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Normaliz :: normaliz

normaliz

Description

This function applies Normaliz to the input data, which can be a matrix specifying a cone and an integer indicating the type for Normaliz or a list consisting of pairs of such a matrix and an integer. The function returns an object of type RationalCone. The type determines how the rows of the matrix are interpreted, see also writeNmzData, and decides what the program will do:

  • type 0: Computes the Hilbert basis of the rational cone generated by the rows with respect to the ambient lattice ℤn;
  • type 1: The same as 0, but with respect to the sublattice of ℤn generated by the rows;
  • type 2: Computes the integral points in the polytope spanned by the rows and its Ehrhart semigroup (the semigroup determined by the polytope);
  • type 3: Computes the integral closure of the Rees algebra of the ideal generated by the monomials with exponent vectors the rows;
  • type 4: Computes the Hilbert basis of the rational cone in ℝm given by the system of homogeneous inequalities mat x ≥ 0;
  • type 5: Computes the Hilbert basis of the rational cone given by the nonnegative solutions of the homogeneous system mat x = 0.
  • type 6: Computes the Hilbert basis of the rational cone given by the nonnegative solutions of the system of congruences defined by the rows as follows: Each row (xi1,…,xin,c) represents a congruence xi1z1+…+xinzn ≡ 0 modc.
  • type 10: Computes the monoid as a quotient of ℤ+n modulo a system of congruences (in the semigroup sense) defined by the rows of the input matrix.

It is possible to combine the types 4,5 and 6, see normaliz(List).

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,5)

o3 = RationalCone{gen => | 1 2 0 0 1 2 2 0 1 |                                        }
                         | 0 2 1 2 1 0 1 0 2 |
                         | 1 1 1 1 1 1 1 1 1 |
                         | 2 0 1 0 1 2 1 2 0 |
                         | 1 0 2 2 1 0 0 2 1 |
                  inv => HashTable{height 1 elements => 5                            }
                                   hilbert basis elements => 5
                                   homogeneous => true
                                   homogeneous weights => (0, 0, 0, 0, 1, 0, 0, 0, 0)
                                   index => 2
                                   multiplicity => 4
                                   number extreme rays => 4
                                   number support hyperplanes => 4
                                   rank => 3

o3 : RationalCone

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).

i4 : arc=normaliz(allComputations=>true,eq,5);
i5 : arc#"gen"

o5 = | 1 2 0 0 1 2 2 0 1 |
     | 0 2 1 2 1 0 1 0 2 |
     | 1 1 1 1 1 1 1 1 1 |
     | 2 0 1 0 1 2 1 2 0 |
     | 1 0 2 2 1 0 0 2 1 |

              5        9
o5 : Matrix ZZ  <--- ZZ
i6 : arc#"ext"

o6 = | 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
o6 : Matrix ZZ  <--- ZZ
i7 : arc#"inv"

o7 = HashTable{height 1 elements => 5                            }
               hilbert basis elements => 5
               homogeneous => true
               homogeneous weights => (0, 0, 0, 0, 1, 0, 0, 0, 0)
               index => 2
               multiplicity => 4
               number extreme rays => 4
               number support hyperplanes => 4
               rank => 3

o7 : HashTable

Ways to use normaliz :