Coefficients.h file

declares all classes that represent rings of coefficients

At present, finite fields of the form are implemented for prime integers , as well as the ring of integers and the field of rational numbers.

Classes

struct Q: the field of rational numbers
template<prime P> struct Z_mod: the field for prime integers . We do not check if P is a prime number.

Typedefs

using int_coeff = int_fast64_t: element of the ring of integers
using prime = int: prime number

Functions

auto coeff_is_0(const int_coeff& n) -> bool: true iff n is equal to 0
auto coeff_is_0(const Q& r) -> bool: true iff n is equal to 0
template<prime P> auto coeff_is_0(const Z_mod& n) -> bool: true iff n is equal to 0
auto coeff_is_inv(const int_coeff& n) -> bool: true iff n is invertible, ie either +1 or -1
auto coeff_is_inv(const Q& r) -> bool: true iff n is invertible, ie non-zero
template<prime P> auto coeff_is_inv(const Z_mod& n) -> bool: true iff n is invertible, ie non-zero
auto coeff_to_string(const int_coeff& i) -> std::string: converts integer to string
template<prime P> auto coeff_to_string(const Z_mod& i) -> std::string: converts the coefficient to string
auto coeff_to_string(Q r) -> std::string: converts the coefficient to string
auto continued_fraction(int p, int q) -> std::vector<int>: computes the continued fraction of such that all coefficients have the same sign
auto gcd(int_coeff p, int_coeff q) -> int_coeff: gcd of and
auto is_id(const int_coeff& n) -> bool: true iff n is equal to +1
auto is_id(const Q& r) -> bool: true iff n is equal to +1
template<prime P> auto is_id(const Z_mod& n) -> bool: true iff n is equal to +1
auto operator!=(const Q& r1, const Q& r2) -> bool: true iff the two sides define the distinct rational numbers
auto operator*(const int_coeff& n1, const Q& r2) -> Q: left multiplication with integer
template<prime P> auto operator*(const int_coeff& n1, const Z_mod& n2) -> Z_mod: left multiplication with integer
auto operator*(const Q& r1, const int_coeff& n2) -> Q: right multiplication with integer
auto operator*(const Q& r1, const Q& r2) -> Q: multiplication
template<prime P> auto operator*(const Z_mod& n1, const int_coeff& n2) -> Z_mod: right multiplication with integer
template<prime P> auto operator*(const Z_mod& n1, const Z_mod& n2) -> Z_mod: multiplication
void operator*=(Q& r, const int_coeff& n): multiplication assignment with scalar
void operator*=(Q& r1, const Q& r2): multiplication assignment
template<prime P> void operator*=(Z_mod& n, const int_coeff& i): multiplication assignment with integer
template<prime P> void operator*=(Z_mod& n1, const Z_mod& n2): multiplication assignment
auto operator+(const Q& r1, const Q& r2) -> Q: addition
template<prime P> auto operator+(const Z_mod& n1, const Z_mod& n2) -> Z_mod: addition
void operator+=(Q& r1, const Q& r2): addition assignment
template<prime P> void operator+=(Z_mod& n1, const Z_mod& n2): addition assignment
auto operator-(const Q& r) -> Q: negative
template<prime P> auto operator-(const Z_mod& n) -> Z_mod: negative
auto operator/(const int_coeff& n1, const Q& r2) -> Q: division; gives error if n2 is 0.
template<prime P> auto operator/(const int_coeff& n1, const Z_mod& n2) -> Z_mod: division; gives error if n2 is 0.
auto operator<(const Q& r1, const Q& r2) -> bool: comparison operator
auto operator<<(std::ostream& out, const Q& r) -> std::ostream&: stream ouput format for Q
template<prime P> auto operator<<(std::ostream& out, const Z_mod& n) -> std::ostream&: stream ouput format for Z_mod
auto operator==(const Q& r1, const Q& r2) -> bool: true iff both sides define the same rational number
auto which_coeff(const int_coeff& i) -> std::string: returns the string Z
auto which_coeff(const Q& r) -> std::string: returns the string Q
template<prime P> auto which_coeff(const Z_mod& n) -> std::string: returns the string Z_mod
template<prime P> auto which_P(const Z_mod& n) -> prime: returns the prime