normaliz(Matrix,String) -- calls Normaliz

Synopsis

Function: normaliz
Usage:

normaliz(mat,nmzType)
Inputs:
- a matrix, mat the input matrix for Normaliz
- a string, nmzType the type
Optional inputs:
- allComputations => ..., default value false
- grading => ..., default value {}
Outputs:
- an instance of the type RationalCone, generators of the integral closure and optional output from Normaliz

Description

This function applies Normaliz to the parameter mat in the type set by nmzType. The function returns an object of type RationalCone defined by the file with suffix gen , if computed, and possibly some additional information. The type determines how the rows of the matrix are interpreted, see also normaliz(List) and writeNmzData.

By default, the cone returned contains only the content of the output file .gen, under the key "gen", i.e. the generators that have been computed, line by line, and the content of the output file .inv, under the key "inv".

i1 : setNmzOption("allf",true);

i2 : eq=matrix {{1, 1, 1, -1, -1, -1,  0,  0,  0}, {1, 1, 1,  0,  0,  0, -1, -1, -1}, {0, 1, 1, -1,  0,  0, -1,  0,  0}, {1, 0, 1,  0, -1,  0,  0, -1,  0}, {1, 1, 0,  0,  0, -1,  0,  0, -1}, {0, 1, 1,  0, -1,  0,  0,  0, -1}, {1, 1, 0,  0, -1,  0, -1,  0,  0}};

              7       9
o2 : Matrix ZZ  <-- ZZ

i3 : rc=normaliz(eq,"equations");

i4 : rc#"gen"

o4 = | 0 2 1 2 1 0 1 0 2 |
     | 1 0 2 2 1 0 0 2 1 |
     | 1 1 1 1 1 1 1 1 1 |
     | 1 2 0 0 1 2 2 0 1 |
     | 2 0 1 0 1 2 1 2 0 |

              5       9
o4 : Matrix ZZ  <-- ZZ

To obtain all the information written by Normaliz set the option allComputations to true (to decide which information shall be written by Normaliz use the options for Normaliz, see setNmzOption). Then the method returns an object of type RationalCone whose keys are the suffixes of all the output files written, with value the content of the corresponding output file, which is an matrix whose rows contain the data computed, except for the suffix inv, for which the type is a HashTable (see also getNumInvs). It can also be used with the option grading.

i5 : arc=normaliz(allComputations=>true,eq,"equations");

i6 : arc#"gen"

o6 = | 0 2 1 2 1 0 1 0 2 |
     | 1 0 2 2 1 0 0 2 1 |
     | 1 1 1 1 1 1 1 1 1 |
     | 1 2 0 0 1 2 2 0 1 |
     | 2 0 1 0 1 2 1 2 0 |

              5       9
o6 : Matrix ZZ  <-- ZZ

i7 : arc#"ext"

o7 = | 0 2 1 2 1 0 1 0 2 |
     | 1 0 2 2 1 0 0 2 1 |
     | 1 2 0 0 1 2 2 0 1 |
     | 2 0 1 0 1 2 1 2 0 |

              4       9
o7 : Matrix ZZ  <-- ZZ

i8 : arc#"inv"

o8 = HashTable{"" => (1, 1, 1)                         }
               "class group" => (1, 2, 2)
               "degree 1 elements" => 5
               "dim max subspace" => 0
               "embedding dim" => 9
               "external index" => 1
               "graded" => true
               "grading denom" => 1
               "grading" => (0, 0, 0, 0, 1, 0, 0, 0, 0)
               "hilbert basis elements" => 5
               "hilbert quasipolynomial denom" => 1
               "hilbert series denom" => (1, 1, 1)
               "hilbert series num" => (1, 2, 1)
               "inhomogeneous" => false
               "multiplicity denom" => 1
               "multiplicity" => 4
               "number extreme rays" => 4
               "number support hyperplanes" => 4
               "rank" => 3
               "size triangulation" => 2
               "sum dets" => 4

o8 : HashTable

Ways to use this method: