Rational tutorial#
rational
is a class that builds upon integer
to provide
an implementation of rational numbers represented as a numerator-denominator
integer
pairs.
Like integer
, rational
is parametrised over
an integral value SSize
which represents the number of limbs that will be
stored directly within both the numerator and the denominator before resorting to dynamic memory allocation
(see the integer tutorial for a detailed explanation of the meaning
of the SSize
parameter).
The rational
class adopts the usual canonical representation of rational numbers:
numerator and denominator are coprime,
the denominator is always strictly positive.
This representation is usually enforced automatically by the rational
API (however,
a couple of low-level functions are available to access and mutate directly the numerator and the denominator
in performance-critical scenarios).
In addition to the features common to all mp++ classes, the rational
API provides
a few additional capabilities:
it is possible to construct a
rational
from an integral pair of arguments representing the numerator and the denominator:rat_t q1{3, 4}; // Construction from int numerator and denominator. rat_t q2{long(-6), short(-8)}; // Construction from mixed numerator and denominator types. assert(q1 == q2); // Construction canonicalises the rational.
it is possible to get const and mutable references to the
integer
objects representing the numerator and the denominator via themppp::rational::get_num()
,mppp::rational::get_den()
,mppp::rational::_get_num()
andmppp::rational::_get_den()
member functions. These accessors allow to use any function from theinteger
API on the numerator and on the denominator:rat_t q1{2, 5}; assert(q1.get_num() == 2); // Const access to numerator and denominator. assert(q1.get_den() == 5); q1._get_num() = 1; // Mutable access to numerator and denominator. q1._get_den() = 3; assert(q1 == rat_t{1, 3});
the
mppp::rational::canonicalise()
member function (and its free function counterpart) can be used to canonicalise arational
. This is normally not necessary, unless the numerator or the denominator have been changed manually via the mutable getters:rat_t q1{1, 6}; q1._get_num() = 9; // Change the numerator using the mutable getter. // WARNING: q1 = 9/6, not in canonical form any more! q1.canonicalise(); // Canonicalise q1. assert(q1 == rat_t{3, 2});
a few additional arithmetic and comparison functions are available;
a few exponentiation and number-theoretic functions are available;
like
integer
,rational
also provides a specialisation ofstd::hash
for use in the standard unordered containers.
Many of these features, which are documented in detail in the rational reference, are available in multiple overloads, often both as free and member functions.
Interacting with the GMP API#
Like integer
, rational
provides facilities for interfacing
with the GMP library. Currently, the interoperability is limited to
rational
objects being able to be constructed and assigned from both mpz_t
and mpq_t
objects:
mpz_t m;
mpz_init_set_si(m, -4); // Init an mpz_t with the value -4.
rat_t q1{m}; // Init a rat_t from the mpz_t.
assert(q1 == -4); // Verify that the value is correct.
mpq_t q;
mpq_init(q);
mpq_set_si(q, 3, 4); // Init an mpq_t with the value 3/4.
rat_t q2{q}; // Init a rat_t from the mpq_t.
assert((q2 == rat_t{3, 4})); // Verify that the value is correct.
rat_t q3;
q3 = m; // Assign the mpz_t to another rat_t.
assert(q3 == -4); // Verify that the value is correct.
rat_t q4;
q4 = q; // Assign the mpq_t to another rat_t.
assert((q4 == rat_t{3, 4})); // Verify that the value is correct.
mpz_clear(m); // Clear the mpz_t.
mpq_clear(q); // Clear the mpq_t.
User-defined literals#
Added in version 0.19.
User-defined literals are available to construct
mppp::rational
instances with 1, 2 and 3
limbs of static storage. The literals
are defined within
the inline namespace mppp::literals
, and they support
binary, octal, decimal and hexadecimal representations:
using namespace mppp::literals;
auto q1 = 123_q1; // q1 has 1 limb of static storage,
// and it contains the value 123.
auto q2 = -0b10011_q2; // q2 has 2 limbs of static storage,
// and it contains the value -19
// (-10011 in base 2).
auto q3 = 0146_q1; // q3 has 1 limb of static storage,
// and it contains the value 102
// (146 in base 8).
auto q4 = 0xfe45_q3; // q4 has 3 limbs of static storage,
// and it contains the value 65093
// (fe45 in base 16).
Note that, due to language limitations, only values with unitary denominator can be constructed via these literals.