Function shessian 

pub fn shessian<V>(
    f: &impl Fn(&V) -> f64,
    x0: &V,
    h: Option<f64>,
) -> V::MatrixNxN
where
    V: Vector<f64>,

Hessian of a multivariate, scalar-valued function using the central difference approximation.

Arguments

• f - Multivariate, scalar-valued function, $f:\mathbb{R}^{n}\to\mathbb{R}$.
• x0 - Evaluation point, $\mathbf{x}_{0}\in\mathbb{R}^{n}$.
• h - Relative step size, $h\in\mathbb{R}$. Defaults to CBRT_EPS.

Returns

Hessian of $f$ with respect to $\mathbf{x}$, evaluated at $\mathbf{x}=\mathbf{x}_{0}$.

$$\mathbf{H}(\mathbf{x}_{0})\in\mathbb{R}^{n\times n}$$

Note

This function performs $2n(n+1)$ evaluations of $f(x)$.

Example

Approximate the Hessian of

$$f(\mathbf{x})=x_{0}^{5}x_{1}+x_{0}\sin^{3}{x_{1}}$$

at $\mathbf{x}=(5,8)^{T}$, and compare the result to the true result of

$$ \begin{aligned} \mathbf{H}\left((5,8)^{T}\right)&= \begin{bmatrix} 20x_{0}^{3}x_{1} & 5x_{0}^{4}+3\sin^{2}{x_{1}}\cos{x_{1}} \\ 5x_{0}^{4}+3\sin^{2}{x_{1}}\cos{x_{1}} & 6x_{0}\sin{x_{1}}\cos^{2}{x_{1}}-3x_{0}\sin^{3}{x_{1}} \end{bmatrix} \bigg\rvert_{\mathbf{x}=(5,8)^{T}} \\ &= \begin{bmatrix} 20(5)^{3}(8) & 5(5)^{4}+3\sin^{2}{(8)}\cos{(8)} \\ 5(5)^{4}+3\sin^{2}{(8)}\cos{(8)} & 6(5)\sin{(8)}\cos^{2}{(8)}-3(5)\sin^{3}{(8)} \end{bmatrix} \\ &= \begin{bmatrix} 20000 & 3125+3\sin^{2}{(8)}\cos{(8)} \\ 3125+3\sin^{2}{(8)}\cos{(8)} & 30\sin{(8)}\cos^{2}{(8)}-15\sin^{3}{(8)} \end{bmatrix} \end{aligned} $$

Using standard vectors

use linalg_traits::{Mat, Matrix};
use numtest::*;

use numdiff::central_difference::shessian;

// Define the function, f(x).
let f = |x: &Vec<f64>| x[0].powi(5) * x[1] + x[0] * x[1].sin().powi(3);

// Define the evaluation point.
let x0 = vec![5.0, 8.0];

// Approximate the Hessian of f(x) at the evaluation point.
let hess: Mat<f64> = shessian(&f, &x0, None);

// True Hessian of f(x) at the evaluation point.
let hess_true: Mat<f64> = Mat::from_row_slice(
    2,
    2,
    &[
        20000.0,
        3125.0 + 3.0 * 8.0_f64.sin().powi(2) * 8.0_f64.cos(),
        3125.0 + 3.0 * 8.0_f64.sin().powi(2) * 8.0_f64.cos(),
        30.0 * 8.0_f64.sin() * 8.0_f64.cos().powi(2) - 15.0 * 8.0_f64.sin().powi(3)
    ]
);

// Check the accuracy of the Hessian approximation.
assert_arrays_equal_to_decimal!(hess, hess_true, 3);

Using other vector types

We can also use other types of vectors, such as nalgebra::SVector, nalgebra::DVector, ndarray::Array1, faer::Mat, or any other type of vector that implements the linalg_traits::Vector trait.

use faer::Mat as FMat;
use linalg_traits::{Mat, Matrix, Vector};
use nalgebra::{dvector, DMatrix, DVector, SMatrix, SVector};
use ndarray::{array, Array1, Array2};
use numtest::*;

use numdiff::central_difference::shessian;

let hess_true: Mat<f64> = Mat::from_row_slice(
    2,
    2,
    &[
        20000.0,
        3125.0 + 3.0 * 8.0_f64.sin().powi(2) * 8.0_f64.cos(),
        3125.0 + 3.0 * 8.0_f64.sin().powi(2) * 8.0_f64.cos(),
        30.0 * 8.0_f64.sin() * 8.0_f64.cos().powi(2) - 15.0 * 8.0_f64.sin().powi(3)
    ]
);

// nalgebra::DVector
let f_dvector = |x: &DVector<f64>| x[0].powi(5) * x[1] + x[0] * x[1].sin().powi(3);
let x0_dvector: DVector<f64> = dvector![5.0, 8.0];
let hess_dvector: DMatrix<f64> = shessian(&f_dvector, &x0_dvector, None);
assert_arrays_equal_to_decimal!(hess_dvector, hess_true, 3);

// nalgebra::SVector
let f_svector = |x: &SVector<f64,2>| x[0].powi(5) * x[1] + x[0] * x[1].sin().powi(3);
let x0_svector: SVector<f64, 2> = SVector::from_row_slice(&[5.0, 8.0]);
let hess_svector: SMatrix<f64, 2, 2> = shessian(&f_svector, &x0_svector, None);
assert_arrays_equal_to_decimal!(hess_svector, hess_true, 3);

// ndarray::Array1
let f_array1 = |x: &Array1<f64>| x[0].powi(5) * x[1] + x[0] * x[1].sin().powi(3);
let x0_array1: Array1<f64> = array![5.0, 8.0];
let hess_array1: Array2<f64> = shessian(&f_array1, &x0_array1, None);
assert_arrays_equal_to_decimal!(hess_array1, hess_true, 3);

// faer::Mat
let f_mat = |x: &FMat<f64>| x[(0, 0)].powi(5) * x[(1, 0)] + x[(0, 0)] * x[(1, 0)].sin().powi(3);
let x0_mat: FMat<f64> = FMat::from_slice(&[5.0, 8.0]);
let hess_mat: FMat<f64> = shessian(&f_mat, &x0_mat, None);
assert_arrays_equal_to_decimal!(hess_mat.as_row_slice(), hess_true, 3);

Modifying the relative step size

We can also modify the relative step size. Choosing a coarser relative step size, we get a worse approximation.

use linalg_traits::{Mat, Matrix};
use numtest::*;

use numdiff::central_difference::shessian;

let f = |x: &Vec<f64>| x[0].powi(5) * x[1] + x[0] * x[1].sin().powi(3);
let x0 = vec![5.0, 8.0];

let hess: Mat<f64> = shessian(&f, &x0, Some(0.001));
let hess_true: Mat<f64> = Mat::from_row_slice(
    2,
    2,
    &[
        20000.0,
        3125.0 + 3.0 * 8.0_f64.sin().powi(2) * 8.0_f64.cos(),
        3125.0 + 3.0 * 8.0_f64.sin().powi(2) * 8.0_f64.cos(),
        30.0 * 8.0_f64.sin() * 8.0_f64.cos().powi(2) - 15.0 * 8.0_f64.sin().powi(3)
    ]
);

assert_arrays_equal_to_decimal!(hess, hess_true, 1);