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remainder -- matrix remainder

Synopsis

Usage:
(q,r) = remainder(f,g)
Inputs:
- f, a matrix
- g, a Groebner basis or a matrix, with the same target as f
Outputs:
- r, the remainder of f upon division by g

Description

This operation is the same as Matrix % GroebnerBasis.

The equation g*q+r == f will hold, where q is the map provided by quotient. The source of f should be a free module.

i1 : R = ZZ[x,y]

o1 = R

o1 : PolynomialRing

i2 : f = random(R^2,R^{2:-1})

o2 = | y     -9x-9y |
     | 5x+2y 7y     |

             2       2
o2 : Matrix R  <--- R

i3 : g = vars R ++ vars R

o3 = | x y 0 0 |
     | 0 0 x y |

             2       4
o3 : Matrix R  <--- R

i4 : remainder(f,g)

o4 = 0

             2       2
o4 : Matrix R  <--- R

i5 : f = f + map(target f, source f, id_(R^2))

o5 = | y+1   -9x-9y |
     | 5x+2y 7y+1   |

             2       2
o5 : Matrix R  <--- R

i6 : remainder(f,g)

o6 = | 1 0 |
     | 0 1 |

             2       2
o6 : Matrix R  <--- R

See also

quotientRemainder -- matrix quotient and remainder
remainder' -- matrix quotient and remainder (opposite)

Ways to use `remainder` :

remainder(Matrix,GroebnerBasis)
remainder(Matrix,Matrix)