geometric_algebra 0.3.0

Generate(d) custom libraries for geometric algebras
Documentation 
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//! Solves polynomials with real valued coefficients up to degree 4

#![allow(clippy::many_single_char_names)]
use crate::{epga1d::*, GeometricProduct, GeometricQuotient, Powf, Reversal, SquaredMagnitude};

/// Represents a complex root as homogeneous coordinates
#[derive(Debug, Clone, Copy)]
pub struct Root {
    /// Complex numerator
    pub numerator: ComplexNumber,
    /// Real denominator
    pub denominator: f32,
}

impl Root {
    /// Creates a new [Root]
    pub fn new(numerator: [f32; 2], denominator: f32) -> Self {
        Self {
            numerator: numerator.into(),
            denominator,
        }
    }
}

/// Finds the discriminant and root of a degree 1 polynomial.
///
/// `0 = coefficients[1] * x + coefficients[0]`
pub fn solve_linear(coefficients: [f32; 2], error_margin: f32) -> (f32, Vec<Root>) {
    if coefficients[1].abs() <= error_margin {
        (0.0, vec![])
    } else {
        (
            1.0,
            vec![Root {
                numerator: ComplexNumber::new(-coefficients[0], 0.0),
                denominator: coefficients[1],
            }],
        )
    }
}

/// Finds the discriminant and roots of a degree 2 polynomial.
///
/// `0 = coefficients[2] * x.powi(2) + coefficients[1] * x + coefficients[0]`
pub fn solve_quadratic(coefficients: [f32; 3], error_margin: f32) -> (f32, Vec<Root>) {
    if coefficients[2].abs() <= error_margin {
        return solve_linear([coefficients[0], coefficients[1]], error_margin);
    }
    // https://en.wikipedia.org/wiki/Quadratic_formula
    let discriminant = coefficients[1].powi(2) - 4.0 * coefficients[2] * coefficients[0];
    let q = discriminant.sqrt();
    let mut solutions = Vec::with_capacity(3);
    for s in [-q, q] {
        let numerator = s - ComplexNumber::new(coefficients[1], 0.0);
        solutions.push(Root {
            numerator,
            denominator: 2.0 * coefficients[2],
        });
    }
    (discriminant, solutions)
}

const ROOTS_OF_UNITY_3: [ComplexNumber; 3] = [
    // 0.8660254037844386467637231707529361834714026269051903140279034897
    // ComplexNumber::from_polar(1.0, -120.0/180.0*std::f32::consts::PI),
    // ComplexNumber::from_polar(1.0, 120.0/180.0*std::f32::consts::PI),
    // ComplexNumber::from_polar(1.0, 0.0),
    ComplexNumber::new(-0.5, -0.8660254),
    ComplexNumber::new(-0.5, 0.8660254),
    ComplexNumber::new(1.0, 0.0),
];

/// Finds the discriminant and roots of a degree 3 polynomial.
///
/// `0 = coefficients[3] * x.powi(3) + coefficients[2] * x.powi(2) + coefficients[1] * x + coefficients[0]`
///
/// Also returns the index of the real root if there are two complex roots and one real root.
pub fn solve_cubic(coefficients: [f32; 4], error_margin: f32) -> (f32, Vec<Root>, usize) {
    if coefficients[3].abs() <= error_margin {
        let (discriminant, roots) = solve_quadratic(
            [coefficients[0], coefficients[1], coefficients[2]],
            error_margin,
        );
        return (discriminant, roots, 2);
    }
    // https://en.wikipedia.org/wiki/Cubic_equation
    let d = [
        coefficients[2].powi(2) - 3.0 * coefficients[3] * coefficients[1],
        2.0 * coefficients[2].powi(3) - 9.0 * coefficients[3] * coefficients[2] * coefficients[1]
            + 27.0 * coefficients[3].powi(2) * coefficients[0],
    ];
    let mut solutions = Vec::with_capacity(3);
    let discriminant = d[1].powi(2) - 4.0 * d[0].powi(3);
    let c = discriminant.sqrt();
    let c = ((c + ComplexNumber::new(if c + d[1] == 0.0 { -d[1] } else { d[1] }, 0.0)) * 0.5)
        .powf(1.0 / 3.0);
    for root_of_unity in &ROOTS_OF_UNITY_3 {
        let ci = c.geometric_product(*root_of_unity);
        let denominator = ci * (3.0 * coefficients[3]);
        let numerator =
            (ci * -coefficients[2] - ci.geometric_product(ci) - ComplexNumber::new(d[0], 0.0))
                .geometric_product(denominator.reversal());
        solutions.push(Root {
            numerator,
            denominator: denominator.squared_magnitude(),
        });
    }
    let real_root =
        (((std::f32::consts::PI - c.arg()) / (std::f32::consts::PI * 2.0 / 3.0)) as usize + 1) % 3;
    (discriminant, solutions, real_root)
}

/// Finds the discriminant and roots of a degree 4 polynomial.
///
/// `0 = coefficients[4] * x.powi(4) + coefficients[3] * x.powi(3) + coefficients[2] * x.powi(2) + coefficients[1] * x + coefficients[0]`
pub fn solve_quartic(coefficients: [f32; 5], error_margin: f32) -> (f32, Vec<Root>) {
    if coefficients[4].abs() <= error_margin {
        let (discriminant, roots, _real_root) = solve_cubic(
            [
                coefficients[0],
                coefficients[1],
                coefficients[2],
                coefficients[3],
            ],
            error_margin,
        );
        return (discriminant, roots);
    }
    // https://en.wikipedia.org/wiki/Quartic_function#Solving_a_quartic_equation
    let p = (8.0 * coefficients[4] * coefficients[2] - 3.0 * coefficients[3].powi(2))
        / (8.0 * coefficients[4].powi(2));
    let q = (coefficients[3].powi(3) - 4.0 * coefficients[4] * coefficients[3] * coefficients[2]
        + 8.0 * coefficients[4].powi(2) * coefficients[1])
        / (8.0 * coefficients[4].powi(3));
    let d = [
        coefficients[2].powi(2) - 3.0 * coefficients[3] * coefficients[1]
            + 12.0 * coefficients[4] * coefficients[0],
        2.0 * coefficients[2].powi(3) - 9.0 * coefficients[3] * coefficients[2] * coefficients[1]
            + 27.0 * coefficients[3].powi(2) * coefficients[0]
            + 27.0 * coefficients[4] * coefficients[1].powi(2)
            - 72.0 * coefficients[4] * coefficients[2] * coefficients[0],
    ];
    let discriminant = d[1].powi(2) - 4.0 * d[0].powi(3);
    let c = discriminant.sqrt();
    let c = ((c + ComplexNumber::new(if c + d[1] == 0.0 { -d[1] } else { d[1] }, 0.0)) * 0.5)
        .powf(1.0 / 3.0);
    let e = ((c + ComplexNumber::new(d[0], 0.0).geometric_quotient(c))
        * (1.0 / (3.0 * coefficients[4]))
        - ComplexNumber::new(p * 2.0 / 3.0, 0.0))
    .powf(0.5)
        * 0.5;
    let mut solutions = Vec::with_capacity(4);
    for i in 0..4 {
        let f = (e.geometric_product(e) * -4.0 - ComplexNumber::new(2.0 * p, 0.0)
            + ComplexNumber::new(if i & 2 == 0 { q } else { -q }, 0.0).geometric_quotient(e))
        .powf(0.5)
            * 0.5;
        let g = ComplexNumber::new(-coefficients[3] / (4.0 * coefficients[4]), 0.0)
            + if i & 2 == 0 { -e } else { e }
            + if i & 1 == 0 { -f } else { f };
        solutions.push(Root {
            numerator: g,
            denominator: 1.0,
        });
    }
    (discriminant / -27.0, solutions)
}