chap3.txt

  
  3 GaussForHomalg methods and functions
  
  
  3.1 The Tools
  
  Please   note   that   there   are  more  tool  functions  you  can  define,
  GaussForHomalg  just  provides  homalg with a sufficient subset. This varies
  with  the  type and complexity of the rings you want to define. On the other
  hand,   ImportMatrix   (3.1-4)  is  a  function  specifically  designed  for
  GaussForHomalg.
  
  3.1-1 ZeroMatrix
  
  ZeroMatrix( C )  function
  Returns:  a sparse matrix
  
  This  returns  a sparse matrix with the same dimensions and base ring as the
  homalg matrix C.
  
  3.1-2 IdentityMatrix
  
  IdentityMatrix( C )  function
  Returns:  a sparse matrix
  
  This returns a sparse n × n identity matrix with the same ring as the homalg
  matrix C, n being the number of rows of C.
  
  3.1-3 CopyMatrix
  
  CopyMatrix( C )  function
  Returns:  a sparse matrix
  
  This  returns  a  sparse matrix which is a shallow copy of the sparse matrix
  stored in the Eval attribute of the homalg matrix C.
  
  3.1-4 ImportMatrix
  
  ImportMatrix( M, R )  function
  Returns:  a sparse matrix
  
  This  returns  the  sparse  version  of the GAP matrix M over the ring R. It
  prevents homalg from calling sparse matrix algorithms on dense GAP matrices.
  Note  that  this  is not a "standard" tool but neccessary because of the new
  data type.
  
  3.1-5 Involution
  
  Involution( M )  function
  Returns:  a sparse matrix
  
  This  returns  a  sparse  matrix which is the transpose of the sparse matrix
  stored in the Eval attribute of the homalg matrix M.
  
  3.1-6 CertainRows
  
  CertainRows( M, plist )  function
  Returns:  a sparse matrix
  
  This  returns  the  rows  in  plist  of the sparse matrix stored in the Eval
  attribute of the homalg matrix M as a new matrix.
  
  3.1-7 CertainColumns
  
  CertainColumns( M, plist )  function
  Returns:  a sparse matrix
  
  This  returns  the  columns in plist of the sparse matrix stored in the Eval
  attribute of the homalg matrix M as a new matrix.
  
  3.1-8 UnionOfRows
  
  UnionOfRows( A, B )  function
  Returns:  a sparse matrix
  
  This  returns  the  sparse  matrix  created by concatenating the rows of the
  sparse  matrices  stored in the Eval attributes of the homalg matrices A and
  B.
  
  3.1-9 UnionOfColumns
  
  UnionOfColumns( A, B )  function
  Returns:  a sparse matrix
  
  This  returns  the sparse matrix created by concatenating the columns of the
  sparse  matrices  stored in the Eval attributes of the homalg matrices A and
  B.
  
  3.1-10 DiagMat
  
  DiagMat( e )  function
  Returns:  a sparse matrix
  
  This  method  takes a list e of homalg matrices and returns the sparse block
  matrix of the matrices stored in the Eval attributes of the matrices in e.
  
  3.1-11 KroneckerMat
  
  KroneckerMat( A, B )  function
  Returns:  a sparse matrix
  
  This  returns the sparse Kronecker matrix of the matrices stored in the Eval
  attributes of the homalg matrices A and B.
  
  3.1-12 Compose
  
  Compose( A, B )  function
  Returns:  a sparse matrix
  
  This  returns  the  matrix product of the sparse matrices stored in the Eval
  attributes of the homalg matrices A and B.
  
  3.1-13 NrRows
  
  NrRows( C )  function
  Returns:  an integer
  
  This  returns  the  number  of  rows of the sparse matrix stored in the Eval
  attribute of the homalg matrix C.
  
  3.1-14 NrColumns
  
  NrColumns( C )  function
  Returns:  an integer
  
  This  returns  the number of columns of the sparse matrix stored in the Eval
  attribute of the homalg matrix C.
  
  3.1-15 IsZeroMatrix
  
  IsZeroMatrix( C )  function
  Returns:  true or false
  
  This  returns  true if the sparse matrix stored in the Eval attribute of the
  homalg matrix C is a zero matrix, and false otherwise.
  
  3.1-16 IsDiagonalMatrix
  
  IsDiagonalMatrix( C )  function
  Returns:  true or false
  
  This  returns  true if the sparse matrix stored in the Eval attribute of the
  homalg matrix C is a diagonal matrix, and false otherwise.
  
  3.1-17 ZeroRows
  
  ZeroRows( C )  function
  Returns:  a list
  
  This  returns  the list of zero rows of the sparse matrix stored in the Eval
  attribute of the homalg matrix C.
  
  3.1-18 ZeroColumns
  
  ZeroColumns( C )  function
  Returns:  a list
  
  This  returns  the  list  of zero columns of the sparse matrix stored in the
  Eval attribute of the homalg matrix C.
  
  
  3.2 The Basic Functions and homalg table creation
  
  3.2-1 DecideZeroRows
  
  DecideZeroRows( A, B )  function
  Returns:  a homalg matrix
  
  This  returns  the homalg matrix you get by row reducing the homalg matrix A
  with the homalg matrix B.
  
  3.2-2 DecideZeroRowsEffectively
  
  DecideZeroRowsEffectively( A, B, T )  function
  Returns:  a homalg matrix M
  
  This returns the homalg matrix M you get by row reducing the homalg matrix A
  with  the  homalg  matrix B. The transformation matrix is stored in the void
  homalg matrix T as a side effect. The matrices satisfy M = A + T * B.
  
  3.2-3 SyzygiesGeneratorsOfRows
  
  SyzygiesGeneratorsOfRows( M )  function
  Returns:  a homalg matrix
  
  This  returns  the  row  syzygies  of the homalg matrix M, again as a homalg
  matrix.
  
  3.2-4 RelativeSyzygiesGeneratorsOfRows
  
  RelativeSyzygiesGeneratorsOfRows( M, N )  function
  Returns:  a homalg matrix
  
  The  row syzygies of M are returned, but now the computation interpretes the
  rows of the homalg matrix N as additional zero relations.
  
  3.2-5 RowReducedEchelonForm
  
  RowReducedEchelonForm( M[, U] )  function
  Returns:  a homalg matrix N
  
  If  one  argument  is  given, this returns the triangular basis (reduced row
  echelon  form)  of the homalg matrix M, again as a homalg matrix. In case of
  two  arguments,  still  only  the triangular basis of M is returned, but the
  transformation  matrix  is  stored  in  the  void  homalg matrix U as a side
  effect. The matrices satisfy N = U * M.
  
  3.2-6 CreateHomalgTable
  
  CreateHomalgTable( R )  function
  Returns:  a homalg table
  
  This returns the homalg table of what will become the homalg ring R (at this
  point  R  is  just  a  homalg  object  with  some  properties for the method
  selection  of  CreateHomalgTable). This method includes the needed functions
  stored   in   the   global   variables   CommonHomalgTableForGaussTools  and
  CommonHomalgTableForGaussBasic,  and  can  add  some more to the record that
  will become the homalg table.
  
  
  3.3 Matrix entry manipulation
  
  This is just support for the sparse matrix data type.
  
  3.3-1 MatElm
  
  MatElm( M, r, c, R )  method
  Returns:  M[r,c]
  
  If  the  Eval  attribute  of  the  homalg matrix M over the homalg ring R is
  sparse, this calls the corresponding Gauss command GetEntry.
  
  3.3-2 SetMatElm
  
  SetMatElm( M, r, c, e, R )  method
  Returns:  nothing
  
  If  the  Eval  attribute  of  the  homalg matrix M over the homalg ring R is
  sparse,  this  calls  the  corresponding  Gauss command GetEntry, to achieve
  M[r,c]:=e.
  
  3.3-3 AddToMatElm
  
  AddToMatElm( M, r, c, e, R )  method
  Returns:  nothing
  
  If  the  Eval  attribute  of  the  homalg matrix M over the homalg ring R is
  sparse,  this  calls  the corresponding Gauss command AddToEntry, to achieve
  M[r,c] := M[r,c] + e.