Struct abstalg::PolynomialAlgebra

pub struct PolynomialAlgebra<A> where
    A: AbelianGroup, { /* private fields */ }

The ring of polynomials over a base ring or field where each element is represented as a vector whose last element, the leading coefficient (if any) must be non-zero. This means that the empty vector is the zero element, and every polynomial has a unique representation. Polynomials can be defined over any abelian group, though only the group operations will be available.

Implementations

impl<A> PolynomialAlgebra<A> where
    A: AbelianGroup, 

pub fn new(base: A) -> Self

Creates a new polynomial algebra from the given base.

pub fn base(&self) -> &A

Returns the base ring from which this ring was created.

pub fn degree(&self, elem: &Vec<A::Elem>) -> Option<usize>

Returns the degree of the given polynomial. The zero polynomial has no degree.

pub fn leading_coef(&self, elem: &Vec<A::Elem>) -> Option<A::Elem>

Returns the leading coefficient of the given polynomial. The zero polynomial has no leading coefficient.

Trait Implementations

impl<A> AbelianGroup for PolynomialAlgebra<A> where
    A: AbelianGroup, 

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

The additive identity element of the ring.

fn is_zero(&self, elem: &Self::Elem) -> bool

Checks if the given element is the additive identity of the ring.

fn neg(&self, elem: &Self::Elem) -> Self::Elem

The additive inverse of the given element.

fn add(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem

The additive sum of the given elements

fn times(&self, num: isize, elem: &Self::Elem) -> Self::Elem

Returns an integer multiple of the given element.

fn neg_assign(&self, elem: &mut Self::Elem)

The element is changed to its additive inverse.

fn add_assign(&self, elem1: &mut Self::Elem, elem2: &Self::Elem)

The second element is added to the first one.

fn double(&self, elem: &mut Self::Elem)

Doubles the given element in place.

fn sub(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem

The difference of the given elements.

fn sub_assign(&self, elem1: &mut Self::Elem, elem2: &Self::Elem)

The second element is subtracted from the first one.

impl<A: Clone> Clone for PolynomialAlgebra<A> where
    A: AbelianGroup, 

fn clone(&self) -> PolynomialAlgebra<A>

Returns a copy of the value.

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

Performs copy-assignment from source.

impl<A: Debug> Debug for PolynomialAlgebra<A> where
    A: AbelianGroup, 

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

Formats the value using the given formatter.

impl<A> Domain for PolynomialAlgebra<A> where
    A: AbelianGroup, 

type Elem = Vec<<A as Domain>::Elem, Global>

The type of the elements of this domain.

fn contains(&self, elem: &Self::Elem) -> bool

Checks if the given object is a member of the domain.

fn equals(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool

Checks if the given objects represent the same element of the set.

impl<A> EuclideanDomain for PolynomialAlgebra<A> where
    A: Field, 

fn quo_rem(
    &self,
    elem1: &Self::Elem,
    elem2: &Self::Elem
) -> (Self::Elem, Self::Elem)

Performs the euclidean division algorithm dividing the first elem with the second one and returning the quotient and the remainder. This method panics if the second element is zero (because there is no unique solution)

fn quo(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem

Performs the division just like the quo_rem method would do and returns the quotient. This method panics if the second element is zero.

fn rem(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem

Performs the division just like the quo_rem method would do and returns the remainder. This method panics if the second element is zero.

fn reduced(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool

Returns true if the second element is zero or the first element has zero quotient by the second one. These are the representative elements of the factor ring by the principal ideal generated by the second element.

fn gcd(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem

Calculates the greatest common divisor of two elements using the Euclidean algorithm.

fn lcm(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem

Calculates the lest common divisor of the two elements.

fn extended_gcd(
    &self,
    elem1: &Self::Elem,
    elem2: &Self::Elem
) -> (Self::Elem, Self::Elem, Self::Elem)

Performs the extended Euclidean algorithm which returns the greatest common divisor, and two elements that multiplied with the inputs gives the greatest common divisor.

fn relative_primes(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool

Checks if the given two elements are relative prime.

impl<A> IntegralDomain for PolynomialAlgebra<A> where
    A: IntegralDomain, 

fn try_div(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Option<Self::Elem>

Checks if the second element divides the first one and returns the unique quotient if it exists. This method panics if the second element is zero (because there would be no unique solution).

fn associate_repr(&self, elem: &Self::Elem) -> Self::Elem

We assume, that among all associates of the given elem there is a well defined unique one (non-negative for integers, zero or monic for polynomials). This method returns that representative element.

fn associate_coef(&self, elem: &Self::Elem) -> Self::Elem

Returns the unique invertible element which is the quotient of the associate representative and the given element. This method panics if the element is zero.

fn divisible(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool

Checks if the first element is a multiple of (or divisible by) the second one.

fn associates(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool

Returns true if the two elements are associates (divide each other)

impl<A> Monoid for PolynomialAlgebra<A> where
    A: AbelianGroup + Monoid, 

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

The multiplicative identity element of the ring.

fn is_one(&self, elem: &Self::Elem) -> bool

Checks if the given element is the multiplicative identity.

fn try_inv(&self, elem: &Self::Elem) -> Option<Self::Elem>

Calculates the multiplicative inverse of the given element if it exists.

fn invertible(&self, elem: &Self::Elem) -> bool

Returns true if the given element has a multiplicative inverse.

impl<A> Semigroup for PolynomialAlgebra<A> where
    A: AbelianGroup + Semigroup, 

fn mul(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem

The multiplicative product of the given elements.

fn mul_assign(&self, elem1: &mut Self::Elem, elem2: &Self::Elem)

The first element is multiplied with the second one in place.

fn square(&self, elem: &mut Self::Elem)

Squares the given element in place.

impl<A> UnitaryRing for PolynomialAlgebra<A> where
    A: UnitaryRing, 

fn int(&self, elem: isize) -> Self::Elem

Returns the integer multiple of the one element in the ring.