The relative smoothers

Diagonal Robmin - modifying the second derivatives.

One dimension

Newton's method in one dimension applied to the problem of minimizing a function with derivatives and involves approximating the first derivative everywhere in terms of its value and that of the second derivative at a single evaluated x coordinate, so that

Equation 1

Then the x coordinate at which f is extremal is the one for which the first derivative is zero. The x coordinate value for which the first derivative of f is zero is easily found from equation 1 as

Equation 2

The classic problem with this method comes from the fact that the second derivative at x₀ is too small

This results in overshooting the location of the minimum and in fact frequently finding a value of f(x) that is worse than the original. After evaluating f at the predicted value we now have . The best estimate of is now

Equation 3

This can be done in terms of a Marquardt parameter such that

Equation 4

Then substituting in equation 2 and redoing the step.

Multi-dImensions

In multi-dimensions, equation 4 is generalized to find

Equation 5

In practice at each successful minimization step the smoothers are moved half way back to 1 on a logarithmic scale. The ratios of these smoothers are on the order of 1 to 10²⁰.