dedekind::algebra::Polynomial< R, M > Class Template Reference

The monoid algebra R[M]: a finitely supported function from the monomials to the coefficients, stored as its support sorted along the monomials' chain, no coefficient zero. A net (Domain the monomials, Codomain the coefficients, p(m) the coefficient of m), a commutative rig over a commutative rig, a ring over a ring (- where R has additive inverses). More...

Public Types
using	coefficient_type = R
using	monomial_type = M
using	Domain = M
using	Codomain = R
using	term_type = std::pair<M, R>

Public Member Functions
constexpr	Polynomial ()=default
	The zero polynomial.
constexpr	Polynomial (R scalar)
	A constant.
constexpr	Polynomial (std::vector< term_type > terms)
	The sum of the given terms, in any order: combined, sorted, zero coefficients dropped.
constexpr R	operator() (const M &m) const
	The net: the coefficient of m, zero off the support.
constexpr const std::vector< term_type > &	support () const noexcept
	The support with its coefficients: the finite function this polynomial is, sorted along the monomials' chain.
constexpr bool	is_zero () const
constexpr auto	degree () const
	The total degree of the leading monomial; 0 for the zero polynomial.
constexpr const R &	leading_coefficient () const
constexpr Polynomial	derive (std::size_t slot=0) const
	The formal partial derivative by the variable of slot: ∂(c·m) = e·c·m/π, the factor e by repeated doubling, so no more than R's addition is asked (a rig suffices).

Static Public Member Functions
static constexpr Polynomial	zero ()
static constexpr Polynomial	one ()
static constexpr Polynomial	variable (std::size_t slot)
	The variable of slot, as a polynomial.

Friends
constexpr bool	operator== (const Polynomial &, const Polynomial &)=default
constexpr Polynomial	operator+ (const Polynomial &a, const Polynomial &b)
	Pointwise: the merge of two sorted supports.
constexpr Polynomial	operator- (const Polynomial &a)
	Coefficientwise additive inverse, where R has one.
constexpr Polynomial	operator- (const Polynomial &a, const Polynomial &b)
constexpr Polynomial	inverse (const Polynomial &a, std::plus< Polynomial >)
	The additive inverse under the category's inverse_v bridge, so is_invertible_v<Polynomial, std::plus<Polynomial>> follows from R's.
constexpr Polynomial	operator* (const Polynomial &a, const Polynomial &b)
	The convolution: every pair of terms multiplied, then combined.

Detailed Description

template<typename R, typename M = Monomial<1>>
requires std::equality_comparable<R> && dedekind::category::IsCommutativeMonoid<M, std::multiplies<M>> && dedekind::order::IsTotallyOrdered<M>
class dedekind::algebra::Polynomial< R, M >

The monoid algebra R[M]: a finitely supported function from the monomials to the coefficients, stored as its support sorted along the monomials' chain, no coefficient zero. A net (Domain the monomials, Codomain the coefficients, p(m) the coefficient of m), a commutative rig over a commutative rig, a ring over a ring (- where R has additive inverses).

Template Parameters

R	the coefficients.
M	the monomials: a commutative monoid that is a chain.

Constructor & Destructor Documentation

Polynomial()

template<typename R, typename M = Monomial<1>>

dedekind::algebra::Polynomial< R, M >::Polynomial ( std::vector< term_type > terms )

inlineexplicitconstexpr

The sum of the given terms, in any order: combined, sorted, zero coefficients dropped.

Parameters

terms

monomial and coefficient pairs.

Member Function Documentation

leading_coefficient()

template<typename R, typename M = Monomial<1>>

const R & dedekind::algebra::Polynomial< R, M >::leading_coefficient ( ) const

inlineconstexpr

Exceptions

std::domain_error

on the zero polynomial.

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The documentation for this class was generated from the following file:

• src/main/modules/dedekind/algebra/polynomial.cppm