The chapter introduces the Levenberg-Marquardt algorithm, a numerical method for solving nonlinear least squares problems. The problem involves finding a local minimizer of the function \( \phi(x) = \frac{1}{2} \| F(x) \|^2 \), where \( F: \mathbb{R}^n \rightarrow \mathbb{R}^m \) is a continuously differentiable function. The algorithm, proposed by Levenberg and Marquardt, is known for its elegance but often lacks robustness and theoretical justification in implementations. This work presents a robust and efficient implementation of the algorithm, emphasizing the use of implicitly scaled variables and the Hebden scheme for parameter selection. The chapter also discusses the notation and derivation of the algorithm, including a linearization argument and the solution of a constrained linear least squares problem. The iteration process is outlined, and the next sections will delve into the implementation details.The chapter introduces the Levenberg-Marquardt algorithm, a numerical method for solving nonlinear least squares problems. The problem involves finding a local minimizer of the function \( \phi(x) = \frac{1}{2} \| F(x) \|^2 \), where \( F: \mathbb{R}^n \rightarrow \mathbb{R}^m \) is a continuously differentiable function. The algorithm, proposed by Levenberg and Marquardt, is known for its elegance but often lacks robustness and theoretical justification in implementations. This work presents a robust and efficient implementation of the algorithm, emphasizing the use of implicitly scaled variables and the Hebden scheme for parameter selection. The chapter also discusses the notation and derivation of the algorithm, including a linearization argument and the solution of a constrained linear least squares problem. The iteration process is outlined, and the next sections will delve into the implementation details.