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pub mod add; pub mod mul; pub mod rem; pub mod sub; /// A wrapper struct around a `Vec<isize>` which treats the entries of the `Vec` as the coefficients /// of a polynomial. /// /// # Examples /// ``` /// use polynomint::{Polynomial, poly}; /// /// let quadratic = poly![1, 2, 1]; // x^2 + 2x + 1 /// let linear = poly![-6, 1]; // x - 6 /// assert_eq!(&quadratic * 5, poly![5, 10, 5]); /// assert_eq!(&quadratic * &linear, poly![-6, -11, -4, 1]); /// /// let mut resultant = &quadratic * &linear; /// resultant %= 5; /// assert_eq!(resultant, poly![-1, -1, -4, 1]); /// /// let resultant2 = (&quadratic * &linear).rem_euclid(5); /// assert_eq!(resultant2, poly![4, 4, 1, 1]); /// ``` #[derive(Debug, Clone, PartialEq, Eq)] pub struct Polynomial { coeffs: Vec<isize>, } impl Polynomial { /// Creates a polynomial with the given coefficients, stored in increasing order, with /// any trailing (higher-degree) zeroes removed. /// /// # Examples /// ``` /// use polynomint::Polynomial; /// /// let quadratic = Polynomial::new(vec![1, 2, 3]); // 3x^2 + 2x + 1 /// let cubic = Polynomial::new(vec![8, 12, 6, 1]); // x^3 + 6x^2 + 12x + 8 /// ``` pub fn new(coeffs: Vec<isize>) -> Self { let mut output = Self { coeffs }; output.reduce(); output } /// Creates the zero polynomial, which is stored internally as an empty vector. pub fn zero() -> Self { Self { coeffs: Vec::new() } } /// Creates a constant polynomial with coefficient equal to the argument passed; /// if the argument passed is zero, it is stored internally as an empty vector /// to match `zero()`. pub fn constant(i: isize) -> Self { if i == 0 { Self::zero() } else { Self { coeffs: vec![i] } } } /// Gives the highest power which has a nonzero coefficient; constants are degree zero, except /// the constant polynomial 0, which has degree -1. /// /// # Examples /// ``` /// use polynomint::{Polynomial, poly}; /// /// let zero = Polynomial::zero(); /// let alt_zero = Polynomial::constant(0); /// let three = Polynomial::constant(3); /// let classic = poly![1, 2, 1, 0, 0]; // x^2 + 2x + 1 /// /// assert_eq!(zero.degree(), -1); /// assert_eq!(alt_zero.degree(), -1); /// assert_eq!(three.degree(), 0); /// assert_eq!(classic.degree(), 2); /// ``` pub fn degree(&self) -> isize { (self.coeffs.len() as isize) - 1 } /// Gives a new polynomial equal to the old one times x. /// /// # Examples /// ``` /// use polynomint::{Polynomial, poly}; /// /// let first = poly![1, 2, 3]; /// let second = first.times_x(); /// /// assert_eq!(second, poly![0, 1, 2, 3]); /// ``` pub fn times_x(&self) -> Self { let mut coeffs = vec![0]; coeffs.append(&mut self.coeffs.clone()); Self { coeffs } } /// Gives a new polynomial equal to the remainder of the old one when taken /// modulo `n`. /// /// # Examples /// ``` /// use polynomint::{Polynomial, poly}; /// /// let poly = poly![6, -5, 3, -7, 4]; /// assert_eq!(poly.rem_euclid(2), poly![0, 1, 1, 1]); /// assert_eq!(poly.rem_euclid(4), poly![2, 3, 3, 1]); /// assert_eq!(poly.rem_euclid(5), poly![1, 0, 3, 3, 4]); /// ``` pub fn rem_euclid(&self, n: isize) -> Self { if self.is_zero() { Polynomial::zero() } else { let mut coeffs = self.coeffs.clone(); for i in 0..=self.degree() { coeffs[i as usize] = coeffs[i as usize].rem_euclid(n); } let mut output = Polynomial { coeffs }; output.reduce(); output } } /// Checks whether a polynomial is the zero polynomial. /// /// # Examples /// ``` /// use polynomint::{Polynomial, poly}; /// /// let zero = Polynomial::zero(); /// let also_zero = Polynomial::constant(0); /// let yet_again_zero = poly![0, 0, 0, 0]; /// let even_more_zero = poly![1, 2, 1] - poly![1, 2, 1]; /// let not_zero = poly![0, 1]; /// /// assert!(zero.is_zero()); /// assert!(also_zero.is_zero()); /// assert!(yet_again_zero.is_zero()); /// assert!(even_more_zero.is_zero()); /// assert!(!not_zero.is_zero()); /// ``` pub fn is_zero(&self) -> bool { self.degree() == -1 } /// Creates a new polynomial which is the derivative of the old one. /// /// # Examples /// ``` /// use polynomint::{Polynomial, poly}; /// /// let poly1 = poly![1, -2, 5, 4]; // 4x^3 + 5x^2 - 2x + 1 /// assert_eq!(poly1.derivative(), poly![-2, 10, 12]); // deriv. is 12x^2 + 10x - 2 /// let poly2 = poly![192, 3, -4, -9, 0, 38]; // 38x^5 - 9x^3 - 4x^2 + 3x + 192 /// assert_eq!(poly2.derivative(), poly![3, -8, -27, 0, 190]); // deriv. is 190x^4 - 27x^2 - 8x + 3 /// ``` pub fn derivative(&self) -> Self { if self.degree() <= 0 { Self::zero() } else { let mut coeffs = Vec::new(); for i in 0..self.degree() { coeffs.push((i + 1) * self.coeffs[i as usize + 1]); } let mut output = Self { coeffs }; output.reduce(); output } } /// Removes trailing zeroes from a polynomial. Used to make sure the API only exposes /// polynomials with no stored zeroes of higher-order, both to keep them as lightweight /// as possible and because this invariant is taken advantage of by functions like /// degree(). fn reduce(&mut self) { while self.coeffs.last() == Some(&0) { self.coeffs.pop(); } } } impl std::fmt::Display for Polynomial { fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { let mut s = String::new(); // if our polynomial is zero, the big-ass else block doesn't do anything, so // we just separately handle that case if self.is_zero() { s += &format!("{}", 0); } else { // this flag is basically "have we written a term yet", so we know whether // to write plus/minus signs and to treat minus signs as operations rather // than as prefixes let mut plus_flag = false; for (n, &i) in self.coeffs.iter().enumerate().rev() { // only display terms with nonzero coefficients if i != 0 { // if we're past the first term, print the appropriate operation // sign first if plus_flag { if i < 0 { s += " - "; } else { s += " + " } } // if our term is constant, if n == 0 { // just display that constant, or its absolute value if we already // wrote a minus sign s += &format!("{}", if plus_flag { i.abs() } else { i }); // if our term is linear, } else if n == 1 { // if it's 1, just write "x"; if i == 1 { s += "x"; // if it's -1, and if we're writing the first term, put a minus sign // in front } else if i == -1 { s += &format!("{}x", if plus_flag { "" } else { "-" }); // otherwise, just display the coefficient, or its absolute value // if we already wrote a minus sign } else { s += &format!("{}x", if plus_flag { i.abs() } else { i }); } // rest of cases as above, but with the powers being displayed as well } else if i == 1 { s += &format!("x^{}", n); } else if i == -1 { s += &format!("{}x^{}", if plus_flag { "" } else { "-" }, n); } else { s += &format!("{}x^{}", if plus_flag { i.abs() } else { i }, n); } plus_flag = true; } } } write!(f, "{}", s) } } /// A convenience macro for writing polynomials; essentially a wrapper around `vec![...]`. #[macro_export] macro_rules! poly { () => ( Polynomial::zero(); ); ($($x:expr),*) => ( Polynomial::new(vec![$($x),*]); ) } #[cfg(test)] mod tests { use crate::{poly, Polynomial}; #[test] fn it_works() { let mut quadratic = poly![1, 2, 1]; // x^2 + 2x + 1 let linear = poly![-6, 1]; // x - 6 assert_eq!(&quadratic + &linear, poly![-5, 3, 1]); assert_eq!(&quadratic - &linear, poly![7, 1, 1]); assert_eq!(&quadratic * &linear, poly![-6, -11, -4, 1]); quadratic -= &linear; assert_eq!(quadratic, poly![7, 1, 1]); quadratic += &linear; assert_eq!(quadratic, poly![1, 2, 1]); quadratic *= &linear; assert_eq!(quadratic, poly![-6, -11, -4, 1]); assert_eq!(quadratic.derivative(), poly![-11, -8, 3]); let mut another = poly![1, 3, 3, 1]; // x^3 + 3x^2 + 3x + 1 let pair = poly![-5, 4, 2]; // 2x^2 + 4x - 5 assert_eq!(&another + &pair, poly![-4, 7, 5, 1]); assert_eq!(&another - &pair, poly![6, -1, 1, 1]); assert_eq!(&another * &pair, poly![-5, -11, -1, 13, 10, 2]); another -= &pair; assert_eq!(another, poly![6, -1, 1, 1]); another += &pair; assert_eq!(another, poly![1, 3, 3, 1]); another *= &pair; assert_eq!(another, poly![-5, -11, -1, 13, 10, 2]); assert_eq!(another.derivative(), poly![-11, -2, 39, 40, 10]); } }