Struct SparsePolyRingBase 

pub struct SparsePolyRingBase<R: RingStore> { /* private fields */ }

The univariate polynomial ring R[X]. Polynomials are stored as sparse vectors of coefficients, thus giving improved performance in the case that most coefficients are zero.

Unless polynomials are very sparse, [DensePolyRing] will provide better performance.

Example

 
let ZZ = StaticRing::<i32>::RING;
let P = SparsePolyRing::new(ZZ, "X");
let x10_plus_1 = P.add(P.pow(P.indeterminate(), 10), P.int_hom().map(1));
let power = P.pow(x10_plus_1, 10);
assert_eq!(0, *P.coefficient_at(&power, 1));

This ring has a CanIsoFromTo to dense_poly::DensePolyRingBase.

 
let ZZ = StaticRing::<i32>::RING;
let P = SparsePolyRing::new(ZZ, "X");
let P2 = DensePolyRing::new(ZZ, "X");
let high_power_of_x = P.pow(P.indeterminate(), 10);
assert_el_eq!(P2, P2.pow(P2.indeterminate(), 10), &P.can_iso(&P2).unwrap().map(high_power_of_x));

Trait Implementations

impl<R, P> CanHomFrom<P> for SparsePolyRingBase<R>

where R: RingStore, R::Type: CanHomFrom<<P::BaseRing as RingStore>::Type>, P: ImplGenericCanIsoFromToMarker,

type Homomorphism = <<<SparsePolyRingBase<R> as RingExtension>::BaseRing as RingStore>::Type as CanHomFrom<<<P as RingExtension>::BaseRing as RingStore>::Type>>::Homomorphism

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

fn has_canonical_hom(&self, from: &P) -> 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: &P, el: P::Element, hom: &Self::Homomorphism, ) -> Self::Element

Evaluates the homomorphism.

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

Evaluates the homomorphism, taking the element by reference.

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

Evaluates the homomorphism on rhs, and multiplies the result to lhs.

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

Evaluates the homomorphism on rhs, taking it by reference, and multiplies the result to lhs.

fn fma_map_in( &self, from: &S, lhs: &Self::Element, rhs: &S::Element, summand: Self::Element, hom: &Self::Homomorphism, ) -> Self::Element

Fused-multiply-add. Computes summand + lhs * rhs, where rhs is mapped into the ring via the homomorphism.

impl<R1, R2> CanHomFrom<SparsePolyRingBase<R1>> for SparsePolyRingBase<R2>

where R1: RingStore, R2: RingStore, R2::Type: CanHomFrom<R1::Type>,

type Homomorphism = <<R2 as RingStore>::Type as CanHomFrom<<R1 as RingStore>::Type>>::Homomorphism

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

fn has_canonical_hom( &self, from: &SparsePolyRingBase<R1>, ) -> 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_ref( &self, from: &SparsePolyRingBase<R1>, el: &SparsePolyRingEl<R1>, hom: &Self::Homomorphism, ) -> Self::Element

Evaluates the homomorphism, taking the element by reference.

fn map_in( &self, from: &SparsePolyRingBase<R1>, el: <SparsePolyRingBase<R1> as RingBase>::Element, hom: &Self::Homomorphism, ) -> Self::Element

Evaluates the homomorphism.

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

Evaluates the homomorphism on rhs, and multiplies the result to lhs.

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

Evaluates the homomorphism on rhs, taking it by reference, and multiplies the result to lhs.

fn fma_map_in( &self, from: &S, lhs: &Self::Element, rhs: &S::Element, summand: Self::Element, hom: &Self::Homomorphism, ) -> Self::Element

Fused-multiply-add. Computes summand + lhs * rhs, where rhs is mapped into the ring via the homomorphism.

impl<R, P> CanIsoFromTo<P> for SparsePolyRingBase<R>

where R: RingStore, R::Type: CanIsoFromTo<<P::BaseRing as RingStore>::Type>, P: ImplGenericCanIsoFromToMarker,

type Isomorphism = <<<SparsePolyRingBase<R> as RingExtension>::BaseRing as RingStore>::Type as CanIsoFromTo<<<P as RingExtension>::BaseRing as RingStore>::Type>>::Isomorphism

Data required to compute a preimage under the canonical homomorphism.

fn has_canonical_iso(&self, from: &P) -> 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: &P, el: Self::Element, iso: &Self::Isomorphism, ) -> P::Element

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

impl<R1, R2> CanIsoFromTo<SparsePolyRingBase<R1>> for SparsePolyRingBase<R2>

where R1: RingStore, R2: RingStore, R2::Type: CanIsoFromTo<R1::Type>,

type Isomorphism = <<R2 as RingStore>::Type as CanIsoFromTo<<R1 as RingStore>::Type>>::Isomorphism

Data required to compute a preimage under the canonical homomorphism.

fn has_canonical_iso( &self, from: &SparsePolyRingBase<R1>, ) -> 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: &SparsePolyRingBase<R1>, el: Self::Element, iso: &Self::Isomorphism, ) -> SparsePolyRingEl<R1>

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

impl<R: RingStore + Clone> Clone for SparsePolyRingBase<R>

fn clone(&self) -> Self

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl<R: RingStore> Debug for SparsePolyRingBase<R>

where R::Type: Debug,

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl<R> DivisibilityRing for SparsePolyRingBase<R>

where R: RingStore, R::Type: DivisibilityRing + 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.

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 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.

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 SparsePolyRingBase<R>

where R: RingStore, R::Type: Field + PolyTFracGCDRing,

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

Computes euclidean division with remainder.

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

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

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> PartialEq for SparsePolyRingBase<R>

where R: RingStore,

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> PolyRing for SparsePolyRingBase<R>

where R: RingStore,

type TermsIterator<'a> = TermIterator<'a, R> where Self: 'a

Type of the iterator over all non-zero terms of a polynomial in this ring, as returned by PolyRing::terms().

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

Returns the indeterminate X generating this polynomial ring.

fn terms<'a>(&'a self, f: &'a Self::Element) -> TermIterator<'a, R>

Returns all the nonzero terms of the given polynomial.

fn add_assign_from_terms<I>(&self, lhs: &mut Self::Element, rhs: I)

where I: IntoIterator<Item = (El<Self::BaseRing>, usize)>,

Adds the given terms to the given polynomial.

fn coefficient_at<'a>( &'a self, f: &'a Self::Element, i: usize, ) -> &'a El<Self::BaseRing>

Returns the coefficient of f that corresponds to the monomial X^i.

fn degree(&self, f: &Self::Element) -> Option<usize>

Returns the degree of the polynomial f, i.e. the value d such that f can be written as f(X) = a0 + a1 * X + a2 * X^2 + ... + ad * X^d. Returns None if f is zero.

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

Compute the euclidean division by a monic polynomial rhs.

fn mul_assign_monomial(&self, lhs: &mut Self::Element, rhs_power: usize)

Multiplies the given polynomial with X^rhs_power.

fn truncate_monomials( &self, lhs: &mut Self::Element, truncated_at_inclusive: usize, )

Truncates the monomials of the given polynomial from the given position on, i.e. computes the remainder of the polynomial division of lhs by X^truncated_at_inclusive.

fn map_terms<P, H>(&self, from: &P, el: &P::Element, hom: H) -> Self::Element

where P: ?Sized + PolyRing, H: Homomorphism<<P::BaseRing as RingStore>::Type, <Self::BaseRing as RingStore>::Type>,

Computes the polynomial whose coefficients are the images of the coefficients of el under the given homomorphism.

fn balance_poly(&self, f: &mut Self::Element) -> Option<El<Self::BaseRing>>

where <Self::BaseRing as RingStore>::Type: DivisibilityRing,

Possibly divides all coefficients of the polynomial by a common factor, in order to make them “smaller”.

fn evaluate<R, H>( &self, f: &Self::Element, value: &R::Element, hom: H, ) -> R::Element

where R: ?Sized + RingBase, H: Homomorphism<<Self::BaseRing as RingStore>::Type, R>,

Evaluates the given polynomial at the given values.

impl<R> PrincipalIdealRing for SparsePolyRingBase<R>

where R: RingStore, R::Type: Field + PolyTFracGCDRing,

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 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 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_with_controller<Controller>( &self, lhs: &Self::Element, rhs: &Self::Element, _: Controller, ) -> Self::Element

where Controller: ComputationController,

As PrincipalIdealRing::ideal_gen(), this computes a generator of the ideal (lhs, rhs). However, it additionally accepts a ComputationController to customize the performed computation.

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: RingStore> RingBase for SparsePolyRingBase<R>

type Element = SparsePolyRingEl<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 sub_assign_ref(&self, lhs: &mut Self::Element, rhs: &Self::Element)

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

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

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

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

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

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

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

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

fn characteristic<I: IntegerRingStore + 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 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 one(&self) -> Self::Element

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

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 fma( &self, lhs: &Self::Element, rhs: &Self::Element, summand: Self::Element, ) -> Self::Element

Fused-multiply-add. This computes summand + lhs * rhs.

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

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

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

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

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

Fused-multiply-add with an integer. This computes summand + lhs * rhs.

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_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: RingStore> RingExtension for SparsePolyRingBase<R>

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 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(&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 fma_base( &self, lhs: &Self::Element, rhs: &El<Self::BaseRing>, summand: Self::Element, ) -> Self::Element

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> Domain for SparsePolyRingBase<R>

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

impl<R> ImplGenericCanIsoFromToMarker for SparsePolyRingBase<R>

where R: RingStore,