next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
AlgebraicSplines :: courantFunctions

courantFunctions -- returns the Courant functions of a simplicial complex

Synopsis

Description

This method returns a matrix with as many rows as facets and as many columns as vertices. Column i of the matrix is the Courant function corresponding to vertex Vi. This is the piecewise linear function which takes the value 1 at the vertex Vi and 0 at all other vertices.

i1 : V={{0,0},{0,1},{-1,-1},{1,0}};
i2 : F={{0,1,2},{0,2,3},{0,1,3}};
i3 : courantFunctions(V,F)

o3 = | 2t_0-t_1+t_2  -t_0+t_1 -t_0 0       |
     | -t_0+2t_1+t_2 0        -t_1 t_0-t_1 |
     | -t_0-t_1+t_2  t_1      0    t_0     |

                            3                      4
o3 : Matrix (QQ[t , t , t ])  <--- (QQ[t , t , t ])
                 0   1   2              0   1   2

If the option Homogenize=>false is given, the Courant function corresponding to a cone vertex (if there is one) will be discarded.

i4 : S=QQ[x,y];
i5 : courantFunctions(V,F,Homogenize=>false,BaseRing=>S)

o5 = | -x+y -x 0   |
     | 0    -y x-y |
     | y    0  x   |

             3       3
o5 : Matrix S  <--- S

The Courant functions are used to construct the Stanley Reisner ring of a simplicial complex. See stanleyReisner.

See also

Ways to use courantFunctions :