﻿ Levenbergâ€“Marquardt algorithm

# Levenbergâ€“Marquardt algorithm

The Levenberg -Marquardt algorithm, named after Kenneth Levenberg Marquardt and Donald, is a numerical optimization algorithm for solving non-linear equalization problems using the method of least squares. The method combines the Gauss-Newton method using a regularization, which forces decreasing function values ​​.

The Levenberg -Marquardt algorithm is more robust than the Gauss- Newton method, ie it converges with high probability even in poor starting conditions, but is also here convergence is not guaranteed. It is also at initial values ​​, which are close to the minimum, often slower.

## Description

For the non-linear function is the least-squares minimization problem ( with a smaller number of independent variables versus the number of the functional components )

Be solved starting from an initial guess.

As the Gauss -Newton method F ( x) at each step is replaced by a linearization and the replacement problem:

Considered. In this case, J is the Jacobian matrix of the function F.

In addition, however, it calls for the Levenberg -Marquardt algorithm that. This additional condition can force a reduction of at each step. The parameter is adjusted accordingly.

## Convergence

The Levenberg -Marquardt method proceeds locally in the Newton method. Thus, the convergence is locally quadratic.

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