# gleastsq

A least squares curve fitting library for Gleam. This library uses the Nx library from Elixir under the hood to perform matrix operations.

## Which method should I use?

The library provides four functions for curve fitting: `least_squares`

, `gauss_newton`

, `levenberg_marquardt`

and `trust_region_reflective`

.

## Least Squares

The `least_squares`

function is just an alias for the `levenberg_marquardt`

function.

## Levenberg-Marquardt

Ideal for non-linear least squares problems, particularly when the initial guess is far from the solution. It combines the benefits of the Gauss-Newton method and gradient descent, making it robust and efficient for various scenarios. However, it requires careful tuning of the damping parameter to balance convergence speed and stability.

## Trust-Region Reflective

Best suited for large-scale problems or those **with constraints (upper and lower bounds for each function parameter)**, this method ensures that each iteration stays within a predefined “trust region,” preventing large, unstable steps. It is reliable and effective for challenging optimization problems but can be computationally intensive.

## Gauss-Newton

Efficient for problems where residuals are small and the initial guess is close to the true solution. It approximates the Hessian matrix, leading to faster convergence for well-behaved problems. However, it may struggle with highly non-linear problems or poor initial guesses, as it lacks the robustness of the Levenberg-Marquardt and trust-region reflective methods.

## Installation and Example

```
gleam add gleastsq
```

```
import gleam/io
import gleastsq
fn parabola(x: Float, params: List(Float)) -> Float {
let assert [a, b, c] = params
a *. x *. x +. b *. x +. c
}
pub fn main() {
let x = [0.0, 1.0, 2.0, 3.0, 4.0, 5.0]
let y = [0.0, 1.0, 4.0, 9.0, 16.0, 25.0]
let initial_guess = [1.0, 1.0, 1.0]
let assert Ok(result) =
gleastsq.least_squares(x, y, parabola, initial_guess, opts: [])
io.debug(result) // [1.0, 0.0, 0.0] (within numerical error)
}
```

Further documentation can be found at https://hexdocs.pm/gleastsq.