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// Copyright © 2026 Mikhail Hogrefe
//
// 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::natural::Natural;
use crate::natural_polynomial::NaturalPolynomial;
use alloc::vec::Vec;
use malachite_base::num::arithmetic::traits::{DivExactAssign, Factorial};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_base::polynomial::{NthDerivative, NthDerivativeAssign};
// The falling factorials i(i - 1)...(i - n + 1), for i from n: the first is n!, and each is carried
// to the next by dividing exactly by i - n and multiplying by i, both of which are small.
fn falling_factorials(n: usize) -> impl FnMut(usize) -> Natural {
let mut f = Natural::factorial(u64::exact_from(n));
move |i| {
if i > n {
f.div_exact_assign(Natural::from(i - n));
f *= Natural::from(i);
}
f.clone()
}
}
// The coefficients of the nth derivative, for n at least 1 and less than `xs.len()`: the
// coefficient of x^i, for i at least n, times i(i - 1)...(i - n + 1), moved to x^(i - n). The
// leading coefficient times its falling factorial is nonzero, so the result is normalized.
//
// This is equivalent to `_fmpz_poly_nth_derivative` from `fmpz_poly/nth_derivative.c`, FLINT 3.6.0.
fn nth_derivative_ref(xs: &[Natural], n: usize) -> Vec<Natural> {
let mut falling = falling_factorials(n);
xs.iter()
.enumerate()
.skip(n)
.map(|(i, c)| c * falling(i))
.collect()
}
fn nth_derivative_in_place(xs: &mut Vec<Natural>, n: u64) {
if n == 0 {
return;
}
if u64::exact_from(xs.len()) <= n {
xs.clear();
return;
}
let n = usize::exact_from(n);
xs.drain(..n);
let mut falling = falling_factorials(n);
for (j, c) in xs.iter_mut().enumerate() {
*c *= falling(j + n);
}
}
impl NthDerivative for NaturalPolynomial {
type Output = Self;
/// Computes the $n$th derivative of a [`NaturalPolynomial`], taking it by value.
///
/// $$
/// f(p, n) = p^{(n)} = \sum_{i=n}^d i^{\underline n}a_ix^{i-n}.
/// $$
///
/// Here $i^{\underline n} = i(i-1)\cdots(i-n+1)$ is a falling factorial, and $d$ is the degree.
/// The zeroth derivative is the polynomial itself, and a polynomial of degree less than $n$ has
/// $n$th derivative zero.
///
/// # Worst-case complexity
/// $T(b, k) = O(k(b + k \log k))$
///
/// $M(b, k) = O(b + k^2 \log k)$
///
/// where $T$ is time, $M$ is additional memory, $b$ is the total number of bits of the
/// coefficients, and $k$ is `self.len()`.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::num::basic::traits::Zero;
/// use malachite_base::polynomial::NthDerivative;
/// use malachite_nz::natural_polynomial::NaturalPolynomial;
///
/// let p = NaturalPolynomial::from_str("x^4+3*x^3+2*x+5").unwrap();
/// assert_eq!(p.clone().nth_derivative(2).to_string(), "12*x^2+18*x");
/// assert_eq!(p.clone().nth_derivative(0).to_string(), "x^4+3*x^3+2*x+5");
/// assert_eq!(p.nth_derivative(5), NaturalPolynomial::ZERO);
/// ```
///
/// This is equivalent to `fmpz_poly_nth_derivative` from `fmpz_poly/nth_derivative.c`, FLINT
/// 3.6.0.
#[inline]
fn nth_derivative(mut self, n: u64) -> Self {
self.nth_derivative_assign(n);
self
}
}
impl NthDerivative for &NaturalPolynomial {
type Output = NaturalPolynomial;
/// Computes the $n$th derivative of a [`NaturalPolynomial`], taking it by reference.
///
/// $$
/// f(p, n) = p^{(n)} = \sum_{i=n}^d i^{\underline n}a_ix^{i-n}.
/// $$
///
/// Here $i^{\underline n} = i(i-1)\cdots(i-n+1)$ is a falling factorial, and $d$ is the degree.
/// The zeroth derivative is the polynomial itself, and a polynomial of degree less than $n$ has
/// $n$th derivative zero.
///
/// # Worst-case complexity
/// $T(b, k) = O(k(b + k \log k))$
///
/// $M(b, k) = O(b + k^2 \log k)$
///
/// where $T$ is time, $M$ is additional memory, $b$ is the total number of bits of the
/// coefficients, and $k$ is `self.len()`.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::num::basic::traits::Zero;
/// use malachite_base::polynomial::NthDerivative;
/// use malachite_nz::natural_polynomial::NaturalPolynomial;
///
/// let p = NaturalPolynomial::from_str("x^4+3*x^3+2*x+5").unwrap();
/// assert_eq!((&p).nth_derivative(2).to_string(), "12*x^2+18*x");
/// assert_eq!((&p).nth_derivative(0).to_string(), "x^4+3*x^3+2*x+5");
/// assert_eq!((&p).nth_derivative(5), NaturalPolynomial::ZERO);
/// ```
///
/// This is equivalent to `fmpz_poly_nth_derivative` from `fmpz_poly/nth_derivative.c`, FLINT
/// 3.6.0.
fn nth_derivative(self, n: u64) -> NaturalPolynomial {
if n == 0 {
return self.clone();
}
if u64::exact_from(self.coefficients.len()) <= n {
return NaturalPolynomial {
coefficients: Vec::new(),
};
}
NaturalPolynomial {
coefficients: nth_derivative_ref(&self.coefficients, usize::exact_from(n)),
}
}
}
impl NthDerivativeAssign for NaturalPolynomial {
/// Replaces a [`NaturalPolynomial`] with its $n$th derivative.
///
/// $$
/// p \gets p^{(n)} = \sum_{i=n}^d i^{\underline n}a_ix^{i-n}.
/// $$
///
/// Here $i^{\underline n} = i(i-1)\cdots(i-n+1)$ is a falling factorial, and $d$ is the degree.
/// The zeroth derivative is the polynomial itself, and a polynomial of degree less than $n$ has
/// $n$th derivative zero.
///
/// # Worst-case complexity
/// $T(b, k) = O(k(b + k \log k))$
///
/// $M(b, k) = O(b + k^2 \log k)$
///
/// where $T$ is time, $M$ is additional memory, $b$ is the total number of bits of the
/// coefficients, and $k$ is `self.len()`.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::num::basic::traits::Zero;
/// use malachite_base::polynomial::NthDerivativeAssign;
/// use malachite_nz::natural_polynomial::NaturalPolynomial;
///
/// let mut p = NaturalPolynomial::from_str("x^4+3*x^3+2*x+5").unwrap();
/// p.nth_derivative_assign(2);
/// assert_eq!(p.to_string(), "12*x^2+18*x");
///
/// let mut p = NaturalPolynomial::from_str("x^4+3*x^3+2*x+5").unwrap();
/// p.nth_derivative_assign(0);
/// assert_eq!(p.to_string(), "x^4+3*x^3+2*x+5");
///
/// let mut p = NaturalPolynomial::from_str("x^4+3*x^3+2*x+5").unwrap();
/// p.nth_derivative_assign(5);
/// assert_eq!(p, NaturalPolynomial::ZERO);
/// ```
///
/// This is equivalent to `fmpz_poly_nth_derivative` from `fmpz_poly/nth_derivative.c`, FLINT
/// 3.6.0.
#[inline]
fn nth_derivative_assign(&mut self, n: u64) {
nth_derivative_in_place(&mut self.coefficients, n);
}
}