Refinement was carried out using the TNT refinement package (Tronrud, 1987), modified to include the new conjugate direction method as an option. (The crystallographic portions of the diagonal elements of the normal matrix were calculated by the method of Agarwal, 1978). All four methods were run with the thermal parameters held constant because the refinement methods which do not use curvatures cannot simultaneously vary both positional and thermal parameters. Separate tests were made to compare GC and CD refinement in which both positional and thermal parameters were allowed to change simultaneously. The only differences between these test runs were the set of parameters varied and the method used. All other aspects, such as weights, were identical. The results of these tests are displayed in Figure 1.
The methods which use curvatures (GC and CD) are superior to the methods which do not (SD and CG). After 15 cycles of refinement the conjugate gradient run is similar to the ``gradient over curvature" method because the diagonal elements of the normal matrix for the positional parameters are all approximately equal to each other and the conjugate gradient method can accommodate their differences relatively quickly. However this is not the case for all types of parameters; shifts in B-factors that are numerically small and numerically large have different effects on the value of the function.
The comparison between the run of gradient over curvature refinement
and the run of conjugate direction refinement in which both XYZ's and B's
were varied shows the clear superiority of the new method. The R-factor
of the model produced by 20 cycles of conjugate direction refinement was
13.2 percent and still dropping, with good geometry (bond length rms error
0.027Å and bond angle rms error 
).
The reason the new method produces a lower value for 
is not because the other methods are stuck in higher local minima. For
either conjugate gradient or conjugate direction to work they must be close
enough to a minimum that the higher order terms of Taylor's series expansion
are insignificant. Each method will proceed to the minimum of the expansion,
which is the local minimum. That minimum is the same for the two methods
because the function itself is unchanged, only the set of directions to
be searched has been altered by the new method. Eventually the conjugate
gradient or steepest descent method will descend as low as the conjugate
direction method; it will simply take many more cycles to get there.
Figure 1 shows that even after 20 cycles the
new method has not reached a minimum either. Methods with even greater
power of convergence should be able to produce parameter sets where
is even lower, using affordable amounts of computer time.