  In this article we show the rough outline of a computer algorithm to generate
lower bounds on the exponential function of (in principle) arbitrary precision.
We implemented this to generate all necessary analytic terms for the Boltzmann
machine partition function thus leading to lower bounds of any order. It turns
out that the extra variational parameters can be optimized analytically. We
show that bounds upto nineth order are still reasonably calculable in practical
situations. The generated terms can also be used as extra correction terms
(beyond TAP) in mean field expansions.
