argmin_testfunctions/sphere.rs
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// Copyright 2018-2024 argmin developers
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://apache.org/licenses/LICENSE-2.0> or the MIT license <LICENSE-MIT or
// http://opensource.org/licenses/MIT>, at your option. This file may not be
// copied, modified, or distributed except according to those terms.
//! # Sphere function
//!
//! Defined as
//!
//! `f(x) = \sum_{i=1}^n x_i^2`
//!
//! where `x_i \in (-\infty, \infty)`
//!
//! The minimum is at `f(x_1, x_2, ..., x_n) = f(0, 0, ..., 0) = 0`.
use num::{Float, FromPrimitive};
use std::iter::Sum;
/// Sphere test function
///
/// Defined as
///
/// `f(x_1, x_2, ..., x_n) = \sum_{i=1}^n x_i^2
///
/// where `x_i \in (-\infty, \infty)` and `n > 0`.
///
/// The global minimum is at `f(x_1, x_2, ..., x_n) = f(0, 0, ..., 0) = 0`.
pub fn sphere<T>(param: &[T]) -> T
where
T: Float + FromPrimitive + Sum,
{
param.iter().map(|x| x.powi(2)).sum()
}
/// Derivative of sphere test function
///
/// Defined as
///
/// `f(x_1, x_2, ..., x_n) = (2 * x_1, 2 * x_2, ... 2 * x_n)`
///
/// where `x_i \in (-\infty, \infty)` and `n > 0`.
pub fn sphere_derivative<T>(param: &[T]) -> Vec<T>
where
T: Float + FromPrimitive,
{
let num2 = T::from_f64(2.0).unwrap();
param.iter().map(|x| num2 * *x).collect()
}
/// Derivative of sphere test function
///
/// Defined as
///
/// `f(x_1, x_2, ..., x_n) = (2 * x_1, 2 * x_2, ... 2 * x_n)`
///
/// where `x_i \in (-\infty, \infty)` and `n > 0`.
///
/// This is the const generics version, which requires the number of parameters to be known
/// at compile time.
pub fn sphere_derivative_const<const N: usize, T>(param: &[T; N]) -> [T; N]
where
T: Float + FromPrimitive,
{
let num2 = T::from_f64(2.0).unwrap();
let mut deriv = [T::from_f64(0.0).unwrap(); N];
for i in 0..N {
deriv[i] = num2 * param[i];
}
deriv
}
/// Hessian of sphere test function
pub fn sphere_hessian<T>(param: &[T]) -> Vec<Vec<T>>
where
T: Float + FromPrimitive,
{
let n = param.len();
let mut hessian = vec![vec![T::from_f64(0.0).unwrap(); n]; n];
for (i, row) in hessian.iter_mut().enumerate().take(n) {
row[i] = T::from_f64(2.0).unwrap();
}
hessian
}
/// Hessian of sphere test function
///
/// This is the const generics version, which requires the number of parameters to be known
/// at compile time.
pub fn sphere_hessian_const<const N: usize, T>(_param: &[T; N]) -> [[T; N]; N]
where
T: Float + FromPrimitive,
{
let mut hessian = [[T::from_f64(0.0).unwrap(); N]; N];
for (i, row) in hessian.iter_mut().enumerate().take(N) {
row[i] = T::from_f64(2.0).unwrap();
}
hessian
}
#[cfg(test)]
mod tests {
use super::*;
use approx::assert_relative_eq;
use finitediff::FiniteDiff;
use proptest::prelude::*;
#[test]
fn test_sphere_optimum() {
assert_relative_eq!(sphere(&[0.0_f32, 0.0_f32]), 0.0, epsilon = f32::EPSILON);
assert_relative_eq!(sphere(&[0.0_f64, 0.0_f64]), 0.0, epsilon = f64::EPSILON);
}
proptest! {
#[test]
fn test_sphere(a in -10.0..10.0,
b in -10.0..10.0,
c in -10.0..10.0,
d in -10.0..10.0,
e in -10.0..10.0,
f in -10.0..10.0,
g in -10.0..10.0,
h in -10.0..10.0) {
let param: [f64; 8] = [a, b, c, d, e, f, g, h];
let v1 = sphere(¶m);
let v2 = a.powi(2) + b.powi(2) + c.powi(2) + d.powi(2) + e.powi(2) + f.powi(2) + g.powi(2) + h.powi(2);
assert_relative_eq!(v1, v2, epsilon = f64::EPSILON);
}
}
proptest! {
#[test]
fn test_sphere_derivative(a in -10.0..10.0,
b in -10.0..10.0,
c in -10.0..10.0,
d in -10.0..10.0,
e in -10.0..10.0,
f in -10.0..10.0,
g in -10.0..10.0,
h in -10.0..10.0) {
let param = [a, b, c, d, e, f, g, h];
let derivative = sphere_derivative(¶m);
let derivative_fd =
[2.0 * a, 2.0 * b, 2.0 * c, 2.0 * d, 2.0 * e, 2.0 * f, 2.0 * g, 2.0 * h];
for i in 0..derivative.len() {
assert_relative_eq!(
derivative[i],
derivative_fd[i],
epsilon = 1e-5,
max_relative = 1e-2
);
}
}
}
proptest! {
#[test]
fn test_sphere_derivative_const(a in -10.0..10.0,
b in -10.0..10.0,
c in -10.0..10.0,
d in -10.0..10.0,
e in -10.0..10.0,
f in -10.0..10.0,
g in -10.0..10.0,
h in -10.0..10.0) {
let param = [a, b, c, d, e, f, g, h];
let derivative = sphere_derivative_const(¶m);
let derivative_fd =
[2.0 * a, 2.0 * b, 2.0 * c, 2.0 * d, 2.0 * e, 2.0 * f, 2.0 * g, 2.0 * h];
for i in 0..derivative.len() {
assert_relative_eq!(
derivative[i],
derivative_fd[i],
epsilon = 1e-5,
max_relative = 1e-2,
);
}
}
}
proptest! {
#[test]
fn test_sphere_derivative_finitediff(a in -10.0..10.0,
b in -10.0..10.0,
c in -10.0..10.0,
d in -10.0..10.0,
e in -10.0..10.0,
f in -10.0..10.0,
g in -10.0..10.0,
h in -10.0..10.0) {
let param = [a, b, c, d, e, f, g, h];
let derivative = sphere_derivative(¶m);
let derivative_fd = Vec::from(param).central_diff(&|x| sphere(&x));
for i in 0..derivative.len() {
assert_relative_eq!(
derivative[i],
derivative_fd[i],
epsilon = 1e-5,
max_relative = 1e-2
);
}
}
}
proptest! {
#[test]
fn test_sphere_derivative_const_finitediff(a in -10.0..10.0,
b in -10.0..10.0,
c in -10.0..10.0,
d in -10.0..10.0,
e in -10.0..10.0,
f in -10.0..10.0,
g in -10.0..10.0,
h in -10.0..10.0) {
let param = [a, b, c, d, e, f, g, h];
let derivative = sphere_derivative_const(¶m);
let derivative_fd = Vec::from(param).central_diff(&|x| sphere(&x));
for i in 0..derivative.len() {
assert_relative_eq!(
derivative[i],
derivative_fd[i],
epsilon = 1e-5,
max_relative = 1e-2
);
}
}
}
proptest! {
#[test]
fn test_sphere_hessian(a in -10.0..10.0,
b in -10.0..10.0,
c in -10.0..10.0,
d in -10.0..10.0,
e in -10.0..10.0,
f in -10.0..10.0,
g in -10.0..10.0,
h in -10.0..10.0) {
let param = [a, b, c, d, e, f, g, h];
let hessian = sphere_hessian(¶m);
for i in 0..hessian.len() {
for j in 0..hessian.len() {
if i == j {
assert_relative_eq!(hessian[i][j], 2.0, epsilon = f64::EPSILON);
} else {
assert_relative_eq!(hessian[i][j], 0.0, epsilon = f64::EPSILON);
}
}
}
}
}
proptest! {
#[test]
fn test_sphere_hessian_const(a in -10.0..10.0,
b in -10.0..10.0,
c in -10.0..10.0,
d in -10.0..10.0,
e in -10.0..10.0,
f in -10.0..10.0,
g in -10.0..10.0,
h in -10.0..10.0) {
let param = [a, b, c, d, e, f, g, h];
let hessian = sphere_hessian_const(¶m);
for i in 0..hessian.len() {
for j in 0..hessian.len() {
if i == j {
assert_relative_eq!(hessian[i][j], 2.0, epsilon = f64::EPSILON);
} else {
assert_relative_eq!(hessian[i][j], 0.0, epsilon = f64::EPSILON);
}
}
}
}
}
proptest! {
#[test]
fn test_sphere_hessian_finitediff(a in -10.0..10.0,
b in -10.0..10.0,
c in -10.0..10.0,
d in -10.0..10.0,
e in -10.0..10.0,
f in -10.0..10.0,
g in -10.0..10.0,
h in -10.0..10.0) {
let param = [a, b, c, d, e, f, g, h];
let hessian = sphere_hessian(¶m);
let hessian_fd = Vec::from(param).central_hessian(&|x| sphere_derivative(&x));
for i in 0..hessian.len() {
for j in 0..hessian.len() {
assert_relative_eq!(
hessian[i][j],
hessian_fd[i][j],
epsilon = 1e-5,
max_relative = 1e-2
);
}
}
}
}
proptest! {
#[test]
fn test_sphere_hessian_const_finitediff(a in -10.0..10.0,
b in -10.0..10.0,
c in -10.0..10.0,
d in -10.0..10.0,
e in -10.0..10.0,
f in -10.0..10.0,
g in -10.0..10.0,
h in -10.0..10.0) {
let param = [a, b, c, d, e, f, g, h];
let hessian = sphere_hessian_const(¶m);
let hessian_fd = Vec::from(param).central_hessian(&|x| sphere_derivative(&x));
for i in 0..hessian.len() {
for j in 0..hessian.len() {
assert_relative_eq!(
hessian[i][j],
hessian_fd[i][j],
epsilon = 1e-5,
max_relative = 1e-2
);
}
}
}
}
}