Field of rational numbers ℚ = { p/q : p, q ∈ ℤ, q ≠ 0 } with selectable precision. More...

#include <fields.hpp>

Inheritance diagram for CECCO::Rationals< T >:

Public Member Functions
	Rationals (int n=0, int d=1)
	Construct n / d, simplified to lowest terms with positive denominator.
	Rationals (const Rationals &other)=default
	Rationals (Rationals &&other)=default
constexpr Rationals &	operator= (int l)
	Assign the integer `l` as l / 1.
constexpr Rationals &	operator= (const Rationals &rhs)=default
Rationals &	operator= (Rationals &&rhs)=default
constexpr bool	operator== (const Rationals< T > &rhs) const
	Cross-multiplication equality.
constexpr Rationals	operator- () const &
	Additive inverse (lvalue overload returns a copy with negated numerator).
constexpr Rationals &	operator- () &&
	Additive inverse (rvalue overload negates in place).
constexpr Rationals &	operator+= (const Rationals &rhs)
	*this += rhs, result kept in lowest terms
constexpr Rationals &	operator-= (const Rationals &rhs)
	*this -= rhs, result kept in lowest terms
constexpr Rationals &	operator*= (const Rationals &rhs)
	this = rhs, result kept in lowest terms
Rationals &	operator/= (const Rationals &rhs)
	*this /= rhs; throws std::invalid_argument if rhs is zero
Rationals &	randomize ()
	Random rational (bounded numerator and denominator), simplified.
Rationals &	randomize_force_change ()
	Like randomize but guaranteed to differ from the current value.
size_t	get_multiplicative_order () const
	Multiplicative order.
size_t	get_additive_order () const noexcept
	Additive order: 1 for zero, 0 (infinite) otherwise (characteristic 0).
constexpr bool	has_positive_sign () const noexcept
	Sign predicate (true iff numerator and denominator share their sign).
constexpr bool	is_zero () const noexcept
	True iff numerator is zero.
constexpr Rationals &	erase () noexcept
	Mark this element as erased.
constexpr Rationals &	unerase () noexcept
	Clear the erasure flag, resetting to additive identity 0/1.
constexpr bool	is_erased () const noexcept
	Test whether this element is currently erased (encoded as denominator == 0).
constexpr const T &	get_numerator () const noexcept
	Numerator (sign carrier).
constexpr const T &	get_denominator () const noexcept
	Denominator (always positive after simplification).
Public Member Functions inherited from CECCO::details::Field< Rationals< InfInt > >
Rationals< InfInt > &	operator= (int l)=delete
	Assign from int — derived must implement.
constexpr bool	operator!= (const Rationals< InfInt > &rhs) const
	Inequality, defined as !(*this == rhs).
constexpr Rationals< InfInt >	operator+ () const &
	Unary + on an lvalue: returns a copy.
Rationals< InfInt >	operator- () const &noexcept=delete
	Additive inverse on an lvalue — derived must implement.
Rationals< InfInt > &	operator+= (const Rationals< InfInt > &rhs) noexcept=delete
	*this += rhs — derived must implement
Rationals< InfInt > &	operator-= (const Rationals< InfInt > &rhs) noexcept=delete
	*this -= rhs — derived must implement
Rationals< InfInt > &	operator*= (const Rationals< InfInt > &rhs) noexcept=delete
	this = rhs — derived must implement
Rationals< InfInt > &	operator/= (const Rationals< InfInt > &rhs)=delete
	*this /= rhs; throws std::invalid_argument if rhs is zero — derived must implement
Field &	randomize ()=delete
	Uniform random element of the field — derived must implement; may return the same value.
Field &	randomize_force_change ()=delete
	Like randomize but guaranteed to differ from the current value — derived must implement.
size_t	get_multiplicative_order () const=delete
	Smallest k > 0 with this^k == 1; throws std::invalid_argument if *this is zero.
size_t	get_additive_order () const=delete
	Smallest k > 0 with k * *this == 0; for finite fields of characteristic p this is p (or 1 for zero); for ℚ it is 1 for zero and 0 (infinite) otherwise.
bool	has_positive_sign () const noexcept=delete
	True if the element is "positive" (always true for finite fields; sign of numerator for ℚ).
bool	is_zero () const noexcept=delete
	True if *this is the additive identity.
Field &	erase () noexcept=delete
	Mark this element as erased (out-of-field marker for erasure decoding).
Field &	unerase () noexcept=delete
	Clear the erasure flag, resetting to additive identity.
bool	is_erased () const noexcept=delete
	Test whether this element is currently erased.

Static Public Member Functions
static const std::string	get_info ()
	Human-readable description: "rational number".
static constexpr size_t	get_characteristic () noexcept
	Characteristic of ℚ: 0.

Additional Inherited Members
Protected Member Functions inherited from CECCO::details::Field< Rationals< InfInt > >
	~Field () noexcept=default
Protected Member Functions inherited from CECCO::details::Base
	Base ()=default

Detailed Description

template<SignedIntType T = InfInt>
class CECCO::Rationals< T >

Field of rational numbers ℚ = { p/q : p, q ∈ ℤ, q ≠ 0 } with selectable precision.

Template Parameters

T	Numerator/denominator type satisfying CECCO::SignedIntType (default InfInt)

Characteristic 0. Values are kept in lowest terms with positive denominator at all times, so equality is numerator_a * denominator_b == numerator_b * denominator_a. Construction with a zero denominator throws std::invalid_argument.

Pick T = InfInt for true ℚ — a fixed-width T (e.g. int, long long) caps numerator and denominator and silently overflows past that range.

Usage_Example

Rationals<> a(3, 4);
Rationals<> b(5, 6);
auto c = a + b;          // 19/12 (auto-simplified)
auto d = a / b;          // 9/10
std::cout << c;          // "19/12"

Definition at line 471 of file fields.hpp.

Constructor & Destructor Documentation

◆ Rationals() [1/3]

template<SignedIntType T>

CECCO::Rationals< T >::Rationals	(	int	n = 0,
		int	d = 1 )

Construct n / d, simplified to lowest terms with positive denominator.

Parameters

n	Numerator (default 0)
d	Denominator (default 1)

Exceptions

std::invalid_argument if d == 0

Definition at line 576 of file fields.hpp.

◆ Rationals() [2/3]

template<SignedIntType T = InfInt>

CECCO::Rationals< T >::Rationals ( const Rationals< T > & other )

default

◆ Rationals() [3/3]

template<SignedIntType T = InfInt>

CECCO::Rationals< T >::Rationals ( Rationals< T > && other )

default

Member Function Documentation

◆ erase()

template<SignedIntType T>

Rationals< T > & CECCO::Rationals< T >::erase ( )

constexprnoexcept

Mark this element as erased.

Warning: Erased elements must not participate in field arithmetic; correct use is the caller's responsibility (cf. CECCO_ERASURE_SUPPORT).

Definition at line 717 of file fields.hpp.

◆ get_additive_order()

template<SignedIntType T>

size_t CECCO::Rationals< T >::get_additive_order ( ) const

noexcept

Additive order: 1 for zero, 0 (infinite) otherwise (characteristic 0).

Definition at line 708 of file fields.hpp.

◆ get_characteristic()

template<SignedIntType T = InfInt>

constexpr size_t CECCO::Rationals< T >::get_characteristic ( )

inlinestaticconstexprnoexcept

Characteristic of ℚ: 0.

Definition at line 536 of file fields.hpp.

◆ get_denominator()

template<SignedIntType T = InfInt>

const T & CECCO::Rationals< T >::get_denominator ( ) const

inlineconstexprnoexcept

Denominator (always positive after simplification).

Definition at line 563 of file fields.hpp.

◆ get_info()

template<SignedIntType T = InfInt>

const std::string CECCO::Rationals< T >::get_info ( )

inlinestatic

Human-readable description: "rational number".

Definition at line 530 of file fields.hpp.

◆ get_multiplicative_order()

template<SignedIntType T>

size_t CECCO::Rationals< T >::get_multiplicative_order ( ) const

Multiplicative order.

Returns: 1 for 1/1, 2 for −1/1, 0 (interpreted as infinite) for everything else

Exceptions

std::invalid_argument if *this is zero

Definition at line 695 of file fields.hpp.

◆ get_numerator()

template<SignedIntType T = InfInt>

const T & CECCO::Rationals< T >::get_numerator ( ) const

inlineconstexprnoexcept

Numerator (sign carrier).

Definition at line 561 of file fields.hpp.

◆ has_positive_sign()

template<SignedIntType T = InfInt>

bool CECCO::Rationals< T >::has_positive_sign ( ) const

inlineconstexprnoexcept

Sign predicate (true iff numerator and denominator share their sign).

Definition at line 539 of file fields.hpp.

◆ is_erased()

template<SignedIntType T = InfInt>

bool CECCO::Rationals< T >::is_erased ( ) const

inlineconstexprnoexcept

Test whether this element is currently erased (encoded as denominator == 0).

Definition at line 557 of file fields.hpp.

◆ is_zero()

template<SignedIntType T = InfInt>

bool CECCO::Rationals< T >::is_zero ( ) const

inlineconstexprnoexcept

True iff numerator is zero.

Definition at line 544 of file fields.hpp.

◆ operator*=()

template<SignedIntType T>

Rationals< T > & CECCO::Rationals< T >::operator*= ( const Rationals< T > & rhs )

constexpr

*this *= rhs, result kept in lowest terms

Definition at line 634 of file fields.hpp.

◆ operator+=()

template<SignedIntType T>

Rationals< T > & CECCO::Rationals< T >::operator+= ( const Rationals< T > & rhs )

constexpr

*this += rhs, result kept in lowest terms

Definition at line 608 of file fields.hpp.

◆ operator-() [1/2]

template<SignedIntType T>

Rationals< T > & CECCO::Rationals< T >::operator- ( ) &&

constexpr

Additive inverse (rvalue overload negates in place).

Definition at line 599 of file fields.hpp.

◆ operator-() [2/2]

template<SignedIntType T>

Rationals< T > CECCO::Rationals< T >::operator- ( ) const &

constexpr

Additive inverse (lvalue overload returns a copy with negated numerator).

Definition at line 589 of file fields.hpp.

◆ operator-=()

template<SignedIntType T>

Rationals< T > & CECCO::Rationals< T >::operator-= ( const Rationals< T > & rhs )

constexpr

*this -= rhs, result kept in lowest terms

Definition at line 621 of file fields.hpp.

◆ operator/=()

template<SignedIntType T>

Rationals< T > & CECCO::Rationals< T >::operator/= ( const Rationals< T > & rhs )

*this /= rhs; throws std::invalid_argument if rhs is zero

Definition at line 647 of file fields.hpp.

◆ operator=() [1/3]

template<SignedIntType T = InfInt>

Rationals & CECCO::Rationals< T >::operator= ( const Rationals< T > & rhs )

constexprdefault

◆ operator=() [2/3]

template<SignedIntType T>

Rationals< T > & CECCO::Rationals< T >::operator= ( int l )

constexpr

Assign the integer l as l / 1.

Definition at line 582 of file fields.hpp.

◆ operator=() [3/3]

template<SignedIntType T = InfInt>

Rationals & CECCO::Rationals< T >::operator= ( Rationals< T > && rhs )

default

◆ operator==()

template<SignedIntType T = InfInt>

bool CECCO::Rationals< T >::operator== ( const Rationals< T > & rhs ) const

inlineconstexpr

Cross-multiplication equality.

Definition at line 492 of file fields.hpp.

◆ randomize()

template<SignedIntType T>

Rationals< T > & CECCO::Rationals< T >::randomize ( )

Random rational (bounded numerator and denominator), simplified.

Definition at line 661 of file fields.hpp.

◆ randomize_force_change()

template<SignedIntType T>

Rationals< T > & CECCO::Rationals< T >::randomize_force_change ( )

Like randomize but guaranteed to differ from the current value.

Definition at line 675 of file fields.hpp.

◆ unerase()

template<SignedIntType T>

Rationals< T > & CECCO::Rationals< T >::unerase ( )

constexprnoexcept

Clear the erasure flag, resetting to additive identity 0/1.

Definition at line 723 of file fields.hpp.

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

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Usage_Example

Constructor & Destructor Documentation

◆ Rationals() [1/3]

◆ Rationals() [2/3]

◆ Rationals() [3/3]

Member Function Documentation

◆ erase()

◆ get_additive_order()

◆ get_characteristic()

◆ get_denominator()

◆ get_info()

◆ get_multiplicative_order()

◆ get_numerator()

◆ has_positive_sign()

◆ is_erased()

◆ is_zero()

◆ operator*=()

◆ operator+=()

◆ operator-() [1/2]

◆ operator-() [2/2]

◆ operator-=()

◆ operator/=()

◆ operator=() [1/3]

◆ operator=() [2/3]

◆ operator=() [3/3]

◆ operator==()

◆ randomize()

◆ randomize_force_change()

◆ unerase()