6885
2018
eng
reportzib
0
--
2018-05-28
--
Piecewise Polynomial Taylor Expansions – The Generalization of Faà di Bruno’s Formula
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.
1438-0064
urn:nbn:de:0297-zib-68859
accepted by Modeling, Simulation and Optimization of Complex Processes HPSC2018 in 2019
Tom Streubel
Tom Streubel
Caren Tischendorf
Andreas Griewank
ZIB-Report
18-24
eng
uncontrolled
generalized Taylor expansion
eng
uncontrolled
implicit generation of splines
eng
uncontrolled
nonsmooth integration of differential algebraic equations (DAE and ODE)
eng
uncontrolled
multistep methods
eng
uncontrolled
generalized hermite interpolation
eng
uncontrolled
algorithmic piecewise differentiation (AD and APD)
eng
uncontrolled
evaluation procedures
eng
uncontrolled
treating absolute values (abs, max and min)
Mathematical Optimization
MODAL-GasLab
Streubel, Tom
MODAL-Gesamt
Energy Network Optimization
https://opus4.kobv.de/opus4-zib/files/6885/Streubel_Tom_genTaylor.pdf