LM Algorithm for Nonlinear Least Squares Curve Fitting

LM Algorithm for Nonlinear Least Squares Curve Fitting

Nonlinear least squares is an effective method for estimating parameters in nonlinear models, particularly useful for curve fitting problems. When fitting experimental data to nonlinear models, this approach becomes especially valuable. The Levenberg-Marquardt (LM) algorithm, as an enhanced implementation of nonlinear least squares, combines the advantages of gradient descent and Gauss-Newton methods, demonstrating excellent performance in both convergence speed and stability.

When implementing the LM algorithm for curve fitting in MATLAB, three key elements must be clearly defined: the model function, initial parameter values, and experimental data. The model function defines the mathematical expression to be fitted, initial parameter values provide the starting point for the algorithm's search, and experimental data represent the target to be fitted.

The core principle of the LM algorithm involves iteratively adjusting parameter values to minimize the sum of squared residuals. During each iteration, the algorithm computes the Jacobian matrix based on current parameter values to determine both the direction and step size for parameter updates. When residuals decrease, the algorithm tends to use the Gauss-Newton method for faster convergence; when residuals increase, it automatically switches to gradient descent to ensure stability.

MATLAB provides multiple approaches for implementing the LM algorithm. The most straightforward method involves using built-in functions like lsqnonlin or lsqcurvefit, which already incorporate the core logic of the LM algorithm. Users simply need to provide the model function and initial parameters to automatically complete the fitting process. For custom implementations, one can manually code the LM algorithm, which requires programming the Jacobian matrix calculation and iterative logic. The Jacobian can be computed numerically using finite differences or analytically if derivative functions are provided.

In practical applications, the performance of the LM algorithm significantly depends on the selection of initial parameters. Well-chosen initial values can dramatically improve convergence speed and success rates. Additionally, when encountering convergence difficulties, appropriately adjusting the damping parameter (lambda) or verifying the合理性 of the model function structure serves as effective troubleshooting approaches. The damping parameter controls the trust region size, balancing between gradient descent and Gauss-Newton updates.