Piecewise Polynomial Taylor Expansions – The Generalization of Faà di Bruno’s Formula

Please always quote using this URN: urn:nbn:de:0297-zib-68859
  • We present an extension of Taylor’s theorem towards nonsmooth evalua- tion procedures incorporating absolute value operaions. Evaluations procedures are computer programs of mathematical functions in closed form expression and al- low a different treatment of smooth operations and calls to the absolute value value function. The well known classical Theorem of Taylor defines polynomial approx- imation of sufficiently smooth functions and is widely used for the derivation and analysis of numerical integrators for systems of ordinary differential or differential algebraic equations, for the construction of solvers for the continuous nonlinear op- timization of finite dimensional objective functions and for root solving of nonlinear systems of equations. The herein provided proof is construtive and allow efficiently designed algorithms for the execution and computation of generalized piecewise polynomial expansions. As a demonstration we will derive a k-step method on the basis of polynomial interpolation and the proposed generalized expansions.

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Author:Tom StreubelORCiD, Caren Tischendorf, Andreas GriewankORCiD
Document Type:ZIB-Report
Tag:algorithmic piecewise differentiation (AD and APD); evaluation procedures; generalized Taylor expansion; generalized hermite interpolation; implicit generation of splines; multistep methods; nonsmooth integration of differential algebraic equations (DAE and ODE); treating absolute values (abs, max and min)
Date of first Publication:2018/05/28
Series (Serial Number):ZIB-Report (18-24)
Published in:accepted by Modeling, Simulation and Optimization of Complex Processes HPSC2018 in 2019