Levenberg-Marquardt Least Squares Fitting Algorithm: Implementation and Applications

In this article, we examine a significant least squares fitting algorithm known as the Levenberg-Marquardt Algorithm (LMA). LMA serves as an optimization technique for nonlinear least squares problems, employing an iterative approach to converge toward optimal solutions. The algorithm dynamically adjusts parameters through a hybrid strategy that combines gradient descent and Gauss-Newton methods, effectively minimizing the sum of squared residuals between model predictions and observed data. Implementation typically involves calculating Jacobian matrices for partial derivatives, with damping factor adjustments controlling step sizes during iterations. Key functions include residual calculation, Jacobian estimation (via finite differences or analytical derivatives), and trust region management. Beyond curve fitting applications, LMA proves valuable in diverse nonlinear optimization scenarios such as signal processing calibration and machine learning model training. The algorithm's robustness against ill-conditioned problems makes it particularly suitable for real-world applications where initial parameter estimates may be suboptimal. Overall, LMA represents a widely-adopted optimization tool with substantial practical impact across scientific and engineering domains.