#### D.13.2.8 coneViaInequalities

Procedure from library `gfan.lib` (see gfan_lib).

Usage:
  coneViaInequalities(IE); intmat IE
  coneViaInequalities(IE,E); intmat IE, intmat E
  coneViaInequalities(IE,E,flags); intmat IE, intmat E, int flags

Return:
  cone

Purpose:
  cone consisting of all points x, such that IE*x >= 0 in each component and (if stated) E*x = 0;
  inequalities and (if stated) equations will be transformed, getting rid of redundancies;
  flags may range between 0,..,3 defining an upper and lower bit (0=0*2+0, 1=0*2+1, 2=1*2+0, 3=1*2+1),
  if higher bit is 1, then program assumes each inequality yields a facet, if lower bit is 1, then program assumes the kernel of E is the span of the cone, if either bit is 0, then program computes the information itself.

Example:

    LIB "gfan.lib";
    // Let's define a cone in R^3 given by the following inequalities:
    intmat IE[6][3]=
    1,3,5,
    1,5,3,
    0,1,-1,
    0,1,1,
    1,0,0,
    -1,0,0;
    cone c=coneViaInequalities(IE);
    c;
    ==> AMBIENT_DIM
    ==> 3
    ==> INEQUALITIES
    ==> 0,1,-1,
    ==> 0,1, 1,
    ==> 1,3, 5,
    ==> 1,5, 3
    ==> LINEAR_SPAN
    ==> -1,0,0
    ==>
    // Note that the last two inequalities yield x1 = 0, hence also possible:
    intmat IE[4][3]=
    ==> // ** redefining IE (intmat IE[4][3]=) ./examples/coneViaInequalities.sin\
       g:13
    0,1,-1,
    0,1,1;
    intmat E[1][3]=
    1,0,0;
    cone c=coneViaInequalities(IE,E);
    ==> // ** redefining c (cone c=coneViaInequalities(IE,E);) ./examples/coneVia\
       Inequalities.sing:18
    c;
    ==> AMBIENT_DIM
    ==> 3
    ==> INEQUALITIES
    ==> 0,1,-1,
    ==> 0,1, 1
    ==> LINEAR_SPAN
    ==> 1,0,0
    ==>
    // each inequalities gives rise to a facet
    intmat IE[2][3]=
    ==> // ** redefining IE (intmat IE[2][3]=) ./examples/coneViaInequalities.sin\
       g:21
    0,1,-1,
    0,1,1;
    intmat E[1][3]=
    ==> // ** redefining E (intmat E[1][3]=) ./examples/coneViaInequalities.sing:\
       24
    1,0,0;
    cone c=coneViaInequalities(IE,E,1);
    ==> // ** redefining c (cone c=coneViaInequalities(IE,E,1);) ./examples/coneV\
       iaInequalities.sing:26
    c;
    ==> AMBIENT_DIM
    ==> 3
    ==> INEQUALITIES
    ==> 0,1,-1,
    ==> 0,1, 1
    ==> LINEAR_SPAN
    ==> 1,0,0
    ==>
    // and the kernel of E is the span of the cone
    intmat IE[2][3]=
    ==> // ** redefining IE (intmat IE[2][3]=) ./examples/coneViaInequalities.sin\
       g:29
    0,1,-1,
    0,1,1;
    intmat E[1][3]=
    ==> // ** redefining E (intmat E[1][3]=) ./examples/coneViaInequalities.sing:\
       32
    1,0,0;
    cone c=coneViaInequalities(IE,E,3);
    ==> // ** redefining c (cone c=coneViaInequalities(IE,E,3);) ./examples/coneV\
       iaInequalities.sing:34
    c;
    ==> AMBIENT_DIM
    ==> 3
    ==> FACETS
    ==> 0,1,-1,
    ==> 0,1, 1
    ==> LINEAR_SPAN
    ==> 1,0,0
    ==>

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