Singular Manual: Matrix orderings

| ### B.2.6 Matrix orderings Let 13#13be an invertible 473#473-matrix with integer coefficients and 588#588 the rows of 13#13. The M-ordering < is defined as follows: 589#589 and 590#590. Thus, 591#591 if and only if 592#592 is smaller than 593#593with respect to the lexicographical ordering. The following matrices represent (for 3 variables) the global and local orderings defined above (note that the matrix is not uniquely determined by the ordering): 594#594 lp: 595#595 dp: 596#596 Dp: 597#597 594#594 wp(1,2,3): 598#598 Wp(1,2,3): 599#599 594#594 ls: 600#600 ds: 601#601 Ds: 602#602 594#594 ws(1,2,3): 603#603 Ws(1,2,3): 604#604 Product orderings (see next section) represented by a matrix: 594#594 (dp(3), wp(1,2,3)): 605#605 594#594 (Dp(3), ds(3)): 606#606 Orderings with extra weight vector (see below) represented by a matrix: 594#594 (dp(3), a(1,2,3),dp(3)): 607#607 594#594 (a(1,2,3,4,5),Dp(3), ds(3)): 608#608 Example:
      ring r = 0, (x,y,z), M(1, 0, 0,   0, 1, 0,   0, 0, 1);
which may also be written as:
      intmat m[3][3]=1, 0, 0, 0, 1, 0, 0, 0, 1;
      m;
    ==> 1,0,0,
    ==> 0,1,0,
    ==> 0,0,1
      ring r = 0, (x,y,z), M(m);
      r;
    ==> // coefficients: QQ
    ==> // number of vars : 3
    ==> //        block   1 : ordering M
    ==> //                  : names    x y z
    ==> //                  : weights  1 0 0
    ==> //                  : weights  0 1 0
    ==> //                  : weights  0 0 1
    ==> //        block   2 : ordering C
If the ring has 17#17variables and the matrix does not contain 234#234entries, an error message is given. WARNING: SINGULAR does not check whether the matrix has full rank. In such a case some computations might not terminate, others may not give a sensible result. Having these matrix orderings SINGULAR can compute standard bases for any monomial ordering which is compatible with the natural semigroup structure. In practice the global and local orderings together with block orderings should be sufficient in most cases. These orderings are faster than the corresponding matrix orderings, since evaluating a matrix product is time consuming.

User manual for Singular version 4.3.2, 2023, generated by texi2html.