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//! Multivariate Newton solver for systems of equations.
//!
//! Provides damped Newton iteration with Armijo backtracking line search
//! for solving `F(x) = 0` given a system of `LoweredOp` expressions.
use crate::error::EmlError;
use crate::linalg;
use crate::lower::LoweredOp;
/// Options for the multivariate Newton solver.
#[derive(Clone, Copy, Debug)]
pub struct SystemOpts {
/// Maximum Newton iterations. Default: 100.
pub max_iter: usize,
/// Convergence tolerance (‖F(x)‖ < tol). Default: 1e-10.
pub tol: f64,
/// Maximum Armijo step halvings. Default: 20.
pub max_halvings: usize,
}
impl Default for SystemOpts {
fn default() -> Self {
Self {
max_iter: 100,
tol: 1e-10,
max_halvings: 20,
}
}
}
/// Evaluate the system F at point x.
fn eval_system(fs: &[LoweredOp], x: &[f64]) -> Vec<f64> {
fs.iter().map(|f| f.eval(x)).collect()
}
/// Squared norm ‖v‖².
fn norm_sq(v: &[f64]) -> f64 {
v.iter().map(|&vi| vi * vi).sum()
}
/// Solve the nonlinear system F(x) = 0 using damped Newton with Armijo backtracking.
///
/// Builds the symbolic Jacobian once, then re-evaluates it at each Newton step.
/// Solves J·Δ = -F via LU decomposition. Uses Armijo condition for step acceptance.
pub fn solve_system_newton(
fs: &[LoweredOp],
x0: &[f64],
opts: SystemOpts,
) -> Result<Vec<f64>, EmlError> {
let n = fs.len();
if n == 0 {
return Err(EmlError::InvalidParameter(
"system must have at least one equation",
));
}
if x0.len() != n {
return Err(EmlError::DimensionMismatch(n, x0.len()));
}
// Build symbolic Jacobian (n rows × n cols)
// jac[i * n + j] = ∂f_i/∂x_j
let jac_exprs: Vec<Vec<LoweredOp>> = fs.iter().map(|fi| fi.jacobian(n)).collect();
let mut x = x0.to_vec();
for iter in 0..opts.max_iter {
let f_val = eval_system(fs, &x);
let f_norm_sq = norm_sq(&f_val);
if f_norm_sq.sqrt() < opts.tol {
return Ok(x);
}
if f_norm_sq.is_nan() {
return Err(EmlError::NanEncountered);
}
// Build Jacobian matrix (row-major)
let mut jac_mat: Vec<f64> = jac_exprs
.iter()
.flat_map(|row| row.iter().map(|e| e.eval(&x)))
.collect();
// RHS: -F(x)
let mut rhs: Vec<f64> = f_val.iter().map(|&v| -v).collect();
// Solve J·Δ = -F for the Newton step Δ
linalg::solve_lu(&mut jac_mat, &mut rhs, n)?;
let delta = rhs; // delta now contains Δx
// Armijo backtracking: find step α s.t.
// ‖F(x + α·Δ)‖² ≤ ‖F(x)‖² · (1 - 0.01·α)
let mut alpha = 1.0_f64;
let armijo_c = 0.01;
let mut accepted = false;
for _ in 0..opts.max_halvings {
let x_new: Vec<f64> = x
.iter()
.zip(delta.iter())
.map(|(&xi, &di)| xi + alpha * di)
.collect();
let f_new = eval_system(fs, &x_new);
let f_new_norm_sq = norm_sq(&f_new);
if f_new_norm_sq <= f_norm_sq * (1.0 - armijo_c * alpha) {
x = x_new;
accepted = true;
break;
}
alpha *= 0.5;
}
if !accepted {
// Take the step anyway with smallest alpha (avoids stagnation)
let x_new: Vec<f64> = x
.iter()
.zip(delta.iter())
.map(|(&xi, &di)| xi + alpha * di)
.collect();
x = x_new;
}
// Check convergence after step
let f_after = eval_system(fs, &x);
if norm_sq(&f_after).sqrt() < opts.tol {
return Ok(x);
}
let _ = iter; // suppress lint
}
// Check final residual
let f_final = eval_system(fs, &x);
if norm_sq(&f_final).sqrt() < opts.tol {
return Ok(x);
}
Err(EmlError::NonConvergence {
method: "solve_system_newton",
iterations: opts.max_iter,
})
}
#[cfg(test)]
mod tests {
use super::*;
use crate::lower::LoweredOp;
use std::sync::Arc;
#[test]
fn test_newton_circle_line() {
// Solve: x² + y² = 1, x - y = 0 → (1/√2, 1/√2) or (-1/√2, -1/√2)
// f0 = x0² + x1² - 1
let f0 = LoweredOp::Sub(
Arc::new(LoweredOp::Add(
Arc::new(LoweredOp::Pow(
Arc::new(LoweredOp::Var(0)),
Arc::new(LoweredOp::Const(2.0)),
)),
Arc::new(LoweredOp::Pow(
Arc::new(LoweredOp::Var(1)),
Arc::new(LoweredOp::Const(2.0)),
)),
)),
Arc::new(LoweredOp::Const(1.0)),
);
// f1 = x0 - x1
let f1 = LoweredOp::Sub(Arc::new(LoweredOp::Var(0)), Arc::new(LoweredOp::Var(1)));
let x0 = vec![0.5, 0.5]; // initial guess near (1/√2, 1/√2)
let sol = solve_system_newton(&[f0, f1], &x0, SystemOpts::default()).unwrap();
let expected = 1.0_f64 / 2.0_f64.sqrt();
assert!((sol[0] - expected).abs() < 1e-8, "x={}", sol[0]);
assert!((sol[1] - expected).abs() < 1e-8, "y={}", sol[1]);
}
}