NZMATH  1.2.0
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nzmath.poly.groebner Namespace Reference

Functions

def s_polynomial (f, g, order)
 
def step_reduce (reducee, reducer, order)
 
def reduce_closure (f, reducers, order)
 
def buchberger (generating, order)
 
def reduce_groebner (gbasis, order)
 
def normal_strategy (generating, order)
 

Detailed Description

Groebner basis

Function Documentation

◆ buchberger()

def nzmath.poly.groebner.buchberger (   generating,
  order 
)
Return a Groebner basis of the ideal generated by given generating
set of polynomials with respect to the order.

Be careful, this implementation is very naive.

Definition at line 48 of file groebner.py.

References nzmath.poly.groebner.reduce_closure(), and nzmath.poly.groebner.s_polynomial().

◆ normal_strategy()

def nzmath.poly.groebner.normal_strategy (   generating,
  order 
)
Return a Groebner basis of the ideal generated by given generating
set of polynomials with respect to the order.

This function uses the 'normal strategy'.

Definition at line 99 of file groebner.py.

References nzmath.bigrange.range(), nzmath.poly.groebner.reduce_closure(), and nzmath.poly.groebner.s_polynomial().

◆ reduce_closure()

def nzmath.poly.groebner.reduce_closure (   f,
  reducers,
  order 
)
Return normalized form of f with respect to reducer, a set of
polynomials, and order.

Definition at line 35 of file groebner.py.

References nzmath.poly.groebner.step_reduce().

Referenced by nzmath.poly.groebner.buchberger(), and nzmath.poly.groebner.normal_strategy().

◆ reduce_groebner()

def nzmath.poly.groebner.reduce_groebner (   gbasis,
  order 
)
Return the reduced Groebner basis constructed from a Groebner
basis.

1) lb(f) divides lb(g) => g is not in reduced Groebner basis
2) monic

Definition at line 75 of file groebner.py.

◆ s_polynomial()

def nzmath.poly.groebner.s_polynomial (   f,
  g,
  order 
)
Return S-polynomial of f and g with respect to the order.

S(f, g) = (lc(g)*T/lb(f))*f - (lc(f)*T/lb(g))*g,
where T = lcm(lb(f), lb(g)).

Definition at line 10 of file groebner.py.

Referenced by nzmath.poly.groebner.buchberger(), and nzmath.poly.groebner.normal_strategy().

◆ step_reduce()

def nzmath.poly.groebner.step_reduce (   reducee,
  reducer,
  order 
)
Return the reduced polynomial of reducee by reducer.  That is, if
one of reducee's bases is divisible by the leading base of reducer
with respect to the order, the returned polynomial is the result
of canceling out the term.

Definition at line 22 of file groebner.py.

Referenced by nzmath.poly.groebner.reduce_closure().