Module separable

Separable least squares for partially linear problems

This module implements variable projection (VARPRO) algorithm for solving separable nonlinear least squares problems where the model has the form:

f(x, α, β) = Σ αᵢ φᵢ(x, β)

where α are linear parameters and β are nonlinear parameters.

Example

use ndarray::{array, Array1, Array2};
use scirs2_optimize::least_squares::separable::{separable_least_squares, SeparableOptions};

// Model: y = α₁ * exp(-β * t) + α₂
// Linear parameters: α = [α₁, α₂]
// Nonlinear parameters: β = [β]

// Basis functions that depend on nonlinear parameters
fn basis_functions(t: &[f64], beta: &[f64]) -> Array2<f64> {
    let n = t.len();
    let mut phi = Array2::zeros((n, 2));
     
    for i in 0..n {
        phi[[i, 0]] = (-beta[0] * t[i]).exp(); // exp(-β*t)
        phi[[i, 1]] = 1.0;                     // constant term
    }
    phi
}

// Jacobian of basis functions w.r.t. nonlinear parameters
fn basis_jacobian(t: &[f64], beta: &[f64]) -> Array2<f64> {
    let n = t.len();
    let mut dphi_dbeta = Array2::zeros((n * 2, 1)); // n*p x q
     
    for i in 0..n {
        // d/dβ(exp(-β*t)) = -t * exp(-β*t)
        dphi_dbeta[[i, 0]] = -t[i] * (-beta[0] * t[i]).exp();
        // d/dβ(1) = 0
        dphi_dbeta[[n + i, 0]] = 0.0;
    }
    dphi_dbeta
}

// Data points
let t_data = array![0.0, 0.5, 1.0, 1.5, 2.0];
let y_data = array![2.0, 1.6, 1.3, 1.1, 1.0];

// Initial guess for nonlinear parameters
let beta0 = array![0.5];

let result = separable_least_squares(
    basis_functions,
    basis_jacobian,
    &t_data,
    &y_data,
    &beta0,
    None
)?;

println!("Nonlinear params: {:?}", result.result.x);
println!("Linear params: {:?}", result.linear_params);

Structs

SeparableOptions

Options for separable least squares

SeparableResult

Result structure extended for separable least squares

Enums

LinearSolver

Methods for solving the linear least squares subproblem

Functions

separable_least_squares

Solve a separable nonlinear least squares problem