Definitions

This section is incomplete, and will be expanded in the near future

Custom Truncation Orders

GTPSA allows for significant customization in the truncation of specific variables within a monomial of the truncated power series (TPS). One can specify individually the truncation orders for each variable in a truncated power series, as well as the maximum truncation order for an entire monomial in the TPS. This is best shown with an example:

Suppose we'd like to express a function $f(x_1,x_2)$ as a truncated power series, and keep only terms that are up to 1st-order in $x_1$ but up to 2nd order in $x_2$; basically, we should not have any monomials where $x_1$ has an exponent > 1, nor any monomials where $x_2$ has an exponent > 2. GTPSA allows one to select the individual truncation orders for variables in a monomial in this manner. The next question to consider is the maximum truncation order for the entire monomial; in the above example, note that the 3rd-order term $x_1x_2^2$ follows the rules we layed out so far. But what if we'd also like to truncate all monomials with order 3 and above, and not allow this monomial? This can be achieved by setting the maximum truncation order equal to 2. When defining a GTPSA, the user must always specify the maximum truncation order, which when specifying individual truncation orders must lie within the range $[\textrm{max}{(\textrm{individual truncation orders})}, \textrm{sum}{(\textrm{individual truncation orders})}]$. If individual truncation orders are not specified, then they are automatically set to the maximum truncation order.

Example: allowed monomials for $f(x_1,x_2)$ with individual variable truncation orders [1,2] and different maximum truncation orders:

Parameters

GTPSA allows one to explicitly distinguish between variables and parameters. Generally, a variable would be a dependent variable in a differential equation, and a parameter would be a variation in something defining or influencing the system (for example, in a harmonic oscillator the restoring constant $k$ would be a parameter). Individual truncation orders can be specified for the parameters in the same way as described for the variables, however there is a special extra truncation order the can be specified for solely the parameters part of the monomial, referred to as the parameter order. The parameter order defines the truncation order for only the parameters part of a monomial.

Example: allowed monomials for $f(x_1,k_1,k_2)$ (one variable, two parameters) with individual variable truncation order [1], individual parameter truncation orders [1,1], maximum order = 3, and different parameter orders:

(Note: many monomials are excluded for brevity in the above table)