Top
Back: Module orderings
Forward: Product orderings
FastBack: Representation of mathematical objects
FastForward: Mathematical background
Up: Monomial orderings
Top: Singular Manual
Contents: Table of Contents
Index: Index
About: About this document

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.


Top Back: Module orderings Forward: Product orderings FastBack: Representation of mathematical objects FastForward: Mathematical background Up: Monomial orderings Top: Singular Manual Contents: Table of Contents Index: Index About: About this document
            User manual for Singular version 4.4.1, 2025, generated by texi2html.