conspire 0.7.7

The Rust interface to conspire.
Documentation 
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use crate::math::Quantity;
use crate::units::Dimensionless;
use std::f64::consts::TAU;

use super::{
    super::{
        Rank2, Tensor, TensorArray, TensorError,
        rank_0::{TensorRank0, list::TensorRank0List},
        rank_1::{TensorRank1, cross::CrossProduct},
    },
    TensorRank2,
};
use crate::{ABS_TOL, math::assert::Assert};

impl<I> TensorRank2<3, I, I, Dimensionless> {
    /// Returns the invariants of the 3x3 symmetric tensor.
    pub fn invariants(&self) -> TensorRank0List<3> {
        TensorRank0List::from([
            self.trace().value(),
            self.second_invariant().value(),
            self.determinant(),
        ])
    }
    /// Returns the eigenvalues and (row-wise) eigenvectors of the 3x3 symmetric tensor.
    ///
    /// Reuse this alongside [`TensorRank2::powm_from_eigen`]/[`TensorRank2::dpowm_from_eigen`]
    /// to evaluate several exponents against the same tensor while sharing one
    /// eigendecomposition, instead of paying for a fresh cubic solve on every
    /// [`TensorRank2::powm`]/[`TensorRank2::dpowm`] call.
    pub fn eigen(&self) -> Result<(TensorRank0List<3>, Self), TensorError> {
        let eigenvalues = solve_cubic_symmetric(self.invariants())?;
        let eigenvectors = find_orthonormal_eigenvectors(&eigenvalues, self);
        Ok((eigenvalues, eigenvectors))
    }
}

/// Shared by [`super::logarithm`] and [`super::power`].
pub(super) fn solve_cubic_symmetric(
    coefficients: TensorRank0List<3>,
) -> Result<TensorRank0List<3>, TensorError> {
    let c2 = coefficients[0];
    let c1 = coefficients[1];
    let c0 = coefficients[2];
    let p = c1 - c2 * c2 / 3.0;
    let q = -(2.0 * c2.powi(3) - 9.0 * c2 * c1 + 27.0 * c0) / 27.0;
    if p.abs() < ABS_TOL {
        let t = (-q).cbrt();
        let lambda = t + c2 / 3.0;
        return Ok(TensorRank0List::from([lambda; _]));
    }
    let discriminant = -4.0 * p * p * p - 27.0 * q * q;
    let scale = (4.0 * p * p * p).abs().max(27.0 * q * q);
    if discriminant.abs() <= 1e-13 * scale {
        let r = (q / 2.0).cbrt();
        let lambda_double = r + c2 / 3.0;
        let lambda_simple = -2.0 * r + c2 / 3.0;
        let lambdas = if lambda_double >= lambda_simple {
            [lambda_double, lambda_double, lambda_simple]
        } else {
            [lambda_simple, lambda_double, lambda_double]
        };
        Ok(TensorRank0List::from(lambdas))
    } else if discriminant > 0.0 {
        let sqrt_term = (-p / 3.0).sqrt();
        let cos_arg = 3.0 * q / (2.0 * p * (-p / 3.0).sqrt());
        let cos_arg = cos_arg.clamp(-1.0, 1.0);
        let theta = cos_arg.acos();
        let mut lambdas = [
            2.0 * sqrt_term * (theta / 3.0).cos() + c2 / 3.0,
            2.0 * sqrt_term * ((theta + TAU) / 3.0).cos() + c2 / 3.0,
            2.0 * sqrt_term * ((theta + 2.0 * TAU) / 3.0).cos() + c2 / 3.0,
        ];
        lambdas.iter_mut().for_each(|lambda| {
            for _ in 0..2 {
                let x = *lambda;
                let f = x * x * x - c2 * x * x + c1 * x - c0;
                let f_prime = 3.0 * x * x - 2.0 * c2 * x + c1;
                if f_prime.abs() < ABS_TOL {
                    break;
                }
                *lambda -= f / f_prime;
            }
        });
        lambdas.sort_by(|a, b| b.partial_cmp(a).unwrap());
        Ok(TensorRank0List::from(lambdas))
    } else {
        Err(TensorError::SymmetricMatrixComplexEigenvalues)
    }
}

/// Shared by [`super::logarithm`] and [`super::power`]; see [`solve_cubic_symmetric`].
pub(super) fn find_orthonormal_eigenvectors<I>(
    eigenvalues: &TensorRank0List<3>,
    tensor: &TensorRank2<3, I, I, Dimensionless>,
) -> TensorRank2<3, I, I, Dimensionless> {
    if Assert::default()
        .eq_within_tols(eigenvalues[0], &eigenvalues[1])
        .is_ok()
    {
        let mut eigenvectors = TensorRank2::zero();
        eigenvectors[2] = eigenvector_symmetric(eigenvalues[2], tensor);
        eigenvectors[0] = orthogonal_unit_vector(&eigenvectors[2]);
        eigenvectors[1] = eigenvectors[2].cross(&eigenvectors[0]);
        eigenvectors
    } else if Assert::default()
        .eq_within_tols(eigenvalues[1], &eigenvalues[2])
        .is_ok()
    {
        let mut eigenvectors = TensorRank2::zero();
        eigenvectors[0] = eigenvector_symmetric(eigenvalues[0], tensor);
        eigenvectors[1] = orthogonal_unit_vector(&eigenvectors[0]);
        eigenvectors[2] = eigenvectors[0].cross(&eigenvectors[1]);
        eigenvectors
    } else {
        let mut eigenvectors = eigenvalues
            .iter()
            .map(|&eigenvalue| eigenvector_symmetric(eigenvalue, tensor))
            .collect::<TensorRank2<3, I, I, Dimensionless>>();
        eigenvectors[0].normalize();
        let proj1 = &eigenvectors[1] * &eigenvectors[0];
        for i in 0..3 {
            let projected = proj1 * eigenvectors[0][i];
            eigenvectors[1][i] -= projected;
        }
        eigenvectors[1].normalize();
        eigenvectors[2] = eigenvectors[0].cross(&eigenvectors[1]);
        eigenvectors
    }
}

fn orthogonal_unit_vector<I>(
    vector: &TensorRank1<3, I, Dimensionless>,
) -> TensorRank1<3, I, Dimensionless> {
    let axis = vector
        .iter()
        .enumerate()
        .min_by(|(_, a), (_, b)| a.abs().partial_cmp(&b.abs()).unwrap())
        .map(|(i, _)| i)
        .unwrap();
    let mut other = TensorRank1::<3, I, Dimensionless>::zero();
    other[axis] = Quantity::new(1.0);
    vector.cross(&other).normalized()
}

fn eigenvector_symmetric<I>(
    eigenvalue: TensorRank0,
    tensor: &TensorRank2<3, I, I, Dimensionless>,
) -> TensorRank1<3, I, Dimensionless> {
    let m = tensor - TensorRank2::identity() * eigenvalue;
    [m[1].cross(&m[2]), m[0].cross(&m[2]), m[0].cross(&m[1])]
        .into_iter()
        .max_by(|a, b| a.norm().partial_cmp(&b.norm()).unwrap())
        .unwrap()
        .normalized()
}

/// Shared by [`super::logarithm`] and [`super::power`]; see [`solve_cubic_symmetric`].
pub(super) fn reconstruct_symmetric<I>(
    eigenvalues: TensorRank0List<3>,
    eigenvectors: TensorRank2<3, I, I, Dimensionless>,
) -> TensorRank2<3, I, I, Dimensionless> {
    let mut tensor = TensorRank2::zero();
    eigenvalues
        .iter()
        .zip(eigenvectors.iter())
        .for_each(|(eigenvalue, eigenvector)| {
            tensor
                .iter_mut()
                .zip(eigenvector.iter())
                .for_each(|(tensor_i, eigenvector_i)| {
                    tensor_i.iter_mut().zip(eigenvector.iter()).for_each(
                        |(tensor_ij, eigenvector_j)| {
                            *tensor_ij += eigenvalue * eigenvector_i * eigenvector_j
                        },
                    )
                })
        });
    tensor
}