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```
``````// Copyright 2018-2022 argmin developers
//
// copied, modified, or distributed except according to those terms.

use crate::core::{ArgminFloat, Error, Gradient, Hessian, IterState, Problem, Solver, KV};
use argmin_math::{ArgminDot, ArgminInv, ArgminScaledSub};
#[cfg(feature = "serde1")]
use serde::{Deserialize, Serialize};
use std::default::Default;

/// # Newton's method
///
/// Newton's method iteratively finds the stationary points of a function f by using a second order
/// approximation of f at the current point.
///
/// The stepsize `gamma` can be adjusted with the [`with_gamma`](`Newton::with_gamma`) method. It
/// must be in `(0, 1])` and defaults to `1`.
///
/// ## Requirements on the optimization problem
///
/// The optimization problem is required to implement [`Gradient`] and [`Hessian`].
///
/// ## Reference
///
/// Jorge Nocedal and Stephen J. Wright (2006). Numerical Optimization.
/// Springer. ISBN 0-387-30303-0.
#[derive(Clone, Copy)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
pub struct Newton<F> {
/// gamma
gamma: F,
}

impl<F> Newton<F>
where
F: ArgminFloat,
{
/// Construct a new instance of [`Newton`]
///
/// # Example
///
/// ```
/// # use argmin::solver::newton::Newton;
/// let newton: Newton<f64> = Newton::new();
/// ```
pub fn new() -> Self {
Newton { gamma: float!(1.0) }
}

/// Set step size gamma
///
/// Gamma must be in `(0, 1]` and defaults to `1`.
///
/// # Example
///
/// ```
/// # use argmin::solver::newton::Newton;
/// # use argmin::core::Error;
/// # fn main() -> Result<(), Error> {
/// let newton: Newton<f64> = Newton::new().with_gamma(0.4)?;
/// # Ok(())
/// # }
/// ```
pub fn with_gamma(mut self, gamma: F) -> Result<Self, Error> {
if gamma <= float!(0.0) || gamma > float!(1.0) {
return Err(argmin_error!(
InvalidParameter,
"Newton: gamma must be in  (0, 1]."
));
}
self.gamma = gamma;
Ok(self)
}
}

impl<F> Default for Newton<F>
where
F: ArgminFloat,
{
fn default() -> Newton<F> {
Newton::new()
}
}

impl<O, P, G, H, F> Solver<O, IterState<P, G, (), H, F>> for Newton<F>
where
O: Gradient<Param = P, Gradient = G> + Hessian<Param = P, Hessian = H>,
P: Clone + ArgminScaledSub<P, F, P>,
H: ArgminInv<H> + ArgminDot<G, P>,
F: ArgminFloat,
{
const NAME: &'static str = "Newton method";

fn next_iter(
&mut self,
problem: &mut Problem<O>,
mut state: IterState<P, G, (), H, F>,
) -> Result<(IterState<P, G, (), H, F>, Option<KV>), Error> {
let param = state.take_param().ok_or_else(argmin_error_closure!(
NotInitialized,
concat!(
"`Newton` requires an initial parameter vector. ",
"Please provide an initial guess via `Executor`s `configure` method."
)
))?;
let hessian = problem.hessian(&param)?;
Ok((state.param(new_param), None))
}
}

#[cfg(test)]
mod tests {
use super::*;
use crate::core::ArgminError;
#[cfg(feature = "_ndarrayl")]
use crate::core::Executor;
use crate::test_trait_impl;
#[cfg(feature = "_ndarrayl")]
use approx::assert_relative_eq;

test_trait_impl!(newton_method, Newton<f64>);

#[test]
fn test_new() {
let solver: Newton<f64> = Newton::new();
assert_eq!(solver.gamma.to_ne_bytes(), 1.0f64.to_ne_bytes());
}

#[test]
fn test_default() {
let solver_new: Newton<f64> = Newton::new();
let solver_def: Newton<f64> = Newton::default();
assert_eq!(
solver_new.gamma.to_ne_bytes(),
solver_def.gamma.to_ne_bytes()
);
}

#[test]
fn test_with_gamma() {
for new_gamma in [f64::EPSILON, 0.1, 0.5, 0.9, 1.0] {
let solver: Newton<f64> = Newton::new().with_gamma(new_gamma).unwrap();
assert_eq!(solver.gamma.to_ne_bytes(), new_gamma.to_ne_bytes());
}

for new_gamma in [1.0 + f64::EPSILON, 2.0, 0.0, -1.0] {
let res = Newton::new().with_gamma(new_gamma);
assert_error!(
res,
ArgminError,
"Invalid parameter: \"Newton: gamma must be in  (0, 1].\""
);
}
}

#[cfg(feature = "_ndarrayl")]
#[test]
fn test_next_iter_param_not_initialized() {
use crate::core::State;
use ndarray::{Array, Array1, Array2};
let mut newton: Newton<f64> = Newton::new();

struct NewtonProblem {}

type Param = Array1<f64>;

Ok(Array1::from_vec(vec![1.0, 2.0]))
}
}

impl Hessian for NewtonProblem {
type Param = Array1<f64>;
type Hessian = Array2<f64>;

fn hessian(&self, _p: &Self::Param) -> Result<Self::Hessian, Error> {
Ok(Array::from_shape_vec((2, 2), vec![1.0f64, 0.0, 0.0, 1.0])?)
}
}

let res = newton.next_iter(&mut Problem::new(NewtonProblem {}), IterState::new());
assert_error!(
res,
ArgminError,
concat!(
"Not initialized: \"`Newton` requires an initial parameter vector. ",
"Please provide an initial guess via `Executor`s `configure` method.\""
)
);
}

#[cfg(feature = "_ndarrayl")]
#[test]
fn test_solver() {
use crate::core::State;
use ndarray::{Array, Array1, Array2};
struct Problem {}

type Param = Array1<f64>;

Ok(Array1::from_vec(vec![1.0, 2.0]))
}
}

impl Hessian for Problem {
type Param = Array1<f64>;
type Hessian = Array2<f64>;

fn hessian(&self, _p: &Self::Param) -> Result<Self::Hessian, Error> {
Ok(Array::from_shape_vec((2, 2), vec![1.0f64, 0.0, 0.0, 1.0])?)
}
}

// Single iteration, starting from [0, 0], gamma = 1
let problem = Problem {};
let solver: Newton<f64> = Newton::new();
let init_param = Array1::from_vec(vec![0.0, 0.0]);

let param = Executor::new(problem, solver)
.configure(|config| config.param(init_param).max_iters(1))
.run()
.unwrap()
.state
.get_best_param()
.unwrap()
.clone();
assert_relative_eq!(param[0], -1.0, epsilon = f64::EPSILON);
assert_relative_eq!(param[1], -2.0, epsilon = f64::EPSILON);

// Two iterations, starting from [0, 0], gamma = 1
let problem = Problem {};
let solver: Newton<f64> = Newton::new();
let init_param = Array1::from_vec(vec![0.0, 0.0]);

let param = Executor::new(problem, solver)
.configure(|config| config.param(init_param).max_iters(2))
.run()
.unwrap()
.state
.get_best_param()
.unwrap()
.clone();
assert_relative_eq!(param[0], -2.0, epsilon = f64::EPSILON);
assert_relative_eq!(param[1], -4.0, epsilon = f64::EPSILON);

// Single iteration, starting from [0, 0], gamma = 0.5
let problem = Problem {};
let solver: Newton<f64> = Newton::new().with_gamma(0.5).unwrap();
let init_param = Array1::from_vec(vec![0.0, 0.0]);

let param = Executor::new(problem, solver)
.configure(|config| config.param(init_param).max_iters(1))
.run()
.unwrap()
.state
.get_best_param()
.unwrap()
.clone();
assert_relative_eq!(param[0], -0.5, epsilon = f64::EPSILON);
assert_relative_eq!(param[1], -1.0, epsilon = f64::EPSILON);

// Two iterations, starting from [0, 0], gamma = 0.5
let problem = Problem {};
let solver: Newton<f64> = Newton::new().with_gamma(0.5).unwrap();
let init_param = Array1::from_vec(vec![0.0, 0.0]);

let param = Executor::new(problem, solver)
.configure(|config| config.param(init_param).max_iters(2))
.run()
.unwrap()
.state
.get_best_param()
.unwrap()
.clone();
assert_relative_eq!(param[0], -1.0, epsilon = f64::EPSILON);
assert_relative_eq!(param[1], -2.0, epsilon = f64::EPSILON);
}
}
``````