numerical_analysis 0.1.2

A collection of algorithms for numerical analysis.
Documentation 
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use core::ops::{Add, AddAssign, Div, Mul, MulAssign, Sub, SubAssign};

use na::Const;
use na::Scalar;
use nalgebra as na;

/// Compute a first order accurate numeric gradient from a function.
///
/// `x` is the N dimensional point at which the gradient is evaluated.
/// `f` : R^N-> R is the function whose gradient is being computed.
/// `epsilon` is the coordinate wise spacing, usually named `h` in most sources.
pub fn gradient<T, const N: usize>(
    x: &na::Matrix<T, Const<N>, Const<1>, na::ArrayStorage<T, N, 1>>,
    f: &dyn Fn(&na::Matrix<T, Const<N>, Const<1>, na::ArrayStorage<T, N, 1>>) -> T,
    epsilon: T,
) -> na::Matrix<T, Const<N>, Const<1>, na::ArrayStorage<T, N, 1>>
where
    T: Scalar
        + Add<Output = T>
        + Sub<Output = T>
        + Div<Output = T>
        + Mul<Output = T>
        + Copy,
    na::ArrayStorage<T, N, 1>: Default,
{
    let mut grad =
        na::Matrix::<T, Const<N>, Const<1>, na::ArrayStorage<T, N, 1>>::default();
    for i in 0..N
    {
        let mut nx = x.clone();
        nx[i] = nx[i] + epsilon;
        let mut px = x.clone();
        px[i] = px[i] - epsilon;
        grad[i] = (f(&nx) - f(&px)) / (epsilon + epsilon);
    }

    grad
}

/// Compute a first order accurate numeric divergence from a function.
///
/// `x` is the N dimensional point at which the divergence is evaluated.
/// `f` : R^N-> R is the function whose divergence is being computed.
/// `epsilon` is the coordinate wise spacing, usually named `h` in most sources.
pub fn divergence<T, const N: usize>(
    x: &na::Matrix<T, Const<N>, Const<1>, na::ArrayStorage<T, N, 1>>,
    f: &dyn Fn(
        &na::Matrix<T, Const<N>, Const<1>, na::ArrayStorage<T, N, 1>>,
    ) -> na::Matrix<T, Const<N>, Const<1>, na::ArrayStorage<T, N, 1>>,
    epsilon: T,
) -> T
where
    T: Scalar
        + Add<Output = T>
        + Sub<Output = T>
        + Div<Output = T>
        + Mul<Output = T>
        + Copy
        + Default,
    na::ArrayStorage<T, N, 1>: Default,
{
    let mut div = T::default();
    for i in 0..N
    {
        let mut nx = x.clone();
        nx[i] = nx[i] + epsilon;
        let mut px = x.clone();
        px[i] = px[i] - epsilon;
        div = div + (f(&nx)[i] - f(&px)[i]) / (epsilon + epsilon);
    }

    div
}

// Taken from:
// https://rodolphe-vaillant.fr/entry/118/curvature-of-a-distance-field-implicit-surface
/// Compute an approximation of the mean curvature of an SDF function in $R^3$.
///
/// `x` is the N dimensional point at which the divergence is evaluated.
/// $f: R^N-> R$ is the function whose curvature is being computed.
/// `epsilon` is the coordinate wise spacing, usually named `h` in most sources.
type Vec3<T> = na::Matrix<T, Const<3>, Const<1>, na::ArrayStorage<T, 3, 1>>;
pub fn mean_curvature<T>(
    x: &na::Matrix<T, Const<3>, Const<1>, na::ArrayStorage<T, 3, 1>>,
    f: &dyn Fn(&na::Matrix<T, Const<3>, Const<1>, na::ArrayStorage<T, 3, 1>>) -> T,
    epsilon: T,
) -> T
where
    T: Scalar
        + Add<Output = T>
        + Sub<Output = T>
        + Div<Output = T>
        + Mul<Output = T>
        + Copy
        + Default
        + AddAssign
        + SubAssign
        + num_traits::Float,
    na::ArrayStorage<T, 3, 1>: Default,
{
    let z = T::from(0.).unwrap();
    let t1 = f(&(x + Vec3::new(epsilon, z, z)));
    let t2 = f(&(x - Vec3::new(epsilon, z, z)));
    let t3 = f(&(x + Vec3::new(z, epsilon, z)));
    let t4 = f(&(x - Vec3::new(z, epsilon, z)));
    let t5 = f(&(x + Vec3::new(z, z, epsilon)));
    let t6 = f(&(x - Vec3::new(z, z, epsilon)));

    (T::from(0.5).unwrap() / (epsilon * epsilon))
        * (t1 + t2 + t3 + t4 + t5 + t6 - T::from(6.).unwrap() * f(x))
}

pub fn gaussian_curvature<T>(
    x: &na::Matrix<T, Const<3>, Const<1>, na::ArrayStorage<T, 3, 1>>,
    f: &dyn Fn(&na::Matrix<T, Const<3>, Const<1>, na::ArrayStorage<T, 3, 1>>) -> T,
    epsilon: T,
) -> T
where
    T: Scalar
        + Add<Output = T>
        + Sub<Output = T>
        + Div<Output = T>
        + Mul<Output = T>
        + Copy
        + Default
        + AddAssign
        + SubAssign
        + num_traits::Float
        + MulAssign,
    na::ArrayStorage<T, 3, 1>: Default,
{
    let g = gradient(x, f, epsilon);
    let h = adjoint_hessian(x, f, epsilon);

    let k = g.transpose() * h * g;
    let n = g.dot(&g);
    let curv = k[(0, 0)] / (n * n);

    curv
}

pub fn hessian<T>(
    x: &na::Matrix<T, Const<3>, Const<1>, na::ArrayStorage<T, 3, 1>>,
    f: &dyn Fn(&na::Matrix<T, Const<3>, Const<1>, na::ArrayStorage<T, 3, 1>>) -> T,
    epsilon: T,
) -> na::Matrix3<T>
where
    T: Scalar
        + Add<Output = T>
        + Sub<Output = T>
        + Div<Output = T>
        + Mul<Output = T>
        + Copy
        + Default
        + AddAssign
        + SubAssign
        + num_traits::Float,
    na::ArrayStorage<T, 3, 1>: Default,
{
    let mut hessian = na::Matrix3::zeros();

    for i in 0..3
    {
        for j in 0..3
        {
            let mut acc = T::from(0.).unwrap();

            for dx in (-1..2).step_by(2)
            {
                let ox = T::from(dx).unwrap() * epsilon;
                for dy in (-1..2).step_by(2)
                {
                    let oy = T::from(dy).unwrap() * epsilon;

                    let mut sample = x.clone();
                    sample[i] += ox;
                    sample[j] += oy;
                    acc += f(&sample);
                }
            }

            hessian[(i, j)] = acc / (T::from(4.).unwrap() * epsilon * epsilon);
        }
    }

    hessian
}

pub fn adjoint_hessian<T>(
    x: &na::Matrix<T, Const<3>, Const<1>, na::ArrayStorage<T, 3, 1>>,
    f: &dyn Fn(&na::Matrix<T, Const<3>, Const<1>, na::ArrayStorage<T, 3, 1>>) -> T,
    epsilon: T,
) -> na::Matrix3<T>
where
    T: Scalar
        + Add<Output = T>
        + Sub<Output = T>
        + Div<Output = T>
        + Mul<Output = T>
        + Copy
        + Default
        + AddAssign
        + SubAssign
        + num_traits::Float,
    na::ArrayStorage<T, 3, 1>: Default,
{
    let h = hessian(x, f, epsilon);
    let mut adjoint = na::Matrix3::zeros();

    adjoint[(0, 0)] = h[(1, 1)] * h[(2, 2)] - h[(1, 2)] * h[(2, 1)];
    adjoint[(0, 1)] = h[(1, 2)] * h[(2, 0)] - h[(1, 0)] * h[(2, 2)];
    adjoint[(0, 2)] = h[(1, 0)] * h[(2, 1)] - h[(1, 1)] * h[(2, 0)];

    adjoint[(1, 0)] = h[(0, 2)] * h[(2, 1)] - h[(0, 1)] * h[(2, 2)];
    adjoint[(1, 1)] = h[(0, 0)] * h[(2, 2)] - h[(0, 2)] * h[(2, 0)];
    adjoint[(1, 2)] = h[(0, 1)] * h[(2, 0)] - h[(0, 0)] * h[(2, 1)];

    adjoint[(2, 0)] = h[(0, 1)] * h[(1, 2)] - h[(0, 2)] * h[(1, 1)];
    adjoint[(2, 1)] = h[(1, 0)] * h[(0, 2)] - h[(0, 0)] * h[(1, 2)];
    adjoint[(2, 2)] = h[(0, 0)] * h[(1, 1)] - h[(0, 1)] * h[(1, 0)];

    adjoint
}