num-dual 0.13.6

use crate::linalg::LU;
use crate::{
    Dual, DualNum, DualNumFloat, DualSVec, DualStruct, DualVec, Gradients, first_derivative,
    jacobian, partial,
};
use nalgebra::allocator::Allocator;
use nalgebra::{DefaultAllocator, Dim, OVector, SVector, U1, U2};
use std::marker::PhantomData;

/// Calculate the derivative of the unary implicit function
///         g(x, args) = 0
/// ```
/// # use num_dual::{implicit_derivative, DualNum, Dual2_64};
/// # use approx::assert_relative_eq;
/// let y = Dual2_64::from(25.0).derivative();
/// let x = implicit_derivative(|x,y| x.powi(2)-y, 5.0f64, &y);
/// assert_relative_eq!(x.re, y.sqrt().re, max_relative=1e-16);
/// assert_relative_eq!(x.v1, y.sqrt().v1, max_relative=1e-16);
/// assert_relative_eq!(x.v2, y.sqrt().v2, max_relative=1e-16);
/// ```
pub fn implicit_derivative<G, D: DualNum<F>, F: DualNumFloat, A: DualStruct<Dual<D, F>, F>>(
    g: G,
    x: F,
    args: &A::Inner,
) -> D
where
    G: Fn(Dual<D, F>, &A) -> Dual<D, F>,
{
    let mut x = D::from(x);
    for _ in 0..D::NDERIV {
        let (f, df) = first_derivative(partial(&g, args), x.clone());
        x -= f / df;
    }
    x
}

/// Calculate the derivative of the binary implicit function
///         g(x, y, args) = 0
/// ```
/// # use num_dual::{implicit_derivative_binary, Dual64};
/// # use approx::assert_relative_eq;
/// let a = Dual64::from(4.0).derivative();
/// let [x, y] =
///     implicit_derivative_binary(|x, y, a| [x * y - a, x + y - a - 1.0], 1.0f64, 4.0f64, &a);
/// assert_relative_eq!(x.re, 1.0, max_relative = 1e-16);
/// assert_relative_eq!(x.eps, 0.0, max_relative = 1e-16);
/// assert_relative_eq!(y.re, a.re, max_relative = 1e-16);
/// assert_relative_eq!(y.eps, a.eps, max_relative = 1e-16);
/// ```
pub fn implicit_derivative_binary<
    G,
    D: DualNum<F>,
    F: DualNumFloat,
    A: DualStruct<DualVec<D, F, U2>, F>,
>(
    g: G,
    x: F,
    y: F,
    args: &A::Inner,
) -> [D; 2]
where
    G: Fn(DualVec<D, F, U2>, DualVec<D, F, U2>, &A) -> [DualVec<D, F, U2>; 2],
{
    let mut x = D::from(x);
    let mut y = D::from(y);
    let args = A::from_inner(args);
    for _ in 0..D::NDERIV {
        let (f, jac) = jacobian(
            |x| {
                let [[x, y]] = x.data.0;
                SVector::from(g(x, y, &args))
            },
            &SVector::from([x.clone(), y.clone()]),
        );
        let [[f0, f1]] = f.data.0;
        let [[j00, j10], [j01, j11]] = jac.data.0;
        let det = (j00.clone() * &j11 - j01.clone() * &j10).recip();
        x -= (j11 * &f0 - j01 * &f1) * &det;
        y -= (j00 * &f1 - j10 * &f0) * &det;
    }
    [x, y]
}

/// Calculate the derivative of the multivariate implicit function
///         g(x, args) = 0
/// ```
/// # use num_dual::{implicit_derivative_vec, Dual64};
/// # use approx::assert_relative_eq;
/// # use nalgebra::SVector;
/// let a = Dual64::from(4.0).derivative();
/// let x = implicit_derivative_vec(
///     |x, a| SVector::from([x[0] * x[1] - a, x[0] + x[1] - a - 1.0]),
///     SVector::from([1.0f64, 4.0f64]),
///     &a,
///     );
/// assert_relative_eq!(x[0].re, 1.0, max_relative = 1e-16);
/// assert_relative_eq!(x[0].eps, 0.0, max_relative = 1e-16);
/// assert_relative_eq!(x[1].re, a.re, max_relative = 1e-16);
/// assert_relative_eq!(x[1].eps, a.eps, max_relative = 1e-16);
/// ```
pub fn implicit_derivative_vec<
    G,
    D: DualNum<F> + Copy,
    F: DualNumFloat,
    A: DualStruct<DualVec<D, F, N>, F>,
    N: Dim,
>(
    g: G,
    x: OVector<F, N>,
    args: &A::Inner,
) -> OVector<D, N>
where
    DefaultAllocator: Allocator<N> + Allocator<N, N> + Allocator<U1, N>,
    G: Fn(OVector<DualVec<D, F, N>, N>, &A) -> OVector<DualVec<D, F, N>, N>,
{
    let mut x = x.map(D::from);
    let args = A::from_inner(args);
    for _ in 0..D::NDERIV {
        let (f, jac) = jacobian(|x| g(x, &args), &x);
        x -= LU::new(jac).unwrap().solve(&f);
    }
    x
}

/// Calculate the derivative of stationary points of the scalar potential
///         g(x, args)
/// ```
/// # use num_dual::{implicit_derivative_sp, Dual64, DualNum, Dual2Vec, HyperDual};
/// # use approx::assert_relative_eq;
/// # use nalgebra::{vector, dvector};
/// let a = Dual64::from(2.0).derivative();
/// let x = implicit_derivative_sp(
///     |x, a: &Dual2Vec<_, _, _>| (a - x[0]).powi(2) + (x[1] - x[0]*x[0]).powi(2)*100.0,
///     vector![2.0f64, 4.0f64],
///     &a,
///     );
/// assert_relative_eq!(x[0].re, a.re, max_relative = 1e-13);
/// assert_relative_eq!(x[0].eps, a.eps, max_relative = 1e-13);
/// assert_relative_eq!(x[1].re, (a*a).re, max_relative = 1e-13);
/// assert_relative_eq!(x[1].eps, (a*a).eps, max_relative = 1e-13);
///
/// let x = implicit_derivative_sp(
///     |x, a: &HyperDual<_, _>| (a - x[0]).powi(2) + (x[1] - x[0]*x[0]).powi(2)*100.0,
///     dvector![2.0f64, 4.0f64],
///     &a,
///     );
/// assert_relative_eq!(x[0].re, a.re, max_relative = 1e-13);
/// assert_relative_eq!(x[0].eps, a.eps, max_relative = 1e-13);
/// assert_relative_eq!(x[1].re, (a*a).re, max_relative = 1e-13);
/// assert_relative_eq!(x[1].eps, (a*a).eps, max_relative = 1e-13);
/// ```
pub fn implicit_derivative_sp<
    G,
    D: DualNum<F> + Copy,
    F: DualNumFloat,
    A: DualStruct<N::Dual2<D, F>, F>,
    N: Gradients,
>(
    g: G,
    x: OVector<F, N>,
    args: &A::Inner,
) -> OVector<D, N>
where
    DefaultAllocator: Allocator<N> + Allocator<N, N> + Allocator<U1, N>,
    G: Fn(OVector<N::Dual2<D, F>, N>, &A) -> N::Dual2<D, F>,
{
    let mut x = x.map(D::from);
    for _ in 0..D::NDERIV {
        let (_, grad, hess) = N::hessian(|x, args| g(x, args), &x, args);
        x -= LU::new(hess).unwrap().solve(&grad);
    }
    x
}

/// An implicit function g(x, args) = 0 for which derivatives of x can be
/// calculated with the [ImplicitDerivative] struct.
pub trait ImplicitFunction<F> {
    /// data type of the parameter struct, needs to implement [DualStruct].
    type Parameters<D>;

    /// data type of the variable `x`, needs to be either `D`, `[D; 2]`, or `SVector<D, N>`.
    type Variable<D>;

    /// implementation of the residual function g(x, args) = 0.
    fn residual<D: DualNum<F> + Copy>(
        x: Self::Variable<D>,
        parameters: &Self::Parameters<D>,
    ) -> Self::Variable<D>;
}

/// Helper struct that stores parameters in dual and real form and provides functions
/// for evaluating real residuals (for external solvers) and implicit derivatives for
/// arbitrary dual numbers.
pub struct ImplicitDerivative<G: ImplicitFunction<F>, D: DualNum<F> + Copy, F: DualNumFloat, V> {
    base: G::Parameters<D::Real>,
    derivative: G::Parameters<D>,
    phantom: PhantomData<V>,
}

impl<G: ImplicitFunction<F>, D: DualNum<F> + Copy, F: DualNum<F> + DualNumFloat>
    ImplicitDerivative<G, D, F, G::Variable<f64>>
where
    G::Parameters<D>: DualStruct<D, F, Real = G::Parameters<F>>,
{
    pub fn new(_: G, parameters: G::Parameters<D>) -> Self {
        Self {
            base: parameters.re(),
            derivative: parameters,
            phantom: PhantomData,
        }
    }

    /// Evaluate the (real) residual for a scalar function.
    pub fn residual(&self, x: G::Variable<F>) -> G::Variable<F> {
        G::residual(x, &self.base)
    }
}

impl<G: ImplicitFunction<F>, D: DualNum<F> + Copy, F: DualNum<F> + DualNumFloat>
    ImplicitDerivative<G, D, F, F>
where
    G::Parameters<D>: DualStruct<D, F, Real = G::Parameters<F>>,
{
    /// Evaluate the implicit derivative for a scalar function.
    pub fn implicit_derivative<A: DualStruct<Dual<D, F>, F, Inner = G::Parameters<D>>>(
        &self,
        x: F,
    ) -> D
    where
        G: ImplicitFunction<F, Variable<Dual<D, F>> = Dual<D, F>, Parameters<Dual<D, F>> = A>,
    {
        implicit_derivative(G::residual::<Dual<D, F>>, x, &self.derivative)
    }
}

impl<G: ImplicitFunction<F>, D: DualNum<F> + Copy, F: DualNum<F> + DualNumFloat>
    ImplicitDerivative<G, D, F, [F; 2]>
where
    G::Parameters<D>: DualStruct<D, F, Real = G::Parameters<F>>,
{
    /// Evaluate the implicit derivative for a bivariate function.
    pub fn implicit_derivative<A: DualStruct<DualVec<D, F, U2>, F, Inner = G::Parameters<D>>>(
        &self,
        x: F,
        y: F,
    ) -> [D; 2]
    where
        G: ImplicitFunction<
                F,
                Variable<DualVec<D, F, U2>> = [DualVec<D, F, U2>; 2],
                Parameters<DualVec<D, F, U2>> = A,
            >,
    {
        implicit_derivative_binary(
            |x, y, args: &A| G::residual::<DualVec<D, F, U2>>([x, y], args),
            x,
            y,
            &self.derivative,
        )
    }
}

impl<G: ImplicitFunction<F>, D: DualNum<F> + Copy, F: DualNum<F> + DualNumFloat, const N: usize>
    ImplicitDerivative<G, D, F, SVector<F, N>>
where
    G::Parameters<D>: DualStruct<D, F, Real = G::Parameters<F>>,
{
    /// Evaluate the implicit derivative for a multivariate function.
    pub fn implicit_derivative<A: DualStruct<DualSVec<D, F, N>, F, Inner = G::Parameters<D>>>(
        &self,
        x: SVector<F, N>,
    ) -> SVector<D, N>
    where
        G: ImplicitFunction<
                F,
                Variable<DualSVec<D, F, N>> = SVector<DualSVec<D, F, N>, N>,
                Parameters<DualSVec<D, F, N>> = A,
            >,
    {
        implicit_derivative_vec(G::residual::<DualSVec<D, F, N>>, x, &self.derivative)
    }
}

#[cfg(test)]
mod test {
    use super::*;
    use nalgebra::SVector;

    struct TestFunction;
    impl ImplicitFunction<f64> for TestFunction {
        type Parameters<D> = D;
        type Variable<D> = D;

        fn residual<D: DualNum<f64> + Copy>(x: D, square: &D) -> D {
            *square - x * x
        }
    }

    struct TestFunction2;
    impl ImplicitFunction<f64> for TestFunction2 {
        type Parameters<D> = (D, D);
        type Variable<D> = [D; 2];

        fn residual<D: DualNum<f64> + Copy>([x, y]: [D; 2], (square_sum, sum): &(D, D)) -> [D; 2] {
            [*square_sum - x * x - y * y, *sum - x - y]
        }
    }

    struct TestFunction3<const N: usize>;
    impl<const N: usize> ImplicitFunction<f64> for TestFunction3<N> {
        type Parameters<D> = D;
        type Variable<D> = SVector<D, N>;

        fn residual<D: DualNum<f64> + Copy>(x: SVector<D, N>, &square_sum: &D) -> SVector<D, N> {
            let mut res = x;
            for i in 1..N {
                res[i] = x[i] - x[i - 1] - D::from(1.0);
            }
            res[0] = square_sum - x.dot(&x);
            res
        }
    }

    #[test]
    fn test() {
        let f: crate::Dual64 = Dual::from(25.0).derivative();
        let func = ImplicitDerivative::new(TestFunction, f);
        println!("{}", func.residual(5.0));
        println!("{}", func.implicit_derivative(5.0));
        println!("{}", f.sqrt());
        assert_eq!(f.sqrt(), func.implicit_derivative(5.0));

        let a: crate::Dual64 = Dual::from(25.0).derivative();
        let b: crate::Dual64 = Dual::from(7.0);
        let func = ImplicitDerivative::new(TestFunction2, (a, b));
        println!("\n{:?}", func.residual([4.0, 3.0]));
        let [x, y] = func.implicit_derivative(4.0, 3.0);
        let xa = (b + (a * 2.0 - b * b).sqrt()) * 0.5;
        let ya = (b - (a * 2.0 - b * b).sqrt()) * 0.5;
        println!("{x}, {y}");
        println!("{xa}, {ya}");
        assert_eq!(x, xa);
        assert_eq!(y, ya);

        let s: crate::Dual64 = Dual::from(30.0).derivative();
        let func = ImplicitDerivative::new(TestFunction3, s);
        println!("\n{:?}", func.residual(SVector::from([1.0, 2.0, 3.0, 4.0])));
        let x = func.implicit_derivative(SVector::from([1.0, 2.0, 3.0, 4.0]));
        let x0 = ((s - 5.0).sqrt() - 5.0) * 0.5;
        println!("{}, {}, {}, {}", x[0], x[1], x[2], x[3]);
        println!("{}, {}, {}, {}", x0 + 1.0, x0 + 2.0, x0 + 3.0, x0 + 4.0);
        assert_eq!(x0 + 1.0, x[0]);
        assert_eq!(x0 + 2.0, x[1]);
        assert_eq!(x0 + 3.0, x[2]);
        assert_eq!(x0 + 4.0, x[3]);
    }
}