Levenberg–Marquardt algorithm: Difference between revisions

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== The problem ==

The primary application of the Levenberg–Marquardt algorithm is in the least-squares curve fitting problem: given a set of <math>m</math> empirical datumdata pairs <math>\left (x_i, y_i\right )</math> of independent and dependent variables, find the parameters {{tmath|\boldsymbol\beta}} of the model curve <math>f\left (x, \boldsymbol\beta\right )</math> so that the sum of the squares of the deviations <math>S\left (\boldsymbol\beta\right )</math> is minimized:

:<math>\hat{\boldsymbol\beta} \in \operatorname{argmin}\limits_{\boldsymbol\beta} S\left (\boldsymbol\beta\right ) \equiv \operatorname{argmin}\limits_{\boldsymbol\beta} \sum_{i=1}^m \left [y_i - f\left (x_i, \boldsymbol\beta\right )\right ]^2,</math> which is assumed to be non-empty.