global convergence of the algorithm suffer from the undesirable properties of steepest descent, in particular, very slow convergence close to the optimum. The absolute values of any choice depend on how well-scaled the initial problem is. Marquardt recommended starting with a value and a factor . Initially setting and computing the residual sum of squares after one step from the starting point with the damping factor of and secondly with . If both of these are worse than the initial point, then the damping is increased by successive multiplication by until a better point is found with a new damping factor of for some . If use of the damping factor results in a reduction in squared residual, then this is taken as the new value of (and the new optimum location is taken as that obtained with this damping factor) and the process continues; if using resulted in a worse residual, but using resulted in a better residual, then is left unchanged and the new optimum is taken as the value obtained with as damping factor. An effective strategy for the control of the damping parameter, called delayed gratification, consists of increasing the parameter by a small amount for each uphill step, and decreasing by a large amount for each downhill step. The idea behind this strategy is to avoid moving downhill too fast in the beginning of optimization, therefore restricting the steps available in future iterations and therefore slowing down convergence. An increase by a factor of 2 and a decrease by a factor of 3 has been shown to be effective in most cases, while for large problems more extreme values can work better, with an increase by a factor of 1.5 and a decrease by a factor of 5. Geodesic acceleration When interpreting the Levenberg–Marquardt step as the velocity along a geodesic path in the parameter space, it is possible to improve the method by adding a second order term that accounts for the acceleration along the geodesic where is the solution of Since this geodesic acceleration term depends only on the directional derivative along the direction of the velocity , it does not require computing the full second order derivative matrix, requiring only a small overhead in terms of computing cost. Since the second order derivative can be a fairly complex expression, it can be convenient to replace it with a finite difference approximation where