Numerical analysis

Many problems are too difficult to be tackled analytically, and often the most appropriate avenue to take is some sort of numerical analysis or random simulation. Even if we are unable to solve a model, it is frequently possible to write a reasonably fast computer algorithm to provide us with the first $N$ coefficients of the generating function, $c_1, c_2, \dots , c_N$, either exactly¹¹ or to a good approximation¹².

It is sometimes possible, from the first $N$ terms (if they are exact), to somehow ``see'' the pattern in the coefficients and hypothesise a solution (which can then be proved by other means). More usual is to use the data to explore the asymptotic behaviour of the model. One can make some assumption about the asymptotic form of the coefficients such as

$$c_n \sim A \mu^n n^{\gamma}$$   as $$n \rightarrow \infty,$$

and then fit the data (exact or approximate) to this form and so obtain estimates of various quantities such as the growth constant or the mean geometric size¹³ of the objects. Indeed for many purposes the asymptotic behaviour of a model is arguably more interesting than the existence of an exact solution (particularly if we are modelling some sort of physical problem). Of course, an exact solution is more useful than a knowledge of the asymptotic behaviour, since one can (usually) extract the asymptotic behaviour from a solution but not vice-versa¹⁴.