The rather advanced Levenberg-Marquardt method uses a combination of the following
two step determinations:
* Steepest descent method: Take simply
AA -t VAF(A)
and use one dimensional optimization to determine the step length t (called a line
search).
* Use a quadratic approximation for F:
F(A + AA) F(A) + AA VF(A) + -AATU (A)AA
2
with R the Hessian matrix for F, and use the step AA which minimized this
approximation AA = -H-1 VF.
We implement all of the statistical calculations and data analysis procedures in
MATLAB@.
4.2 Application to 2D Trf3
A scalar matrix field theory in 1+1 space-time dimensions described by the action
(2-1) can of course be written in the Lightcone World Sheet form, just as was done in
3+1 dimensions. However, with only two dimensions there are no transverse bosonic q
variables living in the bulk of the world sheet. It is also a well known fact that this theory
is ultraviolet finite meaning that we can without considering counter-terms on the world
sheet, proceed directly with simulating the theory using the methods developed in this
chapter. This very simple choice is motivated by this fact and also by the fact that the
theory has been used widely as a toy model and the results can therefore be verified and
compared. The Monte Carlo method can in this context be verified in its own right.
With this preamble, we turn now to the simplest possible case where the only
dynamical variable left in the system are the spins which designate the presence of solid
lines. The b, c ghosts can be eliminated by simply putting in by hand the factors of 1/M
which ,.r v were designed to produce. Although this does introduce a nonlocal effect,