The gauss_newton function performs a basic least squares optimization algorithm. It is used to find the best-fit parameters for a given model function to a set of data points. This function takes as input the data points, the model function, and several optional parameters to control the optimization process.

Parameters

• x (List(Float)) A list of x-values of the data points.
• y (List(Float)) A list of y-values of the data points.
• func (fn(Float, List(Float)) -> Float) The model function that takes an x-value and a list of parameters, and returns the corresponding y-value.
• initial_params (List(Float)) A list of initial guesses for the parameters of the model function.
• opts (List(LeastSquareOptions)) A list of optional parameters to control the optimization process. The available options are:
  • Iterations(Int): The maximum number of iterations to perform. Default is 100.
  • Epsilon(Float): A small value to change x when calculating the derivatives for the function. Default is 0.0001.
  • Tolerance(Float): The convergence tolerance. Default is 0.0001.
  • Damping(Float): The value of the damping parameter. Default is 0.001.

Example

import gleam/io
import gleastsq
import gleastsq/options.{Iterations, Tolerance}

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.gauss_newton(
      x,
      y,
      parabola,
      initial_guess,
      opts: [Iterations(1000), Tolerance(0.001)]
    )

  io.debug(result) // [1.0, 0.0, 0.0] (within numerical error)
}

The least_squares function is an alias for the levenberg_marquardt function. Check the documentation of the levenberg_marquardt function for more information.

The levenberg_marquardt function performs the Levenberg-Marquardt optimization algorithm. It is used to solve non-linear least squares problems. This function takes as input the data points, the model function, and several optional parameters to control the optimization process.

Parameters

• x (List(Float)) A list of x-values of the data points.
• y (List(Float)) A list of y-values of the data points.
• func (fn(Float, List(Float)) -> Float) The model function that takes an x-value and a list of parameters, and returns the corresponding y-value.
• initial_params (List(Float)) A list of initial guesses for the parameters of the model function.
• opts (List(LeastSquareOptions)) A list of optional parameters to control the optimization process. The available options are:
  • Iterations(Int): The maximum number of iterations to perform. Default is 100.
  • Epsilon(Float): A small value to change x when calculating the derivatives for the function. Default is 0.0001.
  • Tolerance(Float): The convergence tolerance. Default is 0.0001.
  • Damping(Float): The initial value of the damping parameter. Default is 0.0001.
  • DampingIncrease(Float): The factor by which the damping parameter is increased when a step fails. Default is 10.0.
  • DampingDecrease(Float): The factor by which the damping parameter is decreased when a step succeeds. Default is 0.1.

Example

import gleam/io
import gleastsq
import gleastsq/options.{Iterations, Tolerance}

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.levenberg_marquardt(
      x,
      y,
      parabola,
      initial_guess,
      opts: [Iterations(1000), Tolerance(0.001)]
    )

  io.debug(result) // [1.0, 0.0, 0.0] (within numerical error)
}

The trust_region_reflective function performs a least squares optimization using the Trust Region Reflective algorithm. It is used to find the best-fit parameters for a given model function to a set of data points. This function takes as input the data points, the model function, and several optional parameters to control the optimization process.

Parameters

• x (List(Float)) A list of x-values of the data points.
• y (List(Float)) A list of y-values of the data points.
• func (fn(Float, List(Float)) -> Float) The model function that takes an x-value and a list of parameters, and returns the corresponding y-value.
• initial_params (List(Float)) A list of initial guesses for the parameters of the model function. Note that a value that is outside of the bounds will be clipped to the bounds.
• lower_bounds (Option(List(Float))) A list of lower bounds for the parameters of the model function. If the lower bound is None, then the parameter is unbounded from below.
• upper_bounds (Option(List(Float))) A list of upper bounds for the parameters of the model function. If the upper bound is None, then the parameter is unbounded from above.
• opts (List(LeastSquareOptions)) A list of optional parameters to control the optimization process. The available options are:
  • Iterations(Int): The maximum number of iterations to perform. Default is 100.
  • Epsilon(Float): A small value to change x when calculating the derivatives for the function. Default is 0.0001.
  • Tolerance(Float): The convergence tolerance. Default is 0.00001.
  • Damping(Float): The value of the damping parameter. Default is 0.001.

Example

import gleam/io
import gleam/option.{Some}
import gleastsq
import gleastsq/options.{Iterations, Tolerance}

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.trust_region_reflective(
      x,
      y,
      parabola,
      initial_guess,
      lower_bounds: Some([0.0, 0.0, 0.0]),
      upper_bounds: Some([2.0, 2.0, 2.0]),
      opts: [Iterations(1000), Tolerance(0.001)]
    )

  io.debug(result) // [1.0, 0.0, 0.0] (within numerical error)
}