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Like other numeric minimization algorithms, the Levenberg–Marquardt algorithm is an [[iteration|iterative]] procedure. To start a minimization, the user has to provide an initial guess for the parameter vector {{tmath|\boldsymbol\beta}}. In cases with only one minimum, an uninformed standard guess like <math>\boldsymbol\beta^\text{T} = \begin{pmatrix}1,\ 1,\ \dots,\ 1\end{pmatrix}</math> will work fine; in cases with [[local minimum|multiple minima]], the algorithm converges to the global minimum only if the initial guess is already somewhat close to the final solution.
In each iteration step, the parameter vector {{tmath|\boldsymbol\beta}} is replaced by a new estimate {{tmath|\boldsymbol\beta + \boldsymbol\delta}}. To determine {{tmath|\boldsymbol\delta}}, the function <math>f\left (x_i, \boldsymbol\beta + \boldsymbol\delta\right )</math> is approximated by its [[Gradient#Linear_approximation_to_a_function|linearization]]:
: <math>f\left (x_i, \boldsymbol\beta + \boldsymbol\delta\right ) \approx f\left (x _i, \boldsymbol\beta\right ) + \mathbf J_i \boldsymbol\delta,</math>
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