Summary

            Normally the weighted sum of the squares of the difference between the data and the approximating function is minimized.  This leads to a relatively constant second derivative array whose inverse is needed.  Minimizing the weighted sum of the 4^th power of the difference between the data and the approximating function makes the second derivative array have smaller contributions from regions with good fits and larger contributions from regions with bad fits.  This is especially usefulful in cases where precision is a problem in poor fit regions. 

Minimizing the extremes

           (1)

                There are only a few changes from the Npow = 1 case discussed in ..\optimization\Extremal.doc htm.  The 2Npow power is necessary to make c² positive definite.   Higher powers suppress the height of the peaks in the fit at the expense of the overall fit.  In those cases where the maximum error is of concern and the fit is to known points with actual errors far less than the e_i in (1).           

                In general this is known, but not a minimum, for some set of constants c₀.  Expand c²(c) about this value so that

(2)

Truncate at the quadratic term to define the predicted c²_P.

(3)

Truncating (2) at the quadratic term, the partial of c² with respect to c_L is approximately given by

   (4).   

At its extreme value, the derivatives of c² with respect to the components of c are equal to zero.  Setting (4) to zero yields

(5)

Multiply both sides of (5) by  (GaussJ.htm) and sum over L

(6)

The sum on the right is d_IJ.  This makes the sum easy to evaluate leading to

(7)

                For Npow>1, even seemingly linear systems are non-linear, thus it is important to keep new constants reasonably close to the old.  In general this is accomplished by adding to (3) a term

             (8)

The S_m is a different smoother for each dimension discussed in ..\optimization\Robmin\Smvals.doc ..\optimization\Robmin\Smvals.htm  The general method is discussed in ..\optimization\Robmin\Robmin.doc .htm.  For our purposes, the important feature is that once the first and second partials of c_p² have been determined this is exactly the same as the Npow = 1 case discussed in ..\optimization\Extremal.doc htm.

Finding the partials of c²

(9)

Note that the error value in (9) appears to the (2Npow-1)’th power.

(10)

The second term in (10) is exactly zero for linear approximating functions, and approximately zero for any approximating function capable of reproducing the data.  This is due to the fact that the leading term is to an odd power and thus contains roughly equal numbers of positive and negative terms.  The first term has a factor that goes to zero in those regions where f_i – f_A.  The second derivative matrix is different for every step as the minimum is approached and is largest for those regions where the fit is poorest.  Especially in those cases where precision is a problem this prevents areas where the fit is already established from dominating the second derivative matrix.