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// SPDX-License-Identifier: Apache-2.0
//! Gauss–Newton weighted batch least squares — the batch differential corrector.
//!
//! This is the estimation core a batch *orbit determination* (or any parameter-fit)
//! uses: given measurements `z` that depend on an unknown state `x` through a model
//! `h(x)`, it linearises `h` about the current estimate (a finite-difference Jacobian
//! `H = ∂h/∂x`), forms and solves the weighted normal equations
//! `(HᵀWH)·Δx = HᵀW·(z − h(x))`, and iterates to convergence.
//!
//! It is deliberately generic over the model closure, so it is independent of any
//! particular measurement set. For orbit determination the state would be the
//! satellite epoch state (and a receiver clock bias) and `h` the range / range-rate /
//! azimuth-elevation model from a ground station; here the engine and its convergence
//! are delivered and tested, while the orbit-specific measurement model, the analytic
//! state-transition matrix, and the published-case validation are follow-ons
//! (see `ROADMAP.md`).
use crate::fusion::ukf::inverse;
/// The outcome of a batch least-squares solve.
#[derive(Clone, Debug)]
pub struct LsqResult {
/// Converged (or last) state estimate.
pub x: Vec<f64>,
/// Iterations actually run.
pub iterations: usize,
/// RMS of the post-fit measurement residual `z − h(x)`.
pub rms_residual: f64,
/// `true` when the step norm fell below `tol` before `max_iter`.
pub converged: bool,
}
/// Central finite-difference Jacobian `H` (`m × n`) of `h` at `x`.
fn fd_jacobian<H>(h: &H, x: &[f64], m: usize) -> Vec<Vec<f64>>
where
H: Fn(&[f64]) -> Vec<f64>,
{
let n = x.len();
let mut jac = vec![vec![0.0; n]; m];
for (p, &xp_val) in x.iter().enumerate() {
let step = 1e-6 * xp_val.abs().max(1.0);
let mut xp = x.to_vec();
let mut xm = x.to_vec();
xp[p] += step;
xm[p] -= step;
let hp = h(&xp);
let hm = h(&xm);
for i in 0..m {
jac[i][p] = (hp[i] - hm[i]) / (2.0 * step);
}
}
jac
}
/// Solve `z ≈ h(x)` by Gauss–Newton iteration with per-measurement weights
/// `weights` (e.g. `1/σ²`), starting from `x0`. Returns `None` on a dimension
/// mismatch or a singular normal matrix (rank-deficient geometry).
///
/// For a *linear* model `h(x) = Ax` this reaches the exact weighted-least-squares
/// solution `(AᵀWA)⁻¹AᵀWz` in a single step (the second iteration then confirms a
/// zero update); for a nonlinear model it converges quadratically near the solution.
pub fn gauss_newton<H>(
h: H,
z: &[f64],
weights: &[f64],
x0: &[f64],
max_iter: usize,
tol: f64,
) -> Option<LsqResult>
where
H: Fn(&[f64]) -> Vec<f64>,
{
let n = x0.len();
let m = z.len();
if weights.len() != m || n == 0 || m < n {
return None;
}
let mut x = x0.to_vec();
let mut iterations = 0;
let mut converged = false;
for it in 0..max_iter {
iterations = it + 1;
let hx = h(&x);
if hx.len() != m {
return None;
}
let r: Vec<f64> = (0..m).map(|i| z[i] - hx[i]).collect();
let jac = fd_jacobian(&h, &x, m);
// Weighted normal equations: a = HᵀWH (n×n), b = HᵀW r (n).
let mut a = vec![vec![0.0; n]; n];
let mut b = vec![0.0; n];
for i in 0..m {
let w = weights[i];
for p in 0..n {
b[p] += jac[i][p] * w * r[i];
for q in 0..n {
a[p][q] += jac[i][p] * w * jac[i][q];
}
}
}
let a_inv = inverse(&a)?;
let dx: Vec<f64> = (0..n)
.map(|p| (0..n).map(|q| a_inv[p][q] * b[q]).sum())
.collect();
for (xp, &d) in x.iter_mut().zip(&dx) {
*xp += d;
}
let dx_norm = dx.iter().map(|v| v * v).sum::<f64>().sqrt();
if dx_norm < tol {
converged = true;
break;
}
}
let hx = h(&x);
let rms = (z
.iter()
.zip(&hx)
.map(|(&zi, &hi)| (zi - hi).powi(2))
.sum::<f64>()
/ m as f64)
.sqrt();
Some(LsqResult {
x,
iterations,
rms_residual: rms,
converged,
})
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn linear_line_fit_recovers_exact_solution() {
// Fit y = a + b·t to points exactly on the line a=2, b=3. A linear model must
// reach the exact weighted-least-squares answer (residual 0).
let ts = [0.0, 1.0, 2.0, 3.0, 4.0];
let z: Vec<f64> = ts.iter().map(|&t| 2.0 + 3.0 * t).collect();
let w = vec![1.0; ts.len()];
let model = move |x: &[f64]| ts.iter().map(|&t| x[0] + x[1] * t).collect::<Vec<_>>();
let r = gauss_newton(model, &z, &w, &[0.0, 0.0], 10, 1e-12).expect("solves");
assert!((r.x[0] - 2.0).abs() < 1e-9, "a = {}", r.x[0]);
assert!((r.x[1] - 3.0).abs() < 1e-9, "b = {}", r.x[1]);
assert!(r.rms_residual < 1e-9 && r.converged);
}
#[test]
fn nonlinear_exponential_fit_converges() {
// Fit y = a·exp(b·t); recover a=1.5, b=0.5 from noise-free samples.
let ts = [0.0f64, 0.5, 1.0, 1.5, 2.0, 2.5];
let z: Vec<f64> = ts.iter().map(|&t| 1.5 * (0.5 * t).exp()).collect();
let w = vec![1.0; ts.len()];
let model = move |x: &[f64]| {
ts.iter()
.map(|&t| x[0] * (x[1] * t).exp())
.collect::<Vec<_>>()
};
let r = gauss_newton(model, &z, &w, &[1.0, 1.0], 50, 1e-12).expect("solves");
assert!((r.x[0] - 1.5).abs() < 1e-6, "a = {}", r.x[0]);
assert!((r.x[1] - 0.5).abs() < 1e-6, "b = {}", r.x[1]);
assert!(r.rms_residual < 1e-6, "rms = {}", r.rms_residual);
}
#[test]
fn range_multilateration_recovers_a_known_position() {
// Orbit-determination flavour: recover a 3-D position from ranges to four known
// stations. Noise-free measurements ⇒ exact recovery (residual ≈ 0).
let stations = [
[0.0, 0.0, 0.0],
[1000.0, 0.0, 0.0],
[0.0, 1000.0, 0.0],
[0.0, 0.0, 1000.0],
[1000.0, 1000.0, 1000.0],
];
let truth = [350.0, -120.0, 640.0];
let range = |s: &[f64; 3], p: &[f64]| {
((p[0] - s[0]).powi(2) + (p[1] - s[1]).powi(2) + (p[2] - s[2]).powi(2)).sqrt()
};
let z: Vec<f64> = stations.iter().map(|s| range(s, &truth)).collect();
let w = vec![1.0; stations.len()];
let model = move |x: &[f64]| stations.iter().map(|s| range(s, x)).collect::<Vec<_>>();
let r = gauss_newton(model, &z, &w, &[0.0, 0.0, 0.0], 50, 1e-10).expect("solves");
for (k, (&got, &want)) in r.x.iter().zip(&truth).enumerate() {
assert!((got - want).abs() < 1e-4, "x[{k}] = {got} vs {want}");
}
assert!(r.rms_residual < 1e-4, "rms = {}", r.rms_residual);
}
#[test]
fn rejects_underdetermined_or_mismatched_inputs() {
let model = |x: &[f64]| vec![x[0], x[0] + x[1]];
// Fewer measurements than unknowns (m < n).
assert!(gauss_newton(model, &[1.0], &[1.0], &[0.0, 0.0], 5, 1e-9).is_none());
// Weight/measurement length mismatch.
assert!(gauss_newton(model, &[1.0, 2.0], &[1.0], &[0.0, 0.0], 5, 1e-9).is_none());
}
}