integer_ring.hpp - Integers

integer_ring.hpp - Integers#

struct IntegerRing#

Implements integer arithmetic for integer coefficients in expressions making up an Element.

There should be no need to instantiate this struct. It’s usually only used as a template parameter: 

#include <exact-real/module.hpp>
#include <exact-real/real_number.hpp>
#include <exact-real/element.hpp>
#include <exact-real/integer_ring.hpp>

auto M = exactreal::Module<exactreal::IntegerRing>::make({
  exactreal::RealNumber::rational(1),
  exactreal::RealNumber::random()});
*M
// -> ℤ-Module(1, ℝ(<...>))

Public Types

typedef mpz_class ElementClass#

The internal representation of element of this ring, i.e., GMP integers.

Public Functions

IntegerRing()#

Create the integer ring.

IntegerRing(const mpz_class&)#

Create “an” integer ring that contains the argument.

This is identical to IntegerRing::IntegerRing() but other rings, say number fields, have different rings and need to be able to construct the smallest ring that contains a certain element.

ElementClass coerce(const ElementClass &x) const#

Convert the integer argument into this integer ring, i.e., just return the integer unchanged.

Number fields need the equivalent of this method to convert elements between different number fields.

bool operator==(const IntegerRing &R) const#

Return whether this integer ring is identical to R, i.e., return true since there is only one integer ring.

Public Static Functions

static IntegerRing compositum(const IntegerRing &lhs, const IntegerRing &rhs)#

Return the integer ring that contains both arguments.

This is just the IntegerRing::IntegerRing() but other rings need such functionality to compute a composite ring.

static bool unit(const ElementClass &x)#

Return whether x is a unit, i.e., whether it is 1 or -1.

static Arb arb(const ElementClass &x, long prec)#

Return an (exact) Arb approximation of the integer x. 

#include <exact-real/arb.hpp>
exactreal::IntegerRing::arb(1, 64)
// -> 1.00000

static mpz_class floor(const ElementClass &x)#

Return the floor of the integer x, i.e., the element unchanged.

static std::optional<mpq_class> rational(const ElementClass &x)#

Return the integer x as a rational number. 

exactreal::IntegerRing::rational(1).value()
// -> 1

template<typename T>
static ElementClass &imul(ElementClass &x, const T &t)#

Multiply the integer x with t.

The argument may be a primitive integer, a GMP integer, or a GMP rational that is an integer.

template<typename T>
static ElementClass &idiv(ElementClass &x, const T &t)#

Divide the integer x by t.

The argument may be a primitive integer, a GMP integer, or a GMP rational. This operation is only supported if the result of the exact division is an integer.

Public Static Attributes

static bool contains_rationals = false#

Whether this ring contains the rational numbers, i.e., false.

static bool isField = false#

Whether this ring is a field, i.e., false.

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