| Safe Haskell | Safe-Infered |
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Algebra.Algorithms.Groebner
- divModPolynomial :: (IsMonomialOrder order, IsPolynomial r n, Field r) => OrderedPolynomial r order n -> [OrderedPolynomial r order n] -> ([(OrderedPolynomial r order n, OrderedPolynomial r order n)], OrderedPolynomial r order n)
- divPolynomial :: (IsPolynomial r n, Field r, IsMonomialOrder order) => OrderedPolynomial r order n -> [OrderedPolynomial r order n] -> [(OrderedPolynomial r order n, OrderedPolynomial r order n)]
- modPolynomial :: (IsPolynomial r n, Field r, IsMonomialOrder order) => OrderedPolynomial r order n -> [OrderedPolynomial r order n] -> OrderedPolynomial r order n
- calcGroebnerBasis :: (Field k, IsPolynomial k n, IsMonomialOrder order) => Ideal (OrderedPolynomial k order n) -> [OrderedPolynomial k order n]
- calcGroebnerBasisWith :: (Field k, IsPolynomial k n, IsMonomialOrder order, IsMonomialOrder order') => order -> Ideal (OrderedPolynomial k order' n) -> [OrderedPolynomial k order n]
- simpleBuchberger :: (Field k, IsPolynomial k n, IsMonomialOrder order) => Ideal (OrderedPolynomial k order n) -> [OrderedPolynomial k order n]
- reduceMinimalGroebnerBasis :: (Field k, IsPolynomial k n, IsMonomialOrder order) => [OrderedPolynomial k order n] -> [OrderedPolynomial k order n]
- minimizeGroebnerBasis :: (Field k, IsPolynomial k n, IsMonomialOrder order) => [OrderedPolynomial k order n] -> [OrderedPolynomial k order n]
- isIdealMember :: (IsPolynomial k n, Field k, IsMonomialOrder o) => OrderedPolynomial k o n -> Ideal (OrderedPolynomial k o n) -> Bool
- intersection :: forall r k n ord. (IsMonomialOrder ord, Field r, IsPolynomial r k, IsPolynomial r n, IsPolynomial r (k :+: n)) => Vector (Ideal (OrderedPolynomial r ord n)) k -> Ideal (OrderedPolynomial r Lex n)
- thEliminationIdeal :: (IsMonomialOrder ord, Field k, IsPolynomial k m, IsPolynomial k (m :-: n), (n :<<= m) ~ True) => SNat n -> Ideal (OrderedPolynomial k ord m) -> Ideal (OrderedPolynomial k Lex (m :-: n))
- quotIdeal :: forall k ord n. (IsPolynomial k n, Field k, IsMonomialOrder ord) => Ideal (OrderedPolynomial k ord n) -> Ideal (OrderedPolynomial k ord n) -> Ideal (OrderedPolynomial k Lex n)
- quotByPrincipalIdeal :: (Field k, IsPolynomial k n, IsMonomialOrder ord) => Ideal (OrderedPolynomial k ord n) -> OrderedPolynomial k ord n -> Ideal (OrderedPolynomial k Lex n)
- saturationIdeal :: forall k ord n. (IsPolynomial k n, Field k, IsMonomialOrder ord) => Ideal (OrderedPolynomial k ord n) -> Ideal (OrderedPolynomial k ord n) -> Ideal (OrderedPolynomial k Lex n)
- saturationByPrincipalIdeal :: (Field k, IsPolynomial k n, IsMonomialOrder ord) => Ideal (OrderedPolynomial k ord n) -> OrderedPolynomial k ord n -> Ideal (OrderedPolynomial k Lex n)
Polynomial division
divModPolynomial :: (IsMonomialOrder order, IsPolynomial r n, Field r) => OrderedPolynomial r order n -> [OrderedPolynomial r order n] -> ([(OrderedPolynomial r order n, OrderedPolynomial r order n)], OrderedPolynomial r order n)
Calculate a polynomial quotient and remainder w.r.t. second argument.
divPolynomial :: (IsPolynomial r n, Field r, IsMonomialOrder order) => OrderedPolynomial r order n -> [OrderedPolynomial r order n] -> [(OrderedPolynomial r order n, OrderedPolynomial r order n)]
A Quotient of given polynomial w.r.t. the second argument.
modPolynomial :: (IsPolynomial r n, Field r, IsMonomialOrder order) => OrderedPolynomial r order n -> [OrderedPolynomial r order n] -> OrderedPolynomial r order n
Remainder of given polynomial w.r.t. the second argument.
Groebner basis
calcGroebnerBasis :: (Field k, IsPolynomial k n, IsMonomialOrder order) => Ideal (OrderedPolynomial k order n) -> [OrderedPolynomial k order n]
Caliculating reduced Groebner basis of the given ideal w.r.t. the graded reversed lexicographic order.
calcGroebnerBasisWith :: (Field k, IsPolynomial k n, IsMonomialOrder order, IsMonomialOrder order') => order -> Ideal (OrderedPolynomial k order' n) -> [OrderedPolynomial k order n]
Caliculating reduced Groebner basis of the given ideal w.r.t. the specified monomial order.
simpleBuchberger :: (Field k, IsPolynomial k n, IsMonomialOrder order) => Ideal (OrderedPolynomial k order n) -> [OrderedPolynomial k order n]
Apply Buchberger's algorithm and calculate Groebner basis for the given ideal.
reduceMinimalGroebnerBasis :: (Field k, IsPolynomial k n, IsMonomialOrder order) => [OrderedPolynomial k order n] -> [OrderedPolynomial k order n]
Reduce minimum Groebner basis into reduced Groebner basis.
minimizeGroebnerBasis :: (Field k, IsPolynomial k n, IsMonomialOrder order) => [OrderedPolynomial k order n] -> [OrderedPolynomial k order n]
Minimize the given groebner basis.
Ideal operations
isIdealMember :: (IsPolynomial k n, Field k, IsMonomialOrder o) => OrderedPolynomial k o n -> Ideal (OrderedPolynomial k o n) -> Bool
Test if the given polynomial is the member of the ideal.
intersection :: forall r k n ord. (IsMonomialOrder ord, Field r, IsPolynomial r k, IsPolynomial r n, IsPolynomial r (k :+: n)) => Vector (Ideal (OrderedPolynomial r ord n)) k -> Ideal (OrderedPolynomial r Lex n)
An intersection ideal of given ideals.
thEliminationIdeal :: (IsMonomialOrder ord, Field k, IsPolynomial k m, IsPolynomial k (m :-: n), (n :<<= m) ~ True) => SNat n -> Ideal (OrderedPolynomial k ord m) -> Ideal (OrderedPolynomial k Lex (m :-: n))
Calculate n-th elimination ideal.
quotIdeal :: forall k ord n. (IsPolynomial k n, Field k, IsMonomialOrder ord) => Ideal (OrderedPolynomial k ord n) -> Ideal (OrderedPolynomial k ord n) -> Ideal (OrderedPolynomial k Lex n)
Ideal quotient by the given ideal.
quotByPrincipalIdeal :: (Field k, IsPolynomial k n, IsMonomialOrder ord) => Ideal (OrderedPolynomial k ord n) -> OrderedPolynomial k ord n -> Ideal (OrderedPolynomial k Lex n)
Ideal quotient by a principal ideals.
saturationIdeal :: forall k ord n. (IsPolynomial k n, Field k, IsMonomialOrder ord) => Ideal (OrderedPolynomial k ord n) -> Ideal (OrderedPolynomial k ord n) -> Ideal (OrderedPolynomial k Lex n)
Saturation ideal
saturationByPrincipalIdeal :: (Field k, IsPolynomial k n, IsMonomialOrder ord) => Ideal (OrderedPolynomial k ord n) -> OrderedPolynomial k ord n -> Ideal (OrderedPolynomial k Lex n)
Saturation by a principal ideal.