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// Copyright © 2026 Mikhail Hogrefe
//
// Uses code adopted from the FLINT Library.
//
// Copyright © 2020 Daniel Schultz
//
// This file is part of Malachite.
//
// Malachite is free software: you can redistribute it and/or modify it under the terms of the GNU
// Lesser General Public License (LGPL) as published by the Free Software Foundation; either version
// 3 of the License, or (at your option) any later version. See <https://www.gnu.org/licenses/>.
use crate::Rational;
use malachite_base::num::arithmetic::traits::{DivExact, ExtendedGcd, Gcd};
use malachite_base::num::basic::traits::Zero;
use malachite_nz::integer::Integer;
// Each impl below computes the same quantities, differing only in which operands are consumed. The
// quotients are those of _fmpq_gcd_cofactors from fmpq/gcd_cofactors.c, FLINT 3.6.0, computed the
// same way: with a = ±p/q and b = ±r/s canonical, g = gcd(p, r)/lcm(q, s), a/g = ±(p/gcd(p,
// r))(s/gcd(q, s)), and b/g = ±(r/gcd(p, r))(q/gcd(q, s)).
impl ExtendedGcd for Rational {
type Gcd = Self;
type Cofactor = Integer;
/// Computes the GCD (greatest common divisor) of two [`Rational`]s $a$ and $b$, and also the
/// integer coefficients $x$ and $y$ in Bézout's identity $ax+by=\gcd(a,b)$. Both [`Rational`]s
/// are taken by value.
///
/// The integer combinations of two rationals are exactly the integer multiples of their GCD, so
/// integer Bézout coefficients exist. They are the coefficients of the integer
/// [`extended_gcd`](malachite_base::num::arithmetic::traits::ExtendedGcd) applied to the
/// coprime integer quotients $a/g$ and $b/g$, and inherit its normalization.
///
/// $f(a, b) = (g, x, y)$, where $g = \gcd(a, b) \geq 0$ and $ax + by = g$, and $f(0, 0) = (0,
/// 0, 0)$.
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::ExtendedGcd;
/// use malachite_base::strings::ToDebugString;
/// use malachite_q::Rational;
///
/// // -1 * 2/3 + -1 * -3/4 = 1/12
/// assert_eq!(
/// Rational::from_signeds(2, 3)
/// .extended_gcd(Rational::from_signeds(-3, 4))
/// .to_debug_string(),
/// "(1/12, -1, -1)"
/// );
/// ```
///
/// This is fmpq_gcd_cofactors from fmpq/gcd_cofactors.c, FLINT 3.6.0, except that FLINT returns
/// the quotients $a/g$ and $b/g$ themselves, while this function takes one more step and
/// returns Bézout coefficients for them.
fn extended_gcd(self, other: Self) -> (Self, Integer, Integer) {
let ng = (&self.numerator).gcd(&other.numerator);
if ng == 0u32 {
// Both numerators are zero, so both rationals are.
return (Self::ZERO, Integer::ZERO, Integer::ZERO);
}
let dg = (&self.denominator).gcd(&other.denominator);
let dxbar = (&self.denominator).div_exact(&dg);
let dybar = other.denominator.div_exact(dg);
let x_over_g =
Integer::from_sign_and_abs(self.sign, self.numerator.div_exact(&ng) * &dybar);
let y_over_g =
Integer::from_sign_and_abs(other.sign, other.numerator.div_exact(&ng) * dxbar);
let g = Self {
sign: true,
denominator: self.denominator * dybar,
numerator: ng,
};
let (one, u, v) = x_over_g.extended_gcd(y_over_g);
debug_assert!(one == 1u32);
(g, u, v)
}
}
impl ExtendedGcd<&Self> for Rational {
type Gcd = Self;
type Cofactor = Integer;
/// Computes the GCD (greatest common divisor) of two [`Rational`]s $a$ and $b$, and also the
/// integer coefficients $x$ and $y$ in Bézout's identity $ax+by=\gcd(a,b)$. The first
/// [`Rational`] is taken by value and the second by reference.
///
/// The integer combinations of two rationals are exactly the integer multiples of their GCD, so
/// integer Bézout coefficients exist. They are the coefficients of the integer
/// [`extended_gcd`](malachite_base::num::arithmetic::traits::ExtendedGcd) applied to the
/// coprime integer quotients $a/g$ and $b/g$, and inherit its normalization.
///
/// $f(a, b) = (g, x, y)$, where $g = \gcd(a, b) \geq 0$ and $ax + by = g$, and $f(0, 0) = (0,
/// 0, 0)$.
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::ExtendedGcd;
/// use malachite_base::strings::ToDebugString;
/// use malachite_q::Rational;
///
/// // -1 * 2/3 + -1 * -3/4 = 1/12
/// assert_eq!(
/// Rational::from_signeds(2, 3)
/// .extended_gcd(&Rational::from_signeds(-3, 4))
/// .to_debug_string(),
/// "(1/12, -1, -1)"
/// );
/// ```
///
/// This is fmpq_gcd_cofactors from fmpq/gcd_cofactors.c, FLINT 3.6.0, except that FLINT returns
/// the quotients $a/g$ and $b/g$ themselves, while this function takes one more step and
/// returns Bézout coefficients for them.
fn extended_gcd(self, other: &Self) -> (Self, Integer, Integer) {
let ng = (&self.numerator).gcd(&other.numerator);
if ng == 0u32 {
// Both numerators are zero, so both rationals are.
return (Self::ZERO, Integer::ZERO, Integer::ZERO);
}
let dg = (&self.denominator).gcd(&other.denominator);
let dxbar = (&self.denominator).div_exact(&dg);
let dybar = (&other.denominator).div_exact(dg);
let x_over_g =
Integer::from_sign_and_abs(self.sign, self.numerator.div_exact(&ng) * &dybar);
let y_over_g =
Integer::from_sign_and_abs(other.sign, (&other.numerator).div_exact(&ng) * dxbar);
let g = Self {
sign: true,
denominator: self.denominator * dybar,
numerator: ng,
};
let (one, u, v) = x_over_g.extended_gcd(y_over_g);
debug_assert!(one == 1u32);
(g, u, v)
}
}
impl ExtendedGcd<Rational> for &Rational {
type Gcd = Rational;
type Cofactor = Integer;
/// Computes the GCD (greatest common divisor) of two [`Rational`]s $a$ and $b$, and also the
/// integer coefficients $x$ and $y$ in Bézout's identity $ax+by=\gcd(a,b)$. The first
/// [`Rational`] is taken by reference and the second by value.
///
/// The integer combinations of two rationals are exactly the integer multiples of their GCD, so
/// integer Bézout coefficients exist. They are the coefficients of the integer
/// [`extended_gcd`](malachite_base::num::arithmetic::traits::ExtendedGcd) applied to the
/// coprime integer quotients $a/g$ and $b/g$, and inherit its normalization.
///
/// $f(a, b) = (g, x, y)$, where $g = \gcd(a, b) \geq 0$ and $ax + by = g$, and $f(0, 0) = (0,
/// 0, 0)$.
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::ExtendedGcd;
/// use malachite_base::strings::ToDebugString;
/// use malachite_q::Rational;
///
/// // -1 * 2/3 + -1 * -3/4 = 1/12
/// assert_eq!(
/// (&Rational::from_signeds(2, 3))
/// .extended_gcd(Rational::from_signeds(-3, 4))
/// .to_debug_string(),
/// "(1/12, -1, -1)"
/// );
/// ```
///
/// This is fmpq_gcd_cofactors from fmpq/gcd_cofactors.c, FLINT 3.6.0, except that FLINT returns
/// the quotients $a/g$ and $b/g$ themselves, while this function takes one more step and
/// returns Bézout coefficients for them.
fn extended_gcd(self, other: Rational) -> (Rational, Integer, Integer) {
let ng = (&self.numerator).gcd(&other.numerator);
if ng == 0u32 {
// Both numerators are zero, so both rationals are.
return (Rational::ZERO, Integer::ZERO, Integer::ZERO);
}
let dg = (&self.denominator).gcd(&other.denominator);
let dxbar = (&self.denominator).div_exact(&dg);
let dybar = other.denominator.div_exact(dg);
let x_over_g =
Integer::from_sign_and_abs(self.sign, (&self.numerator).div_exact(&ng) * &dybar);
let y_over_g =
Integer::from_sign_and_abs(other.sign, other.numerator.div_exact(&ng) * dxbar);
let g = Rational {
sign: true,
denominator: &self.denominator * dybar,
numerator: ng,
};
let (one, u, v) = x_over_g.extended_gcd(y_over_g);
debug_assert!(one == 1u32);
(g, u, v)
}
}
impl ExtendedGcd<&Rational> for &Rational {
type Gcd = Rational;
type Cofactor = Integer;
/// Computes the GCD (greatest common divisor) of two [`Rational`]s $a$ and $b$, and also the
/// integer coefficients $x$ and $y$ in Bézout's identity $ax+by=\gcd(a,b)$. Both [`Rational`]s
/// are taken by reference.
///
/// The integer combinations of two rationals are exactly the integer multiples of their GCD, so
/// integer Bézout coefficients exist. They are the coefficients of the integer
/// [`extended_gcd`](malachite_base::num::arithmetic::traits::ExtendedGcd) applied to the
/// coprime integer quotients $a/g$ and $b/g$, and inherit its normalization.
///
/// $f(a, b) = (g, x, y)$, where $g = \gcd(a, b) \geq 0$ and $ax + by = g$, and $f(0, 0) = (0,
/// 0, 0)$.
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::ExtendedGcd;
/// use malachite_base::strings::ToDebugString;
/// use malachite_q::Rational;
///
/// // -1 * 2/3 + -1 * -3/4 = 1/12
/// assert_eq!(
/// (&Rational::from_signeds(2, 3))
/// .extended_gcd(&Rational::from_signeds(-3, 4))
/// .to_debug_string(),
/// "(1/12, -1, -1)"
/// );
/// ```
///
/// This is fmpq_gcd_cofactors from fmpq/gcd_cofactors.c, FLINT 3.6.0, except that FLINT returns
/// the quotients $a/g$ and $b/g$ themselves, while this function takes one more step and
/// returns Bézout coefficients for them.
fn extended_gcd(self, other: &Rational) -> (Rational, Integer, Integer) {
let ng = (&self.numerator).gcd(&other.numerator);
if ng == 0u32 {
// Both numerators are zero, so both rationals are.
return (Rational::ZERO, Integer::ZERO, Integer::ZERO);
}
let dg = (&self.denominator).gcd(&other.denominator);
let dxbar = (&self.denominator).div_exact(&dg);
let dybar = (&other.denominator).div_exact(dg);
let x_over_g =
Integer::from_sign_and_abs(self.sign, (&self.numerator).div_exact(&ng) * &dybar);
let y_over_g =
Integer::from_sign_and_abs(other.sign, (&other.numerator).div_exact(&ng) * dxbar);
let g = Rational {
sign: true,
denominator: &self.denominator * dybar,
numerator: ng,
};
let (one, u, v) = x_over_g.extended_gcd(y_over_g);
debug_assert!(one == 1u32);
(g, u, v)
}
}