ABSTRACT
The idea of differential algebraic (DA) methods II, 2, 51 is based on the observation that it is possible to extract more information about a function than its mere values on computers. One can introduce an operation T denoting the extraction of the Taylor coefficients of a prespecified order n of the function f E cn(Rv). In mathematical terms, T is an equivalence relation, and the application of T corresponds to the transition from the function f to the equivalence class [!] comprising all those functions with identical Taylor expansion in v variables to order n; the classes are apparently characterized by the collection of Taylor coefficients. Since Taylor coefficients of order n for sums and products of functions as well as scalar products with reals can be computed from those of the summands and factors, the set of equivalence classes of functions can be endowed with well-defined operations, leading to the so-called Truncated Power Series Algebra (TPSA) [6, 7]. More advanced tools address the composition of functions. their inversion, solutions of implicit equations, and the introduction of common elementary functions[ I[. For treatment of ODEs and PDEs, the power of TPSA can be enhanced by the introduction of derivations a and their inverses a-I, corresponding to the differentiation and integration on the space of functions, resulting in the Differential Algebra nDv. This structure allows the direct treatment of many questions connected with differentiation and integration of functions, including the solution of the ODEs di/dt = f(x, t) describing the motion and PDEs describing the fields [5].
