remainder' -- matrix quotient and remainder (opposite)

Synopsis

Usage:

r = remainder'(f,g)
Inputs:
- f, a matrix
- g, a Gröbner basis or a matrix, with the same source as f
Outputs:
- r, the remainder of f upon (opposite) division by g

Description

The equation q*g+r == f will hold, where q is the map provided by quotient'. The sources and targets of the maps should be free modules. This function is obtained from remainder by transposing the inputs and outputs.

i1 : R = ZZ[x,y]

o1 = R

o1 : PolynomialRing

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

o2 = {-1} | 8x+y  8x+3y |
     {-1} | 3x+7y 3x+7y |

             2       2
o2 : Matrix R  <--- R

i3 : g = transpose (vars R ++ vars R)

o3 = {-1} | x 0 |
     {-1} | y 0 |
     {-1} | 0 x |
     {-1} | 0 y |

             4       2
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 = {-1} | 8x+y+1 8x+3y   |
     {-1} | 3x+7y  3x+7y+1 |

             2       2
o5 : Matrix R  <--- R

i6 : remainder'(f,g)

o6 = {-1} | 1 0 |
     {-1} | 0 1 |

             2       2
o6 : Matrix R  <--- R

Code

function remainder': source code not available

Ways to use `remainder'` :

"remainder'(Matrix,Matrix)"

For the programmer

The object remainder' is a method function.