Levenberg–Marquardt algorithm

Minimising a sum
$S({\vec {x}})=\sum _{i=1}^{m}[f_{i}({\vec {x}})]^{2}$,
where $f_{i}({\vec {x}})$ represent the components of function ${\vec {f}}$, may be solved by the Levenberg-Marquardt algorithm according to
$(J^{T}J+\lambda I){\vec {q}}=-J^{T}{\vec {f}}$.
$J$ represents the Jacobian of the function ${\vec {f}}$, $\lambda$ the damping factor, which is altered at every iteration, $I$ the identity, ${\vec {q}}$ the solution to an iteration step.