Struct Polynomial 

pub struct Polynomial {
    pub coeffs: Vec<RnsRational>,
    pub channels: Channels,
}

A univariate polynomial with exact rational coefficients.

coeffs[i] is the coefficient of xⁱ. Trailing zero coefficients are trimmed so the last entry (if any) is the nonzero leading coefficient.

Fields

coeffs: Vec<RnsRational>

channels: Channels

Implementations

impl Polynomial

pub fn new(coeffs: Vec<RnsRational>, channels: Channels) -> Self

Build from coefficients (ascending powers), trimming trailing zeros.

pub fn from_int_coeffs(coeffs: &[i64], channels: Channels) -> Self

Build from integer coefficients (ascending powers).

pub fn zero(channels: Channels) -> Self

The zero polynomial.

pub fn one(channels: Channels) -> Self

The constant polynomial 1.

pub fn constant(c: RnsRational) -> Self

A constant polynomial.

pub fn is_zero(&self) -> bool

true iff this is the zero polynomial.

pub fn degree(&self) -> usize

Degree; the zero polynomial has degree 0 by convention here (callers should also check Polynomial::is_zero).

pub fn leading(&self) -> RnsRational

The leading (highest-power) coefficient, or zero for the zero polynomial.

pub fn eval(&self, x: &RnsRational) -> RnsRational

Evaluate at a rational point via Horner’s scheme.

pub fn sign_at(&self, x: &RnsRational) -> i32

Sign of p(x): -1, 0, or +1.

pub fn derivative(&self) -> Self

Formal derivative.

pub fn scalar_mul(&self, s: &RnsRational) -> Self

Multiply by a rational scalar.

pub fn add(&self, other: &Self) -> Self

Polynomial addition.

pub fn sub(&self, other: &Self) -> Self

Polynomial subtraction.

pub fn mul(&self, other: &Self) -> Self

Polynomial multiplication.

pub fn divmod(&self, divisor: &Self) -> (Self, Self)

Long division returning (quotient, remainder) over the rational field. Panics if divisor is zero.

pub fn rem(&self, divisor: &Self) -> Self

Remainder of polynomial division.

pub fn div_exact(&self, divisor: &Self) -> Self

Exact division (asserts the remainder is zero).

pub fn monic(&self) -> Self

Make monic (leading coefficient 1). The zero polynomial is returned as-is.

pub fn gcd(a: &Self, b: &Self) -> Self

Monic GCD of two polynomials (Euclidean algorithm over ℚ).

pub fn squarefree(&self) -> Self

Square-free part: p / gcd(p, p'), made monic.

pub fn sturm_sequence(&self) -> Vec<Self>

The Sturm sequence s0 = p, s1 = p', s_{i+1} = -rem(s_{i-1}, s_i).

pub fn sign_changes(seq: &[Self], x: &RnsRational) -> usize

Count sign changes of a Sturm sequence evaluated at x (zeros ignored).

pub fn sturm_root_count(&self, a: &RnsRational, b: &RnsRational) -> usize

Number of distinct real roots in the half-open interval (a, b].

pub fn root_bound(&self) -> RnsRational

A Cauchy bound B such that every real root lies in (-B, B).

pub fn isolate_real_roots(&self) -> Vec<(RnsRational, RnsRational)>

Isolate every real root into its own interval (lo, hi], sorted ascending. self should be square-free for clean isolation.

pub fn find_rational_root(&self) -> Option<RnsRational>

Find one rational root via the rational-root theorem, or None.

pub fn factor_over_q(&self) -> Vec<Self>

Factor into monic irreducible-over-ℚ factors (for the low degrees that arise from resultants of quadratics/cubics, this is exact; higher-degree factors that resist rational-root splitting are returned whole).

Trait Implementations

impl Clone for Polynomial

fn clone(&self) -> Polynomial

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for Polynomial

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Eq for Polynomial

impl PartialEq for Polynomial

fn eq(&self, other: &Self) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.