  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  
  [1X1.1 [33X[0;0YWhat is the role of the [5XMatricesForHomalg[105X[101X[1X package in the [5Xhomalg[105X[101X[1X project?[133X[101X
  
  
  [1X1.1-1 [33X[0;0Y[5XMatricesForHomalg[105X[101X[1X provides ...[133X[101X
  
  [33X[0;0YThe package [5XMatricesForHomalg[105X provides:[133X
  
  [30X    [33X[0;6Yrings[133X
  
  [30X    [33X[0;6Yring elements[133X
  
  [30X    [33X[0;6Yring maps[133X
  
  [30X    [33X[0;6Ymatrices[133X
  
  
  [1X1.1-2 [33X[0;0Y[5Xhomalg[105X[101X[1X delegates ...[133X[101X
  
  [33X[0;0YThe package [5Xhomalg[105X [13Xdelegates[113X [13Xall[113X matrix operations as it treats matrices and
  their  rings  as [13Xblack boxes[113X. [5Xhomalg[105X comes with a single predefined class of
  rings  and  a  single  predefined  class of matrices over these rings -- the
  so-called  internal  matrices (--> [14X5.1-2[114X) over so-called internal rings (-->
  [14X3.1-4[114X).  An  internal  matrix  (resp. ring) is simply a wrapper containing a
  [5XGAP[105X-builtin  matrix  (resp.  ring).  [5Xhomalg[105X  allows other packages to define
  further  classes  or  extend existing classes of rings and matrices [13Xtogether[113X
  with their operations. For example:[133X
  
  [30X    [33X[0;6YThe  [5Xhomalg[105X  subpackage  [5XResidueClassRingForHomalg[105X  (-->  Appendix  [14XD[114X)
        defines  the  classes  of  residue  class  rings,  residue  class ring
        elements,  and  matrices  over  residue  class rings. Such a matrix is
        defined  by  a  matrix  over  the  ambient  ring which is nevertheless
        interpreted  modulo  the ring relations, i.e. modulo the generators of
        the defining ideal.[133X
  
  [30X    [33X[0;6YThe  package  [5XGaussForHomalg[105X  extends  the  class of internal matrices
        enabling  it  to  wrap  sparse matrices provided by the package [5XGauss[105X.
        [5XGaussForHomalg[105X delegates the essential part of the matrix creation and
        all matrix operations to [5XGauss[105X.[133X
  
  [30X    [33X[0;6YThe  package  [5XHomalgToCAS[105X  defines  the  classes of so-called external
        rings  and  matrices  and  the  package  [5XRingsForHomalg[105X  delegates the
        essential  part  of  the  matrix creation and all matrix operations to
        external  computer  algebra  systems  like  [5XSingular[105X, [5XMacaulay2[105X, [5XSage[105X,
        [5XMacaulay2[105X,  [5XMAGMA[105X,  [5XMaple[105X,  ... . The package [5Xhomalg[105X accesses external
        matrices via pointers. The pointer of an external matrix is simply its
        name in the external system. [5XHomalgToCAS[105X chooses these names.[133X
  
  [30X    [33X[0;6YThe  package  [5XLocalizeRingForHomalg[105X defines the classes of local(ized)
        rings,  local ring elements, and local matrices. A [5Xhomalg[105X local matrix
        contains  a  [5Xhomalg[105X matrix as a numerator and an element of the global
        ring as a denominator.[133X
  
  [33X[0;0YThe  matrix  operations are divided into two classes called [21XTools[121X and [21XBasic[121X.
  The   [21XTools[121X   operations   include  addition,  subtraction,  multiplication,
  extracting  certain  rows or columns, stacking, and augmenting matrices (-->
  Appendix [14XB[114X). The [21XBasic[121X operations include the two basic operations in linear
  algebra   needed   to   solve  an  inhomogeneous  linear  system  [22XXA=B[122X  with
  coefficients in a not necessarily commutative ring [22XR[122X (--> Appendix [14XA[114X):[133X
  
  [30X    [33X[0;6YEffectively  reducing  [22XB[122X  modulo [22XA[122X, i.e. effectively deciding if a row
        (or a set of rows) [22XB[122X lies in the [22XR[122X-span of the rows of the matrix [22XA[122X.[133X
  
  [30X    [33X[0;6YComputing  an [22XR[122X-generating set of row syzygies (=[22XR[122X-relations among the
        rows)  of  [22XA[122X, i.e. computing an [22XR[122X-generating set of the left kernel of
        [22XA[122X.  This  generating  set  is then given as the rows of a matrix [22XY[122X and
        [22XYA=0[122X.[133X
  
  [33X[0;0YThe  first  operation  is  nothing  but  deciding  the  solvability  of  the
  inhomogeneous  system  [22XXA=B[122X and if solvable to compute a particular solution
  [22XX[122X,  while  the  second is to compute an [22XR[122X-generating set for the homogeneous
  solution  space, i.e. the solution space of the homogeneous system [22XYA=0[122X. The
  above is of course also valid for the column convention.[133X
  
  
  [1X1.1-3 [33X[0;0YThe black box concept[133X[101X
  
  [33X[0;0YNow we address the following concerns: Wouldn't the idea of using algorithms
  like the Gröbnerbasis algorithm(s) as a black box (--> [14X1.1-2[114X) contradict the
  following facts?[133X
  
  [30X    [33X[0;6YIt  is  known  that an efficient Gröbnerbasis algorithm depends on the
        ring  [22XR[122X  under  consideration.  For  example the implementation of the
        algorithm depends on the ground ring (or field) [22Xk[122X.[133X
  
  [30X    [33X[0;6YOften  enough  highly  specialized implementations are used to address
        specific  types  of  linear systems of equations (occuring in specific
        homological  problems)  in  order  to increase the speed or reduce the
        space needed for the computations.[133X
  
  [33X[0;0YThe following should clarify the above concerns.[133X
  
  [30X    [33X[0;6YSince  each  ring  comes  with  its  own black box, the first point is
        automatically resolved.[133X
  
  [30X    [33X[0;6YAllow  the  black  box  coming with each ring to contain the different
        available  implementations  and  make  them  accessible  to [5Xhomalg[105X via
        standarized  names, independent of the computer algebra system used to
        perform computations.[133X
  
  
  [1X1.2 [33X[0;0YThis manual[133X[101X
  
  [33X[0;0YChapter [14X2[114X describes the installation of this package. The remaining chapters
  are  each  devoted  to one of the [5XMatricesForHomalg[105X objects (--> [14X1.1-1[114X) with
  its constructors, properties, attributes, and operations.[133X
  
