Struct FractionFieldImplBase

pub struct FractionFieldImplBase<R>
where
    R: RingStore,
    R::Type: Domain,
{ /* private fields */ }

Availability

This API is marked as unstable and is only available when the unstable-enable crate feature is enabled. This comes with no stability guarantees, and could be changed or removed at any time.

Trait Implementations

impl<R, S> CanHomFrom<FractionFieldImplBase<S>> for FractionFieldImplBase<R>

where R: RingStore, S: RingStore, R::Type: Domain + CanHomFrom<S::Type>, S::Type: Domain,

type Homomorphism = <<R as RingStore>::Type as CanHomFrom<<S as RingStore>::Type>>::Homomorphism

Data required to compute the action of the canonical homomorphism on ring elements.

fn has_canonical_hom( &self, from: &FractionFieldImplBase<S>, ) -> Option<Self::Homomorphism>

If there is a canonical homomorphism from -> self, returns Some(data), where data is additional data that can be used to compute the action of the homomorphism on ring elements. Otherwise, None is returned.

fn map_in( &self, from: &FractionFieldImplBase<S>, el: <FractionFieldImplBase<S> as RingBase>::Element, hom: &Self::Homomorphism, ) -> Self::Element

fn map_in_ref( &self, from: &S, el: &S::Element, hom: &Self::Homomorphism, ) -> Self::Element

fn mul_assign_map_in( &self, from: &S, lhs: &mut Self::Element, rhs: S::Element, hom: &Self::Homomorphism, )

fn mul_assign_map_in_ref( &self, from: &S, lhs: &mut Self::Element, rhs: &S::Element, hom: &Self::Homomorphism, )

impl<R, I> CanHomFrom<RationalFieldBase<I>> for FractionFieldImplBase<R>

where R: RingStore, I: RingStore, R::Type: Domain + CanHomFrom<I::Type>, I::Type: IntegerRing,

type Homomorphism = <<R as RingStore>::Type as CanHomFrom<<I as RingStore>::Type>>::Homomorphism

Data required to compute the action of the canonical homomorphism on ring elements.

fn has_canonical_hom( &self, from: &RationalFieldBase<I>, ) -> Option<Self::Homomorphism>

If there is a canonical homomorphism from -> self, returns Some(data), where data is additional data that can be used to compute the action of the homomorphism on ring elements. Otherwise, None is returned.

fn map_in( &self, from: &RationalFieldBase<I>, el: <RationalFieldBase<I> as RingBase>::Element, hom: &Self::Homomorphism, ) -> Self::Element

fn map_in_ref( &self, from: &S, el: &S::Element, hom: &Self::Homomorphism, ) -> Self::Element

fn mul_assign_map_in( &self, from: &S, lhs: &mut Self::Element, rhs: S::Element, hom: &Self::Homomorphism, )

fn mul_assign_map_in_ref( &self, from: &S, lhs: &mut Self::Element, rhs: &S::Element, hom: &Self::Homomorphism, )

impl<R, S> CanIsoFromTo<FractionFieldImplBase<S>> for FractionFieldImplBase<R>

where R: RingStore, S: RingStore, R::Type: Domain + CanIsoFromTo<S::Type>, S::Type: Domain,

type Isomorphism = <<R as RingStore>::Type as CanIsoFromTo<<S as RingStore>::Type>>::Isomorphism

Data required to compute a preimage under the canonical homomorphism.

fn has_canonical_iso( &self, from: &FractionFieldImplBase<S>, ) -> Option<Self::Isomorphism>

If there is a canonical homomorphism from -> self, and this homomorphism is an isomorphism, returns Some(data), where data is additional data that can be used to compute preimages under the homomorphism. Otherwise, None is returned.

fn map_out( &self, from: &FractionFieldImplBase<S>, el: Self::Element, iso: &Self::Isomorphism, ) -> <FractionFieldImplBase<S> as RingBase>::Element

Computes the preimage of el under the canonical homomorphism from -> self.

impl<R, I> CanIsoFromTo<RationalFieldBase<I>> for FractionFieldImplBase<R>

where R: RingStore, I: RingStore, R::Type: Domain + CanIsoFromTo<I::Type>, I::Type: IntegerRing,

type Isomorphism = <<R as RingStore>::Type as CanIsoFromTo<<I as RingStore>::Type>>::Isomorphism

Data required to compute a preimage under the canonical homomorphism.

fn has_canonical_iso( &self, from: &RationalFieldBase<I>, ) -> Option<Self::Isomorphism>

If there is a canonical homomorphism from -> self, and this homomorphism is an isomorphism, returns Some(data), where data is additional data that can be used to compute preimages under the homomorphism. Otherwise, None is returned.

fn map_out( &self, from: &RationalFieldBase<I>, el: Self::Element, iso: &Self::Isomorphism, ) -> <RationalFieldBase<I> as RingBase>::Element

Computes the preimage of el under the canonical homomorphism from -> self.

impl<R> Clone for FractionFieldImplBase<R>

where R: RingStore + Clone, R::Type: Domain,

fn clone(&self) -> Self

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl<R> DivisibilityRing for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain,

fn checked_left_div( &self, lhs: &Self::Element, rhs: &Self::Element, ) -> Option<Self::Element>

Checks whether there is an element x such that rhs * x = lhs, and returns it if it exists.

fn is_unit(&self, x: &Self::Element) -> bool

Returns whether the given element is a unit, i.e. has an inverse.

fn balance_factor<'a, I>(&self, elements: I) -> Option<Self::Element>

where I: Iterator<Item = &'a Self::Element>, Self: 'a,

Function that computes a “balancing” factor of a sequence of ring elements. The only use of the balancing factor is to increase performance, in particular, dividing all elements in the sequence by this factor should make them “smaller” resp. cheaper to process.

type PreparedDivisorData = ()

Additional data associated to a fixed ring element that can be used to speed up division by this ring element.

fn divides_left(&self, lhs: &Self::Element, rhs: &Self::Element) -> bool

Returns whether there is an element x such that rhs * x = lhs. If you need such an element, consider using DivisibilityRing::checked_left_div().

fn divides(&self, lhs: &Self::Element, rhs: &Self::Element) -> bool

Same as DivisibilityRing::divides_left(), but requires a commutative ring.

fn checked_div( &self, lhs: &Self::Element, rhs: &Self::Element, ) -> Option<Self::Element>

Same as DivisibilityRing::checked_left_div(), but requires a commutative ring.

fn prepare_divisor(&self, _: &Self::Element) -> Self::PreparedDivisorData

“Prepares” an element of this ring for division.

fn checked_left_div_prepared( &self, lhs: &Self::Element, rhs: &Self::Element, _rhs_prep: &Self::PreparedDivisorData, ) -> Option<Self::Element>

Same as DivisibilityRing::checked_left_div() but for a prepared divisor.

fn divides_left_prepared( &self, lhs: &Self::Element, rhs: &Self::Element, _rhs_prep: &Self::PreparedDivisorData, ) -> bool

Same as DivisibilityRing::divides_left() but for a prepared divisor.

fn is_unit_prepared(&self, x: &PreparedDivisor<Self>) -> bool

Same as DivisibilityRing::is_unit() but for a prepared divisor.

fn invert(&self, el: &Self::Element) -> Option<Self::Element>

If the given element is a unit, returns its inverse, otherwise None.

impl<R> EuclideanRing for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain,

fn euclidean_deg(&self, val: &Self::Element) -> Option<usize>

Defines how “small” an element is. For details, see EuclideanRing.

fn euclidean_div_rem( &self, lhs: Self::Element, rhs: &Self::Element, ) -> (Self::Element, Self::Element)

Computes euclidean division with remainder.

fn euclidean_div( &self, lhs: Self::Element, rhs: &Self::Element, ) -> Self::Element

Computes euclidean division without remainder.

fn euclidean_rem( &self, lhs: Self::Element, rhs: &Self::Element, ) -> Self::Element

Computes only the remainder of euclidean division.

impl<R> Field for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain,

fn div(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

Computes the division lhs / rhs, where rhs != 0.

impl<R> FractionField for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain,

fn as_fraction( &self, el: Self::Element, ) -> (El<Self::BaseRing>, El<Self::BaseRing>)

Returns a, b such that the given element is a/b.

fn from_fraction( &self, num: El<Self::BaseRing>, den: El<Self::BaseRing>, ) -> Self::Element

Computes num / den.

impl<R> HashableElRing for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain + IntegerRing + HashableElRing,

We don’t have a canonical representation when the base ring is not an integer ring (even if it is a PID), since we can always multiply numerator/denominator by a unit.

fn hash<H: Hasher>(&self, el: &Self::Element, h: &mut H)

Hashes the given ring element.

impl<R> KaratsubaHint for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain,

default fn karatsuba_threshold(&self) -> usize

Define a threshold from which on KaratsubaAlgorithm will use the Karatsuba algorithm.

impl<R> PartialEq for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain,

fn eq(&self, other: &Self) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl<R> PrincipalIdealRing for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain,

fn checked_div_min( &self, lhs: &Self::Element, rhs: &Self::Element, ) -> Option<Self::Element>

Similar to DivisibilityRing::checked_left_div() this computes a “quotient” q of lhs and rhs, if it exists. However, we impose the additional constraint that this quotient be minimal, i.e. there is no q' with q' | q properly and q' * rhs = lhs.

fn extended_ideal_gen( &self, lhs: &Self::Element, rhs: &Self::Element, ) -> (Self::Element, Self::Element, Self::Element)

Computes a Bezout identity for the generator g of the ideal (lhs, rhs) as g = s * lhs + t * rhs.

fn annihilator(&self, val: &Self::Element) -> Self::Element

Returns the (w.r.t. divisibility) smallest element x such that x * val = 0.

fn create_elimination_matrix( &self, a: &Self::Element, b: &Self::Element, ) -> ([Self::Element; 4], Self::Element)

Creates a matrix A of unit determinant such that A * (a, b)^T = (d, 0). Returns (A, d).

fn ideal_gen(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

Computes a generator g of the ideal (lhs, rhs) = (g), also known as greatest common divisor.

fn lcm(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

Computes a generator of the ideal (lhs) ∩ (rhs), also known as least common multiple.

impl<R> RingBase for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain,

type Element = FractionFieldEl<R>

Type of elements of the ring

fn clone_el(&self, val: &Self::Element) -> Self::Element

fn add_assign_ref(&self, lhs: &mut Self::Element, rhs: &Self::Element)

fn add_assign(&self, lhs: &mut Self::Element, rhs: Self::Element)

fn mul_assign_ref(&self, lhs: &mut Self::Element, rhs: &Self::Element)

fn mul_assign(&self, lhs: &mut Self::Element, rhs: Self::Element)

fn negate_inplace(&self, lhs: &mut Self::Element)

fn eq_el(&self, lhs: &Self::Element, rhs: &Self::Element) -> bool

fn is_zero(&self, value: &Self::Element) -> bool

fn is_one(&self, value: &Self::Element) -> bool

fn is_neg_one(&self, value: &Self::Element) -> bool

fn is_approximate(&self) -> bool

Returns whether this ring computes with approximations to elements. This would usually be the case for rings that are based on f32 or f64, to represent real or complex numbers.

fn is_commutative(&self) -> bool

Returns whether the ring is commutative, i.e. a * b = b * a for all elements a, b. Note that addition is assumed to be always commutative.

fn is_noetherian(&self) -> bool

Returns whether the ring is noetherian, i.e. every ideal is finitely generated.

fn from_int(&self, value: i32) -> Self::Element

fn dbg_within<'a>( &self, value: &Self::Element, out: &mut Formatter<'a>, env: EnvBindingStrength, ) -> Result

Writes a human-readable representation of value to out, taking into account the possible context to place parenthesis as needed.

fn characteristic<I: RingStore + Copy>(&self, ZZ: I) -> Option<El<I>>

where I::Type: IntegerRing,

Returns the characteristic of this ring as an element of the given implementation of ZZ.

fn sub_assign_ref(&self, lhs: &mut Self::Element, rhs: &Self::Element)

fn zero(&self) -> Self::Element

fn one(&self) -> Self::Element

fn neg_one(&self) -> Self::Element

fn dbg<'a>(&self, value: &Self::Element, out: &mut Formatter<'a>) -> Result

Writes a human-readable representation of value to out.

fn square(&self, value: &mut Self::Element)

fn negate(&self, value: Self::Element) -> Self::Element

fn sub_assign(&self, lhs: &mut Self::Element, rhs: Self::Element)

fn mul_assign_int(&self, lhs: &mut Self::Element, rhs: i32)

fn mul_int(&self, lhs: Self::Element, rhs: i32) -> Self::Element

fn mul_int_ref(&self, lhs: &Self::Element, rhs: i32) -> Self::Element

fn sub_self_assign(&self, lhs: &mut Self::Element, rhs: Self::Element)

Computes lhs := rhs - lhs.

fn sub_self_assign_ref(&self, lhs: &mut Self::Element, rhs: &Self::Element)

Computes lhs := rhs - lhs.

fn add_ref(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

fn add_ref_fst(&self, lhs: &Self::Element, rhs: Self::Element) -> Self::Element

fn add_ref_snd(&self, lhs: Self::Element, rhs: &Self::Element) -> Self::Element

fn add(&self, lhs: Self::Element, rhs: Self::Element) -> Self::Element

fn sub_ref(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

fn sub_ref_fst(&self, lhs: &Self::Element, rhs: Self::Element) -> Self::Element

fn sub_ref_snd(&self, lhs: Self::Element, rhs: &Self::Element) -> Self::Element

fn sub(&self, lhs: Self::Element, rhs: Self::Element) -> Self::Element

fn mul_ref(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

fn mul_ref_fst(&self, lhs: &Self::Element, rhs: Self::Element) -> Self::Element

fn mul_ref_snd(&self, lhs: Self::Element, rhs: &Self::Element) -> Self::Element

fn mul(&self, lhs: Self::Element, rhs: Self::Element) -> Self::Element

fn pow_gen<R: RingStore>( &self, x: Self::Element, power: &El<R>, integers: R, ) -> Self::Element

where R::Type: IntegerRing,

Raises x to the power of an arbitrary, nonnegative integer given by a custom integer ring implementation.

fn sum<I>(&self, els: I) -> Self::Element

where I: IntoIterator<Item = Self::Element>,

Sums the elements given by the iterator.

fn prod<I>(&self, els: I) -> Self::Element

where I: IntoIterator<Item = Self::Element>,

Computes the product of the elements given by the iterator.

impl<R> RingExtension for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain,

type BaseRing = R

Type of the base ring;

fn base_ring<'a>(&'a self) -> &'a Self::BaseRing

Returns a reference to the base ring.

fn from(&self, x: El<Self::BaseRing>) -> Self::Element

Maps an element of the base ring into this ring.

fn mul_assign_base(&self, lhs: &mut Self::Element, rhs: &El<Self::BaseRing>)

Computes lhs := lhs * rhs, where rhs is mapped into this ring via RingExtension::from_ref(). Note that this may be faster than self.mul_assign(lhs, self.from_ref(rhs)).

fn from_ref(&self, x: &El<Self::BaseRing>) -> Self::Element

Maps an element of the base ring (given as reference) into this ring.

fn mul_assign_base_through_hom<S: ?Sized + RingBase, H: Homomorphism<S, <Self::BaseRing as RingStore>::Type>>( &self, lhs: &mut Self::Element, rhs: &S::Element, hom: H, )

Computes lhs := lhs * rhs, where rhs is mapped into this ring via the given homomorphism, followed by the inclusion (as specified by RingExtension::from_ref()).

impl<R> StrassenHint for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain,

default fn strassen_threshold(&self) -> usize

Define a threshold from which on StrassenAlgorithm will use the Strassen algorithm.

impl<R> Copy for FractionFieldImplBase<R>

where R: RingStore + Copy, R::Type: Domain, El<R>: Copy,

impl<R> Domain for FractionFieldImplBase<R>

where R: RingStore, R::Type: Domain,