In this thesis, the Gauss–Newton algorithm (GNA) and the gradient descent method,
on which the Levenberg–Marquardt algorithm (LMA) is based, are presented. The
LMA is an iterative method for solving nonlinear equations using the method of
least squares. The damping parameter and the damping matrix play a key role in its
performance. By appropriately adjusting the damping parameter, the LMA switches
between the gradient descent method, which helps prevent parameter evaporation
and the GNA, which enables rapid convergence along a narrow valley of the cost
function. A method for solving nonlinear equations using QR decomposition is
also presented. In the MATLAB programming environment, functions based on
the Gauss–Newton and Levenberg–Marquardt algorithms are implemented. These
functions are used to solve a system of nonlinear equations, and the obtained results
are compared.
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