Singular Manual: Matrix orderings

| ### B.2.6 Matrix orderings Let 13#13be an invertible 475#475-matrix with integer coefficients and 590#590 the rows of 13#13. The M-ordering < is defined as follows: 591#591 and 592#592. Thus, 593#593 if and only if 594#594 is smaller than 595#595with 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): 596#596 lp: 597#597 dp: 598#598 Dp: 599#599 596#596 wp(1,2,3): 600#600 Wp(1,2,3): 601#601 596#596 ls: 602#602 ds: 603#603 Ds: 604#604 596#596 ws(1,2,3): 605#605 Ws(1,2,3): 606#606 Product orderings (see next section) represented by a matrix: 596#596 (dp(3), wp(1,2,3)): 607#607 596#596 (Dp(3), ds(3)): 608#608 Orderings with extra weight vector (see below) represented by a matrix: 596#596 (dp(3), a(1,2,3),dp(3)): 609#609 596#596 (a(1,2,3,4,5),Dp(3), ds(3)): 610#610 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 considered as a field
    ==> // 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 236#236entries, 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.

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