Computational aspects of master equation transformation in terms of moments

The master equation describing the temporal evolution of a gaseous system in contact with a heat bath can be transformed into a system of linear, constant-coefficient, first-order differential equations of moments of the population distribution. While it has the advantage that populations are obtained directly from observables (moments), this system of equations is not too well-conditioned and unless precautions are taken, unsurmountable numerical problems appear. These are principally associated with manipulations (inversion and taking the exponential of a matrix) involving slightly modified Vandermonde matrices whose elements span a very wide range of orders of magnitude. This article discusses ways to avoid these pitfalls which consist principally of a suitable matrix normalization.