These figures and related data values were computed with the hopes of helping guess the functional form for what is denoted "g", p.1381, v72, Mathematics of Computation (2003).  Also see figure 5f, p. 1386.  

I used an extremization algorithm of my own design to find, for fixed k, an approximation to the 'optimal a' to about 5-10 digits of accuracy for k=1 to 300.  Since some of the 'a' values were far from 1, the corresponding values when first plotted may seem far from optimal in some cases.  But when the C=- ()/a   are plotted using the 'optimal a' values, it is evident that the 'a' values appear to be correctly chosen.  Note the apparent regularity when considering which values for 'a' are optimal, for given k.

The C values for the approximately optimal 'a' are stored as 'cv[k]' in the data file.  These are thus approximations to the extremal values in the paper referred to as g(k).  Technically, the g(k) values are taken as sups over all 0<a<=1 of C; the values cv(k) in the data file are probably correct to 5-10 digits.

The file 'gamsavalscvals' houses the approximate optimal a values (which are probably correct to 5-10 digits), the corresponding (needed to high precision due to possible loss of significance when Care computed), and the related cv(k) values .  To repeat, it is only because for each k, the optimal 'a' is not known to more than 5-10 digits of precision that the cv(k) values are only approximations to g(k) to 5-10 digits.

Rick Kreminski
November 10, 2003