NCReductionTwoSided -- Reduces the entries of an Matrix with respect to an ideal

Synopsis

Usage:

L = NCReductionTwoSided(M,I)
Inputs:
- M, a matrix,
- I, an ideal,
Outputs:
- L, a matrix,

Description

This command reduces the entries of a RingElement or Matrix with respect to an Ideal or a single row matrix. A Groebner basis is not computed for the input ideal. If one wishes to use this function for normal forms, one must first call NCGB and pass the result to this function.

i1 : A = QQ<|x,y,z|>

o1 = A

o1 : FreeAlgebra

i2 : f = y*z + z*y - x^2

        2
o2 = - x  + y*z + z*y

o2 : A

i3 : g = x*z + z*x - y^2

            2
o3 = x*z - y  + z*x

o3 : A

i4 : h = z^2 - x*y - y*x

                    2
o4 = - x*y - y*x + z

o4 : A

i5 : I = ideal {f,g,h}

               2                     2                       2
o5 = ideal (- x  + y*z + z*y, x*z - y  + z*x, - x*y - y*x + z )

o5 : Ideal of A

i6 : Igb = NCGB(I,10)
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o6 = | x2-yz-zy xz-y2+zx xy+yx-z2 yz2-z2y y2z-zy2 |

             1      5
o6 : Matrix A  <-- A

i7 : NCReductionTwoSided(x^4,I)

                  2       2
o7 = y*z*y*z + y*z y + z*y z + z*y*z*y

o7 : A

i8 : NCReductionTwoSided(x^4,Igb)

                           2 2
o8 = y*z*y*z + z*y*z*y + 2z y

o8 : A

Ways to use NCReductionTwoSided :

NCReductionTwoSided(Matrix,Ideal)
NCReductionTwoSided(Matrix,List)
NCReductionTwoSided(Matrix,Matrix)
NCReductionTwoSided(RingElement,Ideal)
NCReductionTwoSided(RingElement,List)
NCReductionTwoSided(RingElement,Matrix)

For the programmer

The object NCReductionTwoSided is a method function.