polynomials/lib.rs
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use core::num;
use std::ops::{Add, Mul};
fn modulo(a: i32, n: i32) -> i32 {
((a % n) + n) % n
}
fn mod_inverse(b: i64, p: i64) -> i64 {
let mut a = b;
let mut m = p;
let mut u = 1;
let mut v = 0;
while a != 0 {
let t = m / a;
m -= t * a;
std::mem::swap(&mut a, &mut m);
v -= t * u;
std::mem::swap(&mut u, &mut v);
}
(v + p) % p
}
/// Represents a single term in a polynomial, consisting of an exponent and a coefficient.
#[derive(Debug, Clone, Copy, PartialEq, Eq, PartialOrd, Ord)]
pub struct Monomial {
/// The exponent of the monomial.
pub exponent: u32,
/// The coefficient of the monomial.
pub coefficients: i64,
}
impl Monomial {
/// Creates a new `Monomial` with the given exponent and coefficient.
///
/// # Arguments
///
/// * `exponent` - The exponent of the monomial.
/// * `coefficients` - The coefficient of the monomial.
///
/// # Returns
///
/// A new `Monomial` instance.
pub fn new(exponent: u32, coefficients: i64) -> Monomial {
Monomial {
exponent,
coefficients,
}
}
/// Creates a default `Monomial` with an exponent of 0 and a coefficient of 0.0.
///
/// # Returns
///
/// A default `Monomial` instance.
pub fn default() -> Monomial {
Monomial {
exponent: 0,
coefficients: 0,
}
}
}
/// Represents a polynomial, which is a sum of monomials.
#[derive(Debug, Clone)]
pub struct UnivariatePolynomial {
/// The list of monomials that make up the polynomial.
monomials: Vec<Monomial>,
/// The degree of the polynomial, if known.
pub degree: Option<u32>,
pub prime_modulo: Option<u32>,
}
impl UnivariatePolynomial {
/// Adds a monomial to the polynomial. If a monomial with the same exponent already exists,
/// their coefficients are combined.
///
/// # Arguments
///
/// * `exponent` - The exponent of the monomial to add.
/// * `coefficients` - The coefficient of the monomial to add.
pub fn new(monomials: Vec<Monomial>, prime_modulo: u32) -> UnivariatePolynomial {
UnivariatePolynomial {
monomials,
degree: None,
prime_modulo: Some(prime_modulo),
}
}
/// Creates a default `Polynomial` with no monomials and no degree.
///
/// # Returns
///
/// A default `Polynomial` instance.
pub fn default() -> UnivariatePolynomial {
UnivariatePolynomial {
monomials: Vec::new(),
degree: None,
prime_modulo: Some(1),
}
}
pub fn set_prime_modulo(&mut self, prime_modulo: u32) {
self.prime_modulo = Some(prime_modulo);
}
/// Evaluates the polynomial at a given value of `x`.
///
/// # Arguments
///
/// * `x` - The value at which to evaluate the polynomial.
///
/// # Returns
///
/// The result of evaluating the polynomial at `x`.
pub fn evaluate(&self, x: i64) -> i64 {
let mut result: i64 = 0;
let n = self.monomials.len();
for i in 0..n {
result += self.monomials[i].coefficients * x.pow(self.monomials[i].exponent as u32);
}
return result;
}
/// Returns the degree of the polynomial.
///
/// # Returns
///
/// The degree of the polynomial, if known.
pub fn degree(&mut self) -> Option<u32> {
let n = self.monomials.len();
if self.degree.is_none() {
for i in 0..n {
if self.monomials[i].exponent > self.degree.unwrap_or(0) {
self.degree = Some(self.monomials[i].exponent);
}
}
return self.degree;
} else {
return self.degree;
}
}
/// Performs Lagrange interpolation to find a polynomial that passes through the given points.
///
/// # Arguments
///
/// * `x` - A vector of x-coordinates of the points.
/// * `y` - A vector of y-coordinates of the points.
///
/// # Returns
///
/// A `UnivariatePolynomial` that passes through the given points.
//add prime modulo
pub fn interpolate(x: Vec<i64>, y: Vec<i64>, p: u32) -> UnivariatePolynomial {
let n = x.len();
let mut result = UnivariatePolynomial::default();
result.set_prime_modulo(p);
for i in 0..n {
let mut denominator: i64 = 1;
let mut numerator = UnivariatePolynomial::new(vec![Monomial::new(0, 1)], p);
let mut a = y[i];
for j in 0..n {
if i != j {
let x_n = Monomial::new(1, 1); // x
let x_j =
Monomial::new(0, modulo((-x[j]).try_into().unwrap(), p as i32) as i64);
let temp_poly = UnivariatePolynomial::new(vec![x_n, x_j], p);
numerator = numerator * temp_poly;
denominator *= x[i] - x[j];
denominator = modulo(denominator as i32, p as i32) as i64;
}
}
a = a * mod_inverse(denominator, p as i64);
for monomial in &mut numerator.monomials {
monomial.coefficients = modulo((monomial.coefficients * a) as i32, p as i32) as i64;
}
result = result + numerator;
}
result
}
}
impl Mul for UnivariatePolynomial {
type Output = UnivariatePolynomial;
/// Multiplies two polynomials and returns the result.
///
/// # Arguments
///
/// * `p2` - The polynomial to multiply by.
///
/// # Returns
///
/// A new `Polynomial` representing the product of the two polynomials.
fn mul(self, p2: UnivariatePolynomial) -> Self {
let p1: Vec<Monomial> = self.monomials;
let p2: Vec<Monomial> = p2.monomials;
let p: u32 = self.prime_modulo.unwrap();
let mut polynomial: Vec<Monomial> = Vec::new();
for i in 0..p1.len() {
for j in 0..p2.len() {
let coefficients = modulo(
(modulo(p1[i].coefficients as i32, p as i32) as i64
* modulo(p2[j].coefficients as i32, p as i32) as i64)
as i32,
p as i32,
) as i64;
polynomial.push(Monomial {
coefficients: (coefficients % p as i64 + p as i64) % p as i64,
exponent: p1[i].exponent.wrapping_add(p2[j].exponent),
});
}
}
// Combine monomials with the same exponent
for i in 0..polynomial.len() {
let mut j = i + 1;
while j < polynomial.len() {
if polynomial[i].exponent == polynomial[j].exponent {
polynomial[i].coefficients =
polynomial[i].coefficients + polynomial[j].coefficients;
polynomial[i].coefficients =
modulo((polynomial[i].coefficients) as i32, p as i32) as i64;
polynomial.remove(j);
} else {
j += 1;
}
}
}
polynomial.sort();
UnivariatePolynomial {
monomials: polynomial,
degree: None,
prime_modulo: Some(p),
}
}
}
impl Add for UnivariatePolynomial {
type Output = UnivariatePolynomial;
/// Adds two polynomials and returns the result.
///
/// # Arguments
///
/// * `p2` - The polynomial to add.
///
/// # Returns
///
/// A new `Polynomial` representing the sum of the two polynomials.
fn add(self, p2: UnivariatePolynomial) -> Self {
let p1: Vec<Monomial> = self.monomials;
let p2: Vec<Monomial> = p2.monomials;
let mut polynomial: Vec<Monomial> = Vec::new();
let p: u32 = self.prime_modulo.unwrap();
polynomial = [p1, p2].concat();
// Combine monomials with the same exponent
for i in 0..polynomial.len() {
let mut j = i + 1;
while j < polynomial.len() {
if polynomial[i].exponent == polynomial[j].exponent {
polynomial[i].coefficients =
polynomial[i].coefficients + polynomial[j].coefficients;
polynomial[i].coefficients =
modulo((polynomial[i].coefficients) as i32, p as i32) as i64;
polynomial.remove(j);
} else {
j += 1;
}
}
}
polynomial.sort();
UnivariatePolynomial {
monomials: polynomial,
degree: None,
prime_modulo: Some(p),
}
}
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
/// Tests the `evaluate` method of the `Polynomial` struct.
fn test_evaluate() {
let default = UnivariatePolynomial::default();
let m1 = Monomial::new(2, 3);
let m2 = Monomial::new(1, 2);
let m3 = Monomial::new(0, 5);
let p = UnivariatePolynomial {
monomials: vec![m1, m2, m3],
..default
};
let result = p.evaluate(4);
assert_eq!(result, 61);
}
/// Tests the `degree` method of the `UnivariatePolynomial` struct.
#[test]
fn test_degree() {
let default = UnivariatePolynomial::default();
let m1 = Monomial::new(2, 3);
let m2 = Monomial::new(1, 2);
let mut p = UnivariatePolynomial {
monomials: vec![m1, m2],
..default
};
let result = p.degree();
assert_eq!(result, Some(2));
}
/// Tests the multiplication of two UnivariatePolynomials.
#[test]
fn test_multiplication() {
let default = UnivariatePolynomial::default();
let m1 = Monomial::new(2, 5);
let m2 = Monomial::new(1, -4);
let m5 = Monomial::new(0, 2);
let m3 = Monomial::new(3, 1);
let m4 = Monomial::new(2, -2);
let m6 = Monomial::new(0, 5);
let p1 = UnivariatePolynomial {
monomials: vec![m1, m2, m5],
prime_modulo: Some(6),
..default
};
let p2 = UnivariatePolynomial {
monomials: vec![m3, m4, m6],
..default
};
let result = p1 * p2;
assert_eq!(result.monomials[5].coefficients, 5);
assert_eq!(result.monomials[4].coefficients, 4);
assert_eq!(result.monomials[3].coefficients, 4);
assert_eq!(result.monomials[2].coefficients, 3);
assert_eq!(result.monomials[1].coefficients, 4);
assert_eq!(result.monomials[0].coefficients, 4);
assert_eq!(result.monomials[5].exponent, 5);
assert_eq!(result.monomials[4].exponent, 4);
assert_eq!(result.monomials[3].exponent, 3);
assert_eq!(result.monomials[2].exponent, 2);
assert_eq!(result.monomials[1].exponent, 1);
assert_eq!(result.monomials[0].exponent, 0);
}
/// Tests the Lagrange interpolation method.
#[test]
fn test_interpolate() {
let x = vec![0, -2, 2];
let y = vec![4, 1, 3];
let p = 5;
let result = UnivariatePolynomial::interpolate(x, y, p);
assert_eq!(result.monomials[0].coefficients, 4);
assert_eq!(result.monomials[0].exponent, 0);
assert_eq!(result.monomials[1].coefficients, 3);
assert_eq!(result.monomials[1].exponent, 1);
assert_eq!(result.monomials[2].coefficients, 2);
assert_eq!(result.monomials[2].exponent, 2);
}
}