Struct AlgebraicNumber 

pub struct AlgebraicNumber { /* private fields */ }

An algebraic number represented by a polynomial and an isolating interval.

An algebraic number α is represented by:

• A polynomial p(x) such that p(α) = 0
• An isolating interval (a, b) that contains exactly one root of p

This representation allows for exact comparisons and arithmetic on algebraic numbers.

Implementations

impl AlgebraicNumber

pub fn new( polynomial: Polynomial, var: Var, lower: BigRational, upper: BigRational, ) -> Self

Create a new algebraic number from a polynomial and an isolating interval.

Panics

Panics if the interval doesn’t contain exactly one root of the polynomial.

pub fn from_rational(r: BigRational) -> Self

Create an algebraic number from a rational number.

pub fn sqrt(n: &BigRational) -> Option<Self>

Create an algebraic number representing √n for a non-negative rational n.

Returns None if n is negative.

pub fn polynomial(&self) -> &Polynomial

Get the polynomial defining this algebraic number.

pub fn interval(&self) -> (&BigRational, &BigRational)

Get the isolating interval as (lower, upper).

pub fn var(&self) -> Var

Get the variable used in the polynomial.

pub fn refine(&mut self)

Refine the isolating interval by bisection.

This makes the interval smaller, improving precision for approximations.

pub fn approximate(&self) -> BigRational

Get an approximation of the algebraic number as a rational.

This returns the midpoint of the isolating interval.

pub fn approximate_with_precision( &mut self, epsilon: &BigRational, ) -> BigRational

Get an approximation with a specified precision.

Refines the interval until its width is less than epsilon.

pub fn is_zero(&self) -> bool

Check if this algebraic number is definitely zero.

pub fn is_positive(&self) -> bool

Check if this algebraic number is definitely positive.

pub fn is_negative(&self) -> bool

Check if this algebraic number is definitely negative.

pub fn is_rational(&self) -> bool

Check if this algebraic number is actually a rational number.

Returns true if the polynomial is linear (degree 1) which means the number is rational.

pub fn sign(&self) -> Option<i8>

Get the sign of this algebraic number.

Returns:

• Some(1) if definitely positive
• Some(-1) if definitely negative
• Some(0) if definitely zero
• None if sign is unknown (shouldn’t happen with a proper isolating interval)

pub fn cmp_algebraic(&mut self, other: &mut AlgebraicNumber) -> Ordering

Compare this algebraic number with another.

pub fn cmp_rational(&mut self, r: &BigRational) -> Ordering

Compare this algebraic number with a rational.

pub fn negate(&self) -> AlgebraicNumber

Negate this algebraic number.

If α is a root of p(x), then -α is a root of p(-x).

pub fn add_rational(&self, r: &BigRational) -> AlgebraicNumber

Add this algebraic number with a rational.

pub fn mul_rational(&self, r: &BigRational) -> AlgebraicNumber

Multiply this algebraic number by a rational.

pub fn sub_rational(&self, r: &BigRational) -> AlgebraicNumber

Subtract a rational from this algebraic number.

pub fn inverse(&self) -> Option<AlgebraicNumber>

Compute the multiplicative inverse of this algebraic number.

Returns None if the number is zero.

pub fn div_rational(&self, r: &BigRational) -> Option<AlgebraicNumber>

Divide this algebraic number by a rational.

Returns None if the divisor is zero.

pub fn pow(&self, n: i32) -> Option<AlgebraicNumber>

Raise this algebraic number to an integer power.

pub fn add_algebraic(&mut self, other: &mut AlgebraicNumber) -> AlgebraicNumber

Add two algebraic numbers using resultant-based computation.

Given α (root of p(x)) and β (root of q(y)), computes α + β. For now, this uses interval arithmetic for robustness. A full resultant-based implementation requires careful handling of multivariate polynomials and root isolation, which is complex.

Reference: Standard symbolic computation textbooks (e.g., “Algorithms in Real Algebraic Geometry”)

pub fn mul_algebraic(&mut self, other: &mut AlgebraicNumber) -> AlgebraicNumber

Multiply two algebraic numbers using resultant-based computation.

Given α (root of p(x)) and β (root of q(y)), computes α * β. For now, this uses interval arithmetic for robustness. A full resultant-based implementation requires careful handling of multivariate polynomials and root isolation, which is complex.

Reference: Standard symbolic computation textbooks (e.g., “Algorithms in Real Algebraic Geometry”)

Trait Implementations

impl Clone for AlgebraicNumber

fn clone(&self) -> AlgebraicNumber

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for AlgebraicNumber

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

Formats the value using the given formatter.