ARBITRARY ORDER VECTOR REVERSION OF SERIES AND IMPLICIT FUNCTION THEOREM
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High-order modeling and optimization methods are required for handling challenging applications where nonlinear behaviors are important. Vector-valued Taylor series math models provide the core analysis tools for analyzing and optimizing the performance of complex mechanical systems. Two specialized mathematical operations frequently appear, including: (1) successive approximation techniques, and (2) implicit function theorem calculations. Both of these analysis techniques require calculations for vector-valued composite function calculations, where implicit rate calculations represent a specialized composite function calculation. For the special case of scalar function calculations, since 1857, composite function calculations have been handled by invoking the combinatorically motivated mathematical identity of Fa di Bruno. Vector-valued generalizations of di Bruno's formula have been proposed, but the resulting algorithms are very difficult to apply in real-world applications. This paper presents reformulation of the vector-valued di Bruno formula that allows arbitrary order calculations to be recursively generated. Three steps are required for developing algorithms for handling arbitrary order vector generalizations for composite and implicit function calculations. First, Fa di Bruno's identity is replaced with an algorithmically simpler series solution discovered by George Scott in 1861. Second, abstract compound data structures are introduced for managing calculations, which leads to a generalized matrix operator where the indexed object components represent tensors of various orders. Third, generalized product operators are introduced for recursively generating the tensor math models required by Scott's formula. Recursive algorithms are comprehensively addressed for both composite function and implicit rate calculations. Several numerical examples are presented implicit rate calculations, as well as closed-form solutions for Lagrange's implicit function series expansions. The resulting algorithms are recursive, exact, very fact, and scale to arbitrary order.
