bacon_sci_1/polynomial/

mod.rs

use crate::roots::newton_polynomial;
use nalgebra::{ComplexField, RealField};
use num_complex::Complex;
use num_traits::{FromPrimitive, One, Zero};
use std::collections::VecDeque;
use std::{any::TypeId, f64, iter::FromIterator, ops};

/// Polynomial on a ComplexField.
#[derive(Debug, Clone)]
#[cfg_attr(feature = "serialize", derive(Serialize, Deserialize))]
pub struct Polynomial<N: ComplexField> {
    // Index 0 is constant, 1 is linear, etc.
    coefficients: Vec<N>,
    tolerance: <N as ComplexField>::RealField,
}

#[macro_export]
macro_rules! polynomial {
  ( $( $x:expr ),* ) => {
    $crate::polynomial::Polynomial::from_slice(&[$($x),*])
  }
}

impl<N: ComplexField> Polynomial<N> {
    /// Returns the zero polynomial on a given field
    pub fn new() -> Self {
        Polynomial {
            coefficients: vec![N::from_f64(0.0).unwrap()],
            tolerance: N::RealField::from_f64(1e-10).unwrap(),
        }
    }

    pub fn with_tolerance(tolerance: <N as ComplexField>::RealField) -> Result<Self, String> {
        if !tolerance.is_sign_positive() {
            return Err("Polynomial with_tolerance: Tolerance must be positive".to_owned());
        }
        Ok(Polynomial {
            coefficients: vec![N::from_f64(0.0).unwrap()],
            tolerance,
        })
    }

    /// Returns the zero polynomial on a given field with preallocated memory
    pub fn with_capacity(capacity: usize) -> Self {
        let mut coefficients = Vec::with_capacity(capacity);
        coefficients.push(N::zero());
        coefficients.iter().copied().collect()
    }

    /// Create a polynomial from a slice, with the first element of the slice being the highest power
    pub fn from_slice(data: &[N]) -> Self {
        if data.is_empty() {
            return Polynomial {
                coefficients: vec![N::zero()],
                tolerance: N::RealField::from_f64(1e-10).unwrap(),
            };
        }
        Polynomial {
            coefficients: data.iter().rev().copied().collect(),
            tolerance: N::RealField::from_f64(1e-10).unwrap(),
        }
    }

    pub fn set_tolerance(
        &mut self,
        tolerance: <N as ComplexField>::RealField,
    ) -> Result<(), String> {
        if !tolerance.is_sign_positive() {
            return Err("Polynomial set_tolerance: tolerance must be positive".to_owned());
        }

        self.tolerance = tolerance;
        Ok(())
    }

    pub fn get_tolerance(&self) -> <N as ComplexField>::RealField {
        self.tolerance
    }

    /// Get the order of the polynomial
    pub fn order(&self) -> usize {
        self.coefficients.len() - 1
    }

    /// Returns the coefficients in the correct order to recreate the polynomial with Polynomial::from_slice(data: &[N]);
    pub fn get_coefficients(&self) -> Vec<N> {
        let mut cln = self.coefficients.clone();
        cln.reverse();
        cln
    }

    /// Get the coefficient of a power
    pub fn get_coefficient(&self, ind: usize) -> N {
        if ind >= self.coefficients.len() {
            N::zero()
        } else {
            self.coefficients[ind]
        }
    }

    /// Make a polynomial complex
    pub fn make_complex(&self) -> Polynomial<Complex<<N as ComplexField>::RealField>> {
        let mut coefficients: Vec<Complex<N::RealField>> =
            Vec::with_capacity(self.coefficients.len());
        for val in &self.coefficients {
            coefficients.push(Complex::<N::RealField>::new(val.real(), val.imaginary()));
        }
        Polynomial {
            coefficients,
            tolerance: self.tolerance,
        }
    }

    /// Evaluate a polynomial at a value
    pub fn evaluate(&self, x: N) -> N {
        let mut acc = *self.coefficients.last().unwrap();
        for val in self.coefficients.iter().rev().skip(1) {
            acc *= x;
            acc += *val;
        }

        acc
    }

    /// Evaluate a polynomial and its derivative at a value
    pub fn evaluate_derivative(&self, x: N) -> (N, N) {
        if self.coefficients.len() == 1 {
            return (self.coefficients[0], N::zero());
        }
        // Start with biggest coefficients
        let mut acc_eval = *self.coefficients.last().unwrap();
        let mut acc_deriv = *self.coefficients.last().unwrap();
        // For every coefficient except the constant and largest
        for val in self.coefficients.iter().skip(1).rev().skip(1) {
            acc_eval = acc_eval * x + *val;
            acc_deriv = acc_deriv * x + acc_eval;
        }
        // Do the constant for the polynomial evaluation
        acc_eval = x * acc_eval + self.coefficients[0];

        (acc_eval, acc_deriv)
    }

    /// Set a coefficient of a power in the polynomial
    pub fn set_coefficient(&mut self, power: u32, coefficient: N) {
        while (power + 1) > self.coefficients.len() as u32 {
            self.coefficients.push(N::from_f64(0.0).unwrap());
        }
        self.coefficients[power as usize] = coefficient;
    }

    /// Remove the coefficient of a power in the polynomial
    pub fn purge_coefficient(&mut self, power: usize) {
        match self.coefficients.len() {
            len if len == power && len != 1 => {
                self.coefficients.pop();
            }
            _ => {
                self.coefficients[power] = N::from_f64(0.0).unwrap();
            }
        };
    }

    /// Remove all leading 0 coefficients
    pub fn purge_leading(&mut self) {
        while self.coefficients.len() > 1
            && self.coefficients.last().unwrap().real().abs() <= self.tolerance
            && self.coefficients.last().unwrap().imaginary().abs() <= self.tolerance
        {
            self.coefficients.pop();
        }
    }

    /// Get the derivative of the polynomial
    pub fn derivative(&self) -> Self {
        if self.coefficients.len() == 1 {
            return Polynomial {
                coefficients: vec![N::from_f64(0.0).unwrap()],
                tolerance: self.tolerance,
            };
        }

        let mut deriv_coeff = Vec::with_capacity(self.coefficients.len() - 1);

        for (i, val) in self.coefficients.iter().enumerate().skip(1) {
            deriv_coeff.push(N::from_f64(i as f64).unwrap() * *val);
        }

        Polynomial {
            coefficients: deriv_coeff,
            tolerance: self.tolerance,
        }
    }

    /// Get the antiderivative of the polynomial with specified constant
    pub fn antiderivative(&self, constant: N) -> Self {
        let mut coefficients = Vec::with_capacity(self.coefficients.len() + 1);
        coefficients.push(constant);
        for (ind, val) in self.coefficients.iter().enumerate() {
            coefficients.push(*val * N::from_f64(1.0 / (ind + 1) as f64).unwrap());
        }
        Polynomial {
            coefficients,
            tolerance: self.tolerance,
        }
    }

    /// Integrate this polynomial between to starting points
    pub fn integrate(&self, lower: N, upper: N) -> N {
        let poly_anti = self.antiderivative(N::zero());
        poly_anti.evaluate(upper) - poly_anti.evaluate(lower)
    }

    /// Divide this polynomial by another, getting a quotient and remainder, using tol to check for 0
    pub fn divide(&self, divisor: &Polynomial<N>) -> Result<(Self, Self), String> {
        if divisor.coefficients.len() == 1
            && divisor.coefficients[0].real().abs() < self.tolerance
            && divisor.coefficients[0].imaginary().abs() < self.tolerance
        {
            return Err("Polynomial division: Can not divide by 0".to_owned());
        }

        let mut quotient = Polynomial::with_tolerance(self.tolerance)?;
        let mut remainder = Polynomial::from_iter(self.coefficients.iter().copied());
        remainder.tolerance = self.tolerance;
        remainder.purge_leading();
        let mut temp = Polynomial::new();

        if divisor.coefficients.len() == 1 {
            let idivisor = N::from_f64(1.0).unwrap() / divisor.coefficients[0];
            return Ok((
                remainder
                    .coefficients
                    .iter()
                    .map(|c| *c * idivisor)
                    .collect(),
                Polynomial::new(),
            ));
        }

        while remainder.coefficients.len() >= divisor.coefficients.len()
            && !(remainder.coefficients.len() == 1
                && remainder.coefficients[0].real().abs() < self.tolerance
                && remainder.coefficients[0].imaginary().abs() < self.tolerance)
        {
            // Get the power left over from dividing lead terms
            let order = remainder.coefficients.len() - divisor.coefficients.len();
            // Make a vector that is just the lead coefficients divided at the right power
            temp.coefficients = vec![N::zero(); order + 1];
            temp.coefficients[order] =
                *remainder.coefficients.last().unwrap() / *divisor.coefficients.last().unwrap();
            // Add the division to the quotient
            quotient += &temp;
            // Get the amount to shift divisor by
            let padding = temp.coefficients.len() - 1;
            // Multiply every coefficient in divisor by temp's coefficient
            temp = divisor
                .coefficients
                .iter()
                .map(|c| *c * *temp.coefficients.last().unwrap())
                .collect();
            // Shift the coefficients to multiply by the right power of x
            for _ in 0..padding {
                temp.coefficients.insert(0, N::zero());
            }
            // remainder -= temp x d;
            remainder -= &temp;
            while remainder.coefficients.len() > 1
                && remainder.coefficients.last().unwrap().real().abs() < self.tolerance
                && remainder.coefficients.last().unwrap().imaginary().abs() < self.tolerance
            {
                remainder.coefficients.pop();
            }
        }

        Ok((quotient, remainder))
    }

    /// Get the n (possibly including repeats) of the polynomial given n using Laguerre's method
    pub fn roots(
        &self,
        tol: <N as ComplexField>::RealField,
        n_max: usize,
    ) -> Result<VecDeque<Complex<<N as ComplexField>::RealField>>, String> {
        if self.coefficients.len() > 1
            && self.coefficients.last().unwrap().real().abs() < tol
            && self.coefficients.last().unwrap().imaginary().abs() < tol
        {
            return Err("Polynomial roots: Leading 0 coefficient!".to_owned());
        }

        match self.coefficients.len() {
            1 => {
                // Only constant, root only if constant is 0
                if self.coefficients[0].real().abs() < tol
                    && self.coefficients[0].imaginary().abs() < tol
                {
                    return Ok(VecDeque::from(vec![Complex::<N::RealField>::zero()]));
                }
                return Err("Polynomial roots: Non-zero constant has no root".to_owned());
            }
            2 => {
                // Linear term, root easy
                let division = -self.coefficients[0] / self.coefficients[1];
                return Ok(VecDeque::from(vec![Complex::<N::RealField>::new(
                    division.real(),
                    division.imaginary(),
                )]));
            }
            3 => {
                // Use quadratic formula and return in right order
                let determinant = self.coefficients[1].powi(2)
                    - N::from_f64(4.0).unwrap() * self.coefficients[2] * self.coefficients[0];
                let determinant =
                    Complex::<N::RealField>::new(determinant.real(), determinant.imaginary())
                        .sqrt();
                let leading = self.coefficients[2];
                let leading = Complex::<N::RealField>::new(leading.real(), leading.imaginary());
                let leading = leading
                    * Complex::<N::RealField>::new(
                        N::from_f64(2.0).unwrap().real(),
                        N::zero().real(),
                    );
                let secondary = self.coefficients[1];
                let secondary =
                    Complex::<N::RealField>::new(secondary.real(), secondary.imaginary());
                let positive = (-secondary + determinant) / leading;
                let negative = (-secondary - determinant) / leading;
                return Ok(VecDeque::from(vec![positive, negative]));
            }
            _ => {}
        }

        let complex = self.make_complex();
        let derivative = complex.derivative();

        let mut guess = Complex::<N::RealField>::zero();
        let mut k = 0;
        'out: while k < n_max {
            let val = complex.evaluate(guess);
            if val.abs() < tol {
                break 'out;
            }
            let (deriv, second_deriv) = derivative.evaluate_derivative(guess);
            let deriv_quotient = deriv / val;
            let g_sq = deriv_quotient.powi(2);
            let second_deriv_quotient = g_sq - second_deriv / val;
            let order = Complex::<N::RealField>::from_usize(self.coefficients.len() - 1).unwrap();
            let sqrt = ((order - Complex::<N::RealField>::one())
                * (order * second_deriv_quotient - g_sq))
                .sqrt();
            let plus = deriv_quotient + sqrt;
            let minus = deriv_quotient - sqrt;
            let a = if plus.abs() > minus.abs() {
                order / plus
            } else {
                order / minus
            };
            guess -= a;
            k += 1;
        }
        if k == n_max {
            return Err("Polynomial roots: maximum iterations exceeded".to_owned());
        }

        let divisor = polynomial![Complex::<N::RealField>::one(), -guess];
        let (quotient, _) = complex.divide(&divisor)?;
        let mut roots = quotient.roots(tol, n_max)?;
        roots.push_front(guess);

        let mut corrected_roots = VecDeque::with_capacity(roots.len());
        for root in roots.iter() {
            corrected_roots.push_back(newton_polynomial(*root, &complex, tol, n_max)?);
        }

        Ok(corrected_roots)
    }

    // Pad to the smallest power of two less than or equal to size
    fn pad_power_of_two(&mut self, size: usize) {
        let mut power: usize = 1;
        while power < size {
            power <<= 1;
        }
        while self.coefficients.len() < power {
            self.coefficients.push(N::zero());
        }
    }

    /// Get the polynomial in point form evaluated at roots of unity at k points
    /// where k is the smallest power of 2 greater than or equal to size
    pub fn dft(&self, size: usize) -> Vec<Complex<<N as ComplexField>::RealField>> {
        let mut poly = self.make_complex();
        poly.pad_power_of_two(size);
        let mut working = bit_reverse_copy(&poly.coefficients);
        let len = working.len();
        for s in 1..(len as f64).log2() as usize + 1 {
            let m = 1 << s;
            let angle = 2.0 * f64::consts::PI / m as f64;
            let angle = N::RealField::from_f64(angle).unwrap();
            let root_of_unity = Complex::<N::RealField>::new(angle.cos(), angle.sin());
            let mut w = Complex::<N::RealField>::new(N::RealField::one(), N::RealField::zero());
            for j in 0..m / 2 {
                for k in (j..len).step_by(m) {
                    let temp = w * working[k + m / 2];
                    let u = working[k];
                    working[k] = u + temp;
                    working[k + m / 2] = u - temp;
                }
                w *= root_of_unity;
            }
        }
        working
    }

    // Assumes power of 2
    pub fn idft(
        vec: &[Complex<<N as ComplexField>::RealField>],
        tol: <N as ComplexField>::RealField,
    ) -> Self {
        let mut working = bit_reverse_copy(vec);
        let len = working.len();
        for s in 1..(len as f64).log2() as usize + 1 {
            let m = 1 << s;
            let angle = -2.0 * f64::consts::PI / m as f64;
            let angle = N::RealField::from_f64(angle).unwrap();
            let root_of_unity = Complex::<N::RealField>::new(angle.cos(), angle.sin());
            let mut w = Complex::<N::RealField>::new(N::RealField::one(), N::RealField::zero());
            for j in 0..m / 2 {
                for k in (j..len).step_by(m) {
                    let temp = w * working[k + m / 2];
                    let u = working[k];
                    working[k] = u + temp;
                    working[k + m / 2] = u - temp;
                }
                w *= root_of_unity;
            }
        }
        let ilen = Complex::<N::RealField>::new(
            N::from_f64(1.0 / len as f64).unwrap().real(),
            N::zero().real(),
        );
        for val in &mut working {
            *val *= ilen;
        }
        let coefficients = if TypeId::of::<N::RealField>() == TypeId::of::<N>() {
            working
                .iter()
                .map(|c| N::from_real(c.re))
                .collect::<Vec<_>>()
        } else {
            working
                .iter()
                .map(|c| N::from_real(c.re) + (-N::one()).sqrt() * N::from_real(c.im))
                .collect::<Vec<_>>()
        };

        let mut poly = Polynomial {
            coefficients,
            tolerance: tol,
        };
        poly.purge_leading();
        poly
    }
}

fn bit_reverse(mut k: usize, num_bits: usize) -> usize {
    let mut result: usize = 0;
    for _ in 0..num_bits {
        result |= k & 1;
        result <<= 1;
        k >>= 1;
    }
    result >>= 1;
    result
}

// Assumes vec is a power of 2 length
fn bit_reverse_copy<N: RealField>(vec: &[Complex<N>]) -> Vec<Complex<N>> {
    let len = vec.len();
    let mut result = vec![Complex::new(N::zero(), N::zero()); len];
    let num_bits = (len as f64).log2() as usize;
    for k in 0..len {
        result[bit_reverse(k, num_bits)] = vec[k];
    }
    result
}

impl<N: ComplexField> FromIterator<N> for Polynomial<N> {
    fn from_iter<I: IntoIterator<Item = N>>(iter: I) -> Polynomial<N> {
        Polynomial {
            coefficients: Vec::from_iter(iter),
            tolerance: N::RealField::from_f64(1e-10).unwrap(),
        }
    }
}

impl<N: ComplexField> Default for Polynomial<N> {
    fn default() -> Self {
        Self::new()
    }
}

impl<N: ComplexField> Zero for Polynomial<N> {
    fn zero() -> Polynomial<N> {
        Polynomial::new()
    }

    fn is_zero(&self) -> bool {
        for val in &self.coefficients {
            if !val.is_zero() {
                return false;
            }
        }
        true
    }
}

// Operator overloading

impl<N: ComplexField> ops::Add<N> for Polynomial<N> {
    type Output = Polynomial<N>;

    fn add(mut self, rhs: N) -> Polynomial<N> {
        self.coefficients[0] += rhs;
        self
    }
}

impl<N: ComplexField> ops::Add<N> for &Polynomial<N> {
    type Output = Polynomial<N>;

    fn add(self, rhs: N) -> Polynomial<N> {
        let mut coefficients = Vec::from(self.coefficients.as_slice());
        coefficients[0] += rhs;
        Polynomial {
            coefficients,
            tolerance: self.tolerance,
        }
    }
}

impl<N: ComplexField> ops::Add<Polynomial<N>> for Polynomial<N> {
    type Output = Polynomial<N>;

    fn add(mut self, rhs: Polynomial<N>) -> Polynomial<N> {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        for (ind, val) in self.coefficients.iter_mut().take(min_order).enumerate() {
            *val += rhs.coefficients[ind];
        }

        for val in rhs.coefficients.iter().skip(min_order) {
            self.coefficients.push(*val);
        }

        self
    }
}

impl<N: ComplexField> ops::Add<&Polynomial<N>> for Polynomial<N> {
    type Output = Polynomial<N>;

    fn add(mut self, rhs: &Polynomial<N>) -> Polynomial<N> {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        for (ind, val) in self.coefficients.iter_mut().take(min_order).enumerate() {
            *val += rhs.coefficients[ind];
        }

        // Will only run if rhs has higher order
        for val in rhs.coefficients.iter().skip(min_order) {
            self.coefficients.push(*val);
        }

        self
    }
}

impl<N: ComplexField> ops::Add<Polynomial<N>> for &Polynomial<N> {
    type Output = Polynomial<N>;

    fn add(self, rhs: Polynomial<N>) -> Polynomial<N> {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        let mut coefficients =
            Vec::with_capacity(self.coefficients.len().max(rhs.coefficients.len()));
        for (ind, val) in self.coefficients.iter().take(min_order).enumerate() {
            coefficients.push(*val + rhs.coefficients[ind]);
        }

        // Only one loop will run
        for val in self.coefficients.iter().skip(min_order) {
            coefficients.push(*val);
        }

        for val in rhs.coefficients.iter().skip(min_order) {
            coefficients.push(*val);
        }

        Polynomial {
            coefficients,
            tolerance: self.tolerance,
        }
    }
}

impl<N: ComplexField> ops::Add<&Polynomial<N>> for &Polynomial<N> {
    type Output = Polynomial<N>;

    fn add(self, rhs: &Polynomial<N>) -> Polynomial<N> {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        let mut coefficients =
            Vec::with_capacity(self.coefficients.len().max(rhs.coefficients.len()));
        for (ind, val) in self.coefficients.iter().take(min_order).enumerate() {
            coefficients.push(*val + rhs.coefficients[ind]);
        }

        // Only one loop will run
        for val in self.coefficients.iter().skip(min_order) {
            coefficients.push(*val);
        }

        for val in rhs.coefficients.iter().skip(min_order) {
            coefficients.push(*val);
        }

        Polynomial {
            coefficients,
            tolerance: self.tolerance,
        }
    }
}

impl<N: ComplexField> ops::AddAssign<N> for Polynomial<N> {
    fn add_assign(&mut self, rhs: N) {
        self.coefficients[0] += rhs;
    }
}

impl<N: ComplexField> ops::AddAssign<Polynomial<N>> for Polynomial<N> {
    fn add_assign(&mut self, rhs: Polynomial<N>) {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        for (ind, val) in self.coefficients.iter_mut().take(min_order).enumerate() {
            *val += rhs.coefficients[ind];
        }

        for val in rhs.coefficients.iter().skip(min_order) {
            self.coefficients.push(*val);
        }
    }
}

impl<N: ComplexField> ops::AddAssign<&Polynomial<N>> for Polynomial<N> {
    fn add_assign(&mut self, rhs: &Polynomial<N>) {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        for (ind, val) in self.coefficients.iter_mut().take(min_order).enumerate() {
            *val += rhs.coefficients[ind];
        }

        for val in rhs.coefficients.iter().skip(min_order) {
            self.coefficients.push(*val);
        }
    }
}

impl<N: ComplexField> ops::Sub<N> for Polynomial<N> {
    type Output = Polynomial<N>;

    fn sub(mut self, rhs: N) -> Polynomial<N> {
        self.coefficients[0] -= rhs;
        self
    }
}

impl<N: ComplexField> ops::Sub<N> for &Polynomial<N> {
    type Output = Polynomial<N>;

    fn sub(self, rhs: N) -> Polynomial<N> {
        let mut coefficients = Vec::from(self.coefficients.as_slice());
        coefficients[0] -= rhs;
        Polynomial {
            coefficients,
            tolerance: self.tolerance,
        }
    }
}

impl<N: ComplexField> ops::Sub<Polynomial<N>> for Polynomial<N> {
    type Output = Polynomial<N>;

    fn sub(mut self, rhs: Polynomial<N>) -> Polynomial<N> {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        for (ind, val) in self.coefficients.iter_mut().take(min_order).enumerate() {
            *val -= rhs.coefficients[ind];
        }

        for val in rhs.coefficients.iter().skip(min_order) {
            self.coefficients.push(-*val);
        }

        self
    }
}

impl<N: ComplexField> ops::Sub<Polynomial<N>> for &Polynomial<N> {
    type Output = Polynomial<N>;

    fn sub(self, rhs: Polynomial<N>) -> Polynomial<N> {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        let mut coefficients =
            Vec::with_capacity(self.coefficients.len().max(rhs.coefficients.len()));
        for (ind, val) in self.coefficients.iter().take(min_order).enumerate() {
            coefficients.push(*val - rhs.coefficients[ind]);
        }

        // Only one for loop runs
        for val in self.coefficients.iter().skip(min_order) {
            coefficients.push(*val);
        }

        for val in rhs.coefficients.iter().skip(min_order) {
            coefficients.push(-*val);
        }

        Polynomial {
            coefficients,
            tolerance: self.tolerance,
        }
    }
}

impl<N: ComplexField> ops::Sub<&Polynomial<N>> for Polynomial<N> {
    type Output = Polynomial<N>;

    fn sub(mut self, rhs: &Polynomial<N>) -> Polynomial<N> {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        for (ind, val) in self.coefficients.iter_mut().take(min_order).enumerate() {
            *val -= rhs.coefficients[ind];
        }

        for val in rhs.coefficients.iter().skip(min_order) {
            self.coefficients.push(-*val);
        }

        self
    }
}

impl<N: ComplexField> ops::Sub<&Polynomial<N>> for &Polynomial<N> {
    type Output = Polynomial<N>;

    fn sub(self, rhs: &Polynomial<N>) -> Polynomial<N> {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        let mut coefficients =
            Vec::with_capacity(self.coefficients.len().max(rhs.coefficients.len()));
        for (ind, val) in self.coefficients.iter().take(min_order).enumerate() {
            coefficients.push(*val - rhs.coefficients[ind]);
        }

        // Only one for loop runs
        for val in self.coefficients.iter().skip(min_order) {
            coefficients.push(*val);
        }

        for val in rhs.coefficients.iter().skip(min_order) {
            coefficients.push(-*val);
        }

        Polynomial {
            coefficients,
            tolerance: self.tolerance,
        }
    }
}

impl<N: ComplexField> ops::SubAssign<N> for Polynomial<N> {
    fn sub_assign(&mut self, rhs: N) {
        self.coefficients[0] -= rhs;
    }
}

impl<N: ComplexField> ops::SubAssign<Polynomial<N>> for Polynomial<N> {
    fn sub_assign(&mut self, rhs: Polynomial<N>) {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        for (ind, val) in self.coefficients.iter_mut().take(min_order).enumerate() {
            *val -= rhs.coefficients[ind];
        }

        for val in rhs.coefficients.iter().skip(min_order) {
            self.coefficients.push(-*val);
        }
    }
}

impl<N: ComplexField> ops::SubAssign<&Polynomial<N>> for Polynomial<N> {
    fn sub_assign(&mut self, rhs: &Polynomial<N>) {
        let min_order = self.coefficients.len().min(rhs.coefficients.len());
        for (ind, val) in self.coefficients.iter_mut().take(min_order).enumerate() {
            *val -= rhs.coefficients[ind];
        }

        for val in rhs.coefficients.iter().skip(min_order) {
            self.coefficients.push(-*val);
        }
    }
}

impl<N: ComplexField> ops::Mul<N> for Polynomial<N> {
    type Output = Polynomial<N>;

    fn mul(mut self, rhs: N) -> Polynomial<N> {
        for val in &mut self.coefficients {
            *val *= rhs;
        }
        self
    }
}

impl<N: ComplexField> ops::Mul<N> for &Polynomial<N> {
    type Output = Polynomial<N>;

    fn mul(self, rhs: N) -> Polynomial<N> {
        let mut coefficients = Vec::with_capacity(self.coefficients.len());
        for val in &self.coefficients {
            coefficients.push(*val * rhs);
        }
        Polynomial {
            coefficients,
            tolerance: self.tolerance,
        }
    }
}

fn multiply<N: ComplexField>(lhs: &Polynomial<N>, rhs: &Polynomial<N>) -> Polynomial<N> {
    // Do scalar multiplication if one side has no powers
    if rhs.coefficients.len() == 1 {
        return lhs * rhs.coefficients[0];
    }
    if lhs.coefficients.len() == 1 {
        return rhs * lhs.coefficients[0];
    }

    // Special case linear term multiplication
    if rhs.coefficients.len() == 2 {
        let mut shifted = lhs * rhs.coefficients[1];
        shifted.coefficients.insert(0, N::zero());
        return shifted + lhs * rhs.coefficients[0];
    }
    if lhs.coefficients.len() == 2 {
        let mut shifted = rhs * lhs.coefficients[1];
        shifted.coefficients.insert(0, N::zero());
        return shifted + rhs * lhs.coefficients[0];
    }

    let bound = lhs.coefficients.len().max(rhs.coefficients.len()) * 2;
    let left_points = lhs.dft(bound);
    let right_points = rhs.dft(bound);
    let product_points: Vec<_> = left_points
        .iter()
        .zip(right_points.iter())
        .map(|(l_p, r_p)| *l_p * r_p)
        .collect();
    Polynomial::<N>::idft(&product_points, lhs.tolerance)
}

impl<N: ComplexField> ops::Mul<Polynomial<N>> for Polynomial<N> {
    type Output = Polynomial<N>;

    fn mul(self, rhs: Polynomial<N>) -> Polynomial<N> {
        multiply(&self, &rhs)
    }
}

impl<N: ComplexField> ops::Mul<&Polynomial<N>> for Polynomial<N> {
    type Output = Polynomial<N>;

    fn mul(self, rhs: &Polynomial<N>) -> Polynomial<N> {
        multiply(&self, &rhs)
    }
}

impl<N: ComplexField> ops::Mul<Polynomial<N>> for &Polynomial<N> {
    type Output = Polynomial<N>;

    fn mul(self, rhs: Polynomial<N>) -> Polynomial<N> {
        multiply(&self, &rhs)
    }
}

impl<N: ComplexField> ops::Mul<&Polynomial<N>> for &Polynomial<N> {
    type Output = Polynomial<N>;

    fn mul(self, rhs: &Polynomial<N>) -> Polynomial<N> {
        multiply(&self, &rhs)
    }
}

impl<N: ComplexField> ops::MulAssign<N> for Polynomial<N> {
    fn mul_assign(&mut self, rhs: N) {
        for val in self.coefficients.iter_mut() {
            *val *= rhs;
        }
    }
}

impl<N: ComplexField> ops::MulAssign<Polynomial<N>> for Polynomial<N> {
    fn mul_assign(&mut self, rhs: Polynomial<N>) {
        self.coefficients = multiply(&self, &rhs).coefficients;
    }
}

impl<N: ComplexField> ops::MulAssign<&Polynomial<N>> for Polynomial<N> {
    fn mul_assign(&mut self, rhs: &Polynomial<N>) {
        self.coefficients = multiply(&self, &rhs).coefficients;
    }
}

impl<N: ComplexField> ops::Div<N> for Polynomial<N> {
    type Output = Polynomial<N>;

    fn div(mut self, rhs: N) -> Polynomial<N> {
        for val in &mut self.coefficients {
            *val /= rhs;
        }
        self
    }
}

impl<N: ComplexField> ops::Div<N> for &Polynomial<N> {
    type Output = Polynomial<N>;

    fn div(self, rhs: N) -> Polynomial<N> {
        let mut coefficients = Vec::from(self.coefficients.as_slice());
        for val in &mut coefficients {
            *val /= rhs;
        }
        Polynomial {
            coefficients,
            tolerance: self.tolerance,
        }
    }
}

impl<N: ComplexField> ops::DivAssign<N> for Polynomial<N> {
    fn div_assign(&mut self, rhs: N) {
        for val in &mut self.coefficients {
            *val /= rhs;
        }
    }
}

impl<N: ComplexField> ops::Neg for Polynomial<N> {
    type Output = Polynomial<N>;

    fn neg(mut self) -> Polynomial<N> {
        for val in &mut self.coefficients {
            *val = -*val;
        }
        self
    }
}

impl<N: ComplexField> ops::Neg for &Polynomial<N> {
    type Output = Polynomial<N>;

    fn neg(self) -> Polynomial<N> {
        Polynomial {
            coefficients: self.coefficients.iter().map(|c| -*c).collect(),
            tolerance: self.tolerance,
        }
    }
}

impl<N: ComplexField> From<N> for Polynomial<N> {
    fn from(n: N) -> Polynomial<N> {
        polynomial![n]
    }
}

impl<N: RealField> From<Polynomial<N>> for Polynomial<Complex<N>> {
    fn from(poly: Polynomial<N>) -> Polynomial<Complex<N>> {
        poly.make_complex()
    }
}