1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
// 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::rational_polynomial::RationalPolynomial;
use core::mem::replace;
use malachite_base::num::basic::traits::{One, Zero};
use malachite_base::polynomial::{NthDerivative, NthDerivativeAssign};
use malachite_nz::integer_polynomial::IntegerPolynomial;
use malachite_nz::natural::Natural;
impl NthDerivative for RationalPolynomial {
type Output = Self;
/// Computes the $n$th derivative of a [`RationalPolynomial`], 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.
///
/// The numerator is differentiated, and since the falling factorials can share factors with the
/// denominator, the result is then brought back to lowest terms.
///
/// # 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_q::rational_polynomial::RationalPolynomial;
///
/// let p = RationalPolynomial::from_str("1/12*x^4+1/2*x^2+1").unwrap();
/// assert_eq!(p.clone().nth_derivative(2).to_string(), "x^2+1");
/// assert_eq!(
/// p.clone().nth_derivative(0).to_string(),
/// "1/12*x^4+1/2*x^2+1"
/// );
/// assert_eq!(p.nth_derivative(5), RationalPolynomial::ZERO);
/// ```
///
/// This is equivalent to `fmpq_poly_nth_derivative` from `fmpq_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 &RationalPolynomial {
type Output = RationalPolynomial;
/// Computes the $n$th derivative of a [`RationalPolynomial`], 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.
///
/// The numerator is differentiated, and since the falling factorials can share factors with the
/// denominator, the result is then brought back to lowest terms.
///
/// # 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_q::rational_polynomial::RationalPolynomial;
///
/// let p = RationalPolynomial::from_str("1/12*x^4+1/2*x^2+1").unwrap();
/// assert_eq!((&p).nth_derivative(2).to_string(), "x^2+1");
/// assert_eq!((&p).nth_derivative(0).to_string(), "1/12*x^4+1/2*x^2+1");
/// assert_eq!((&p).nth_derivative(5), RationalPolynomial::ZERO);
/// ```
///
/// This is equivalent to `fmpq_poly_nth_derivative` from `fmpq_poly/nth_derivative.c`, FLINT
/// 3.6.0.
fn nth_derivative(self, n: u64) -> RationalPolynomial {
if n == 0 {
return self.clone();
}
RationalPolynomial::canonicalize(
(&self.numerator).nth_derivative(n),
self.denominator.clone(),
)
}
}
impl NthDerivativeAssign for RationalPolynomial {
/// Replaces a [`RationalPolynomial`] 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.
///
/// The numerator is differentiated, and since the falling factorials can share factors with the
/// denominator, the result is then brought back to lowest terms.
///
/// # 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_q::rational_polynomial::RationalPolynomial;
///
/// let mut p = RationalPolynomial::from_str("1/12*x^4+1/2*x^2+1").unwrap();
/// p.nth_derivative_assign(2);
/// assert_eq!(p.to_string(), "x^2+1");
///
/// let mut p = RationalPolynomial::from_str("1/12*x^4+1/2*x^2+1").unwrap();
/// p.nth_derivative_assign(0);
/// assert_eq!(p.to_string(), "1/12*x^4+1/2*x^2+1");
///
/// let mut p = RationalPolynomial::from_str("1/12*x^4+1/2*x^2+1").unwrap();
/// p.nth_derivative_assign(5);
/// assert_eq!(p, RationalPolynomial::ZERO);
/// ```
///
/// This is equivalent to `fmpq_poly_nth_derivative` from `fmpq_poly/nth_derivative.c`, FLINT
/// 3.6.0.
fn nth_derivative_assign(&mut self, n: u64) {
if n == 0 {
return;
}
let mut numerator = replace(&mut self.numerator, IntegerPolynomial::ZERO);
numerator.nth_derivative_assign(n);
let denominator = replace(&mut self.denominator, Natural::ONE);
*self = Self::canonicalize(numerator, denominator);
}
}