaussTools.gi. Like all "Tools"
  files  it  just includes one global variable CommonHomalgTableForGaussTools,
  which is a record of functions with homalg matrices as arguments. The return
  values  of  the  GaussForHomalg tools are documented in 3 and have to be the
  same for each tools table.
  
  In this particular case, the file also includes the following code:
  
  
    if IsBound( HOMALG.OtherInternalMatrixTypes ) then
        Add( HOMALG.OtherInternalMatrixTypes, IsSparseMatrix );
    else
        HOMALG.OtherInternalMatrixTypes := [ IsSparseMatrix ];
    fi;
  
  
  This  is a specialty to explain to homalg that Gauss introduces a new matrix
  type in GAP. Usually, there should not be a need for this.
  
  The  next "general" file is GaussBasic.gi. This includes the basic functions
  based     on     [BR08],    again    stored    in    the    global    record
  CommonHomalgTableForGaussBasic. Preceding this record are some small methods
  to make sure GaussForHomalg works with sparse as well as with dense matrices
  - just like above, these should not be neccessary in general.
  
  In   GaussForHomalg.gi   the  methods  for  matrix  entry  manipulation  are
  installed.
  
  Finally,  we  come  to  the  most important files, making sense of the basic
  functions   and   tools   defined  above.  Depending  on  the  functionality
  (especially concerning function names) of the system you will need different
  files for different rings. In this case, functionality for ℤ / n ℤ is stored
  in  GaussFQI.gi (Finite Quotients of the Integers), while the Rationals have
  their  own  file,  GaussRationals.gi.  Note that both files include only one
  method,  CreateHomalgTable,  using  method  selection  to create the correct
  table.  Depending  on  the  properties  of the ring, the basic functions are
  loaded  and  some more "specific" functions can be defined, in this case for
  example  the  function  RowReducedEchelonForm  (3.2-5), both in a one- and a
  two-argument  version. The tools should be universal enough to be loaded for
  every  possible  ring.  If it is neccessary to overwrite a tool, this is the
  place  to  do  it. An example for this could be Involution (3.1-5), which is
  generally just a matrix transposition, but could be overwritten to be a true
  involution when creating the homalg table for noncommutative rings.