symbolica 3.0.0

A blazing fast computer algebra system
Documentation 
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//! Tests for core univariate-polynomial arithmetic.

use crate::{
    atom::AtomCore,
    domains::{
        Ring, RingOps, SampleableRing,
        finite_field::{FiniteFieldCore, Zp, Zp64},
        integer::Z,
        rational::Q,
    },
    parse,
    poly::univariate::{
        DenseFiniteFieldRootContext, UnivariatePolynomialRing, UnivariatePolynomialSamplingPolicy,
    },
};
use rand::{SeedableRng, rngs::StdRng};

#[test]
fn positive_root_bound_does_not_underflow_or_overflow() {
    let tiny = parse!("2^2000*x-1")
        .to_polynomial::<_, u16>(&Z, None)
        .to_univariate_from_univariate(0);
    let large = parse!("x-2^2000")
        .to_polynomial::<_, u16>(&Z, None)
        .to_univariate_from_univariate(0);
    let tiny_bound = tiny.max_real_root_bound();
    let large_bound = large.max_real_root_bound();
    assert!(tiny_bound > crate::domains::rational::Rational::zero());
    assert!(tiny.map_coeff(|c| c.into(), Q).evaluate(&tiny_bound) > 0);
    assert!(large.map_coeff(|c| c.into(), Q).evaluate(&large_bound) > 0);
    assert_eq!(tiny.isolate_real_root_intervals().len(), 1);
    assert_eq!(large.isolate_real_root_intervals().len(), 1);
}

fn polynomial_with_roots(
    field: &Zp64,
    roots: &[u64],
) -> Vec<<Zp64 as crate::domains::Set>::Element> {
    let mut polynomial = vec![field.one()];
    for root in roots {
        let root = field.to_element(*root);
        let mut product = vec![field.zero(); polynomial.len() + 1];
        for (degree, coefficient) in polynomial.iter().enumerate() {
            field.sub_mul_assign(&mut product[degree], coefficient, &root);
            field.add_assign(&mut product[degree + 1], coefficient);
        }
        polynomial = product;
    }
    polynomial
}

#[test]
fn dense_finite_field_root_context_finds_distinct_nonzero_roots() {
    // Hu-Monagan starts its smooth-prime search here, above i64::MAX.
    let field = Zp64::new(10_030_613_004_288_000_001);
    let expected = [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377];
    let polynomial = polynomial_with_roots(&field, &expected);
    let mut context = DenseFiniteFieldRootContext::new(&field);

    let mut actual = context
        .find_distinct_nonzero_roots(&polynomial)
        .unwrap()
        .iter()
        .map(|root| field.from_element(root))
        .collect::<Vec<_>>();
    actual.sort_unstable();

    assert_eq!(actual, expected);
}

#[test]
fn dense_finite_field_root_context_uses_u32_fields() {
    let field = Zp::new(1_088_391_169);
    let expected = [1u32, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377];
    let mut polynomial = vec![field.one()];
    for root in expected {
        let root = field.to_element(root);
        let mut product = vec![field.zero(); polynomial.len() + 1];
        for (degree, coefficient) in polynomial.iter().enumerate() {
            field.sub_mul_assign(&mut product[degree], coefficient, &root);
            field.add_assign(&mut product[degree + 1], coefficient);
        }
        polynomial = product;
    }

    let mut context = DenseFiniteFieldRootContext::new(&field);
    let mut actual = context
        .find_distinct_nonzero_roots(&polynomial)
        .unwrap()
        .iter()
        .map(|root| field.from_element(root))
        .collect::<Vec<_>>();
    actual.sort_unstable();

    assert_eq!(actual, expected);
}

#[test]
fn dense_finite_field_root_context_rejects_repeated_roots() {
    let field = Zp64::new(17);
    let polynomial = polynomial_with_roots(&field, &[3, 3]);
    let mut context = DenseFiniteFieldRootContext::new(&field);

    assert!(context.find_distinct_nonzero_roots(&polynomial).is_none());
}

#[test]
fn dense_finite_field_root_context_rejects_zero_roots() {
    let field = Zp64::new(17);
    let polynomial = polynomial_with_roots(&field, &[0, 3]);
    let mut context = DenseFiniteFieldRootContext::new(&field);

    assert!(context.find_distinct_nonzero_roots(&polynomial).is_none());
}

#[test]
fn dense_finite_field_root_context_rejects_nonsplit_polynomials() {
    let field = Zp64::new(17);
    let polynomial = vec![field.to_element(3), field.zero(), field.one()];
    let mut context = DenseFiniteFieldRootContext::new(&field);

    assert!(context.find_distinct_nonzero_roots(&polynomial).is_none());
}

#[test]
fn samples_with_degree_and_coefficient_policies() {
    let template = parse!("x")
        .to_polynomial::<_, u8>(&Z, None)
        .to_univariate_from_univariate(0);
    let ring = UnivariatePolynomialRing::from_polynomial(&template);
    let policy = UnivariatePolynomialSamplingPolicy {
        degree: 3..=3,
        coefficient: 1.into()..=1.into(),
    };
    let mut rng = StdRng::seed_from_u64(1);

    let sample = ring.sample(&mut rng, &policy);

    assert_eq!(sample.degree(), 3);
    assert!(
        sample
            .coefficients()
            .iter()
            .all(|coefficient| coefficient == &1)
    );
}

#[test]
fn exact_division_with_nonunit_constant_term() {
    let p = parse!("6*x^2+15*x+6")
        .to_polynomial::<_, u16>(&Z, None)
        .to_univariate_from_univariate(0);
    let d = parse!("2*x+4")
        .to_polynomial::<_, u16>(&Z, None)
        .to_univariate_from_univariate(0);
    let q = parse!("3*x+3/2")
        .to_polynomial::<_, u16>(&Q, None)
        .to_univariate_from_univariate(0);
    assert!(p.try_div(&d).is_none());
    assert_eq!(
        p.map_coeff(|c| c.into(), Q)
            .try_div(&d.map_coeff(|c| c.into(), Q)),
        Some(q)
    );
    let p = parse!("6*x^2+18*x+12")
        .to_polynomial::<_, u16>(&Z, None)
        .to_univariate_from_univariate(0);
    let q = parse!("3*x+3")
        .to_polynomial::<_, u16>(&Z, None)
        .to_univariate_from_univariate(0);
    assert_eq!(p.try_div(&d), Some(q));
}

#[test]
fn derivative_and_integral_are_inverses() {
    let polynomial = parse!("x^2+5x+x^7+3")
        .to_polynomial::<_, u8>(&Q, None)
        .to_univariate_from_univariate(0);

    assert_eq!(polynomial.integrate().derivative(), polynomial);
}

#[test]
fn arithmetic_and_evaluation() {
    let a = parse!("x^2+5x+x^7+3")
        .to_polynomial::<_, u8>(&Z, None)
        .to_univariate_from_univariate(0);
    let b = parse!("x^2 + 6")
        .to_polynomial::<_, u8>(&Z, None)
        .to_univariate_from_univariate(0);

    let expected_sum = parse!("9+5*x+2*x^2+x^7")
        .to_polynomial::<_, u8>(&Z, None)
        .to_univariate_from_univariate(0);
    let expected_product = parse!("18+30*x+9*x^2+5*x^3+x^4+6*x^7+x^9")
        .to_polynomial::<_, u8>(&Z, None)
        .to_univariate_from_univariate(0);
    let expected_quotient = parse!("1+36*x+-6*x^3+x^5")
        .to_polynomial::<_, u8>(&Z, None)
        .to_univariate_from_univariate(0);
    let expected_remainder = parse!("-3+-211*x")
        .to_polynomial::<_, u8>(&Z, None)
        .to_univariate_from_univariate(0);

    assert_eq!(&a + &b, expected_sum);
    assert_eq!(&a * &b, expected_product);
    assert_eq!(a.quot_rem(&b), (expected_quotient, expected_remainder));
    assert_eq!(a.evaluate(&5.into()), 78178);
}

#[test]
fn pseudo_remainder_does_not_divide_coefficients() {
    let factor = parse!("2*x^2+3*x+5").to_polynomial::<_, u8>(&Z, None);
    let variables = factor.variables().clone();
    let cofactor = parse!("7*x^2-x+4")
        .to_polynomial::<_, u8>(&Z, Some(variables.clone()))
        .to_univariate_from_univariate(0);
    let factor = factor.to_univariate_from_univariate(0);
    let product = &factor * &cofactor;
    let non_factor = parse!("2*x+1")
        .to_polynomial::<_, u8>(&Z, Some(variables))
        .to_univariate_from_univariate(0);

    assert!(product.pseudo_remainder(&factor).is_zero());
    assert!(!product.pseudo_remainder(&non_factor).is_zero());
}

#[test]
fn gcd_over_rationals_uses_polynomial_gcd() {
    let a = parse!("(x^12+x+1)*(x^8-3*x^2+2)")
        .to_polynomial::<_, u16>(&Q, None)
        .to_univariate_from_univariate(0);
    let b = parse!("(x^10-x+3)*(x^8-3*x^2+2)")
        .to_polynomial::<_, u16>(&Q, None)
        .to_univariate_from_univariate(0);
    let expected = parse!("x^8-3*x^2+2")
        .to_polynomial::<_, u16>(&Q, None)
        .to_univariate_from_univariate(0);

    assert_eq!(a.gcd(&b), expected);
    assert_eq!(a.gcd_euclidean(&b), expected);
}