Levenberg–Marquardt algorithm

In mathematics and statistics, ainimising a sum

${\displaystyle S({\vec {x}})=\sum _{i=1}^{m}[f_{i}({\vec {x}})]^{2}}$

where

${\displaystyle f_{i}({\vec {x}})}$

represent the components of the function

${\displaystyle {\vec {f}}}$,

may be solved by the Levenberg-Marquardt algorithm according to

${\displaystyle (J^{T}J+\lambda I){\vec {q}}=-J^{T}{\vec {f}}}$.

Here ${\displaystyle J}$ represents the Jacobian of the function ${\displaystyle {\vec {f}}}$, ${\displaystyle \lambda }$ the damping factor, which is altered at every iteration, ${\displaystyle I}$ the identity matrix, and ${\displaystyle {\vec {q}}}$ the solution to an iteration step.