QEtaLinearReduction(C, F, grade, coefficient)ΒΆ
qetalinred.spad line 81 [edit on github]
- F: QEtaAlgebra C 
- grade: F -> Integer 
- coefficient: (F, Integer) -> C 
QEtaLinearReduction
- greaterGrade?: (F, F) -> Boolean
- greaterGrade?(x, y)returns grade(- x)>grade(- y).
- linearTransform: (List F, Matrix C -> Matrix C) -> List F
- linearTransform(bas,trf)returns linearTransform(- bas,- trf,0).
- linearTransform: (List F, Matrix C -> Matrix C, Integer) -> List F
- linearTransform(bas,trf,mn)considers the coefficients of the elements (highest grade down to- mn) as the rows of a matrix, attaches a unit matrix to the right-hand side and applies- trfto that matrix. The recorded transformations steps on this matrix are then performed on- bas. By using linearTransform(- bas,rowEchelon) one can achieve a rowEcholon reduction of- bas.
- reduce: (F, List F) -> F
- reduce(u, bas)returns an element- vsuch that not(reducible?(- u,- b)) for any- bin bas.
- reducible?: (F, F) -> Boolean
- reducible?(u, b)returns- trueiff n=grade(- b) and not(zero?(coefficient(- u,- n))) and there exists- fin- Csuch that for v=u-- f*b, c=coefficient(- u,- n), and d=coefficient(- v,- n) zero?- dholds or euclideanSize(- c)>euclideanSize(- d).
- rowEchelon: List F -> List F
- rowEchelon(bas)returns linearTransform(- bas,rowEchelon$- M,0) where- M- ==>Matrix(- C).
- tailReduce: (F, List F) -> F
- tailReduce(u,bas)returns an element- vsuch that grade(- u)=grade(- v) and not(reducible?(- u,- b)) for any- bin- basand n<grade(- u).
- topReduce: (F, List F) -> F
- topReduce(u,bas)returns an element- vsuch that not(grade(- u)=grade(- b) and reducible?(- u,- b)) for any- bin- bas.