ARBITRARY ORDER VECTOR REVERSION OF SERIES AND IMPLICIT FUNCTION THEOREM

摘要

High-order modeling and optimization methods are required for handling challengingapplications where nonlinear behaviors are important. Vector-valuedTaylor series math models provide the core analysis tools for analyzing andoptimizing the performance of complex mechanical systems. Two specializedmathematical operations frequently appear, including: (1) successive approximationtechniques, and (2) implicit function theorem calculations. Both ofthese analysis techniques require calculations for vector-valued compositefunction calculations, where implicit rate calculations represent a specializedcomposite function calculation. For the special case of scalar function calculations,since 1857, composite function calculations have been handled by invokingthe combinatorically motivated mathematical identity of Faá di Bruno.Vector-valued generalizations of di Bruno’s formula have been proposed, butthe resulting algorithms are very difficult to apply in real-world applications.This paper presents reformulation of the vector-valued di Bruno formula thatallows arbitrary order calculations to be recursively generated. Three steps arerequired for developing algorithms for handling arbitrary order vector generalizationsfor composite and implicit function calculations. First, Faá di Bruno’sidentity is replaced with an algorithmically simpler series solution discoveredby George Scott in 1861. Second, abstract compound data structures are introducedfor managing calculations, which leads to a generalized matrix operatorwhere the indexed object components represent tensors of various orders.Third, generalized product operators are introduced for recursively generatingthe tensor math models required by Scott’s formula. Recursive algorithms arecomprehensively addressed for both composite function and implicit rate calculations.Several numerical examples are presented implicit rate calculations, aswell as closed-form solutions for Lagrange’s implicit function series expansions.The resulting algorithms are recursive, exact, very fact, and scale to arbitraryorder.