Curve fitting

The Nautilus.CurveFit module provides a single export: lm_scalar_1param, a Levenberg-Marquardt optimizer for single-parameter curve fitting.

lm_scalar_1param

Signature:

lm_scalar_1param(
  model: f32 -> f32 -> f32,    // model(x, theta) -> predicted y
  dmodel: f32 -> f32 -> f32,   // dmodel(x, theta) -> d(predicted_y)/d(theta)
  xs: tensor[n, f32],          // input data points
  ys: tensor[n, f32],          // observed y-values
  theta0: f32,                 // initial parameter guess
  lambda0: f32,                // initial damping factor
  tol: f32,                    // convergence tolerance on |delta_theta|
  max_iters: int64             // iteration cap
) -> f32                       // fitted theta

The optimizer iterates the damped Gauss-Newton update:

theta_next = theta + (J^T J * (1 + lambda))^{-1} * J^T r

where J is the Jacobian (here a scalar dmodel(x, theta) per data point) and r is the residual vector y - model(x, theta). Convergence is declared when |theta_next - theta| < tol. Returns NaN for empty data.

Example: fitting a linear model

import Nautilus.CurveFit (lm_scalar_1param)

def linear_model(x: f32, theta: f32) -> f32 = mul(theta, x)
def linear_dmodel(x: f32, theta: f32) -> f32 = x

def fit_slope[n](xs: tensor[n, f32], ys: tensor[n, f32]) -> f32 =
  lm_scalar_1param(
    linear_model, linear_dmodel, xs, ys,
    cast(0.0, f32),      // initial guess
    cast(0.01, f32),      // damping
    cast(1.0e-8, f32),    // tolerance
    cast(100, int64)      // max iterations
  )

For a dataset generated by y = 2*x + noise, this converges to approximately 2.0.

Limitations

• Single parameter only. Multi-parameter Levenberg-Marquardt (using inv_2x2/inv_3x3 for the normal equations) is structurally possible but not yet exported.
• The caller must supply the analytical derivative dmodel. There is no automatic differentiation fallback in this function.
• The damping factor lambda0 is held constant (no adaptive lambda scheduling as in full LM implementations).