Computation of higher order Lie derivatives on the Infinity Computer

Novel first and second order numerical differentiation techniques and their application to nonlinear analysis of Kirchhoff–Love shells

“…The Adi algorithm overrides the arithmetic operations in the software code of the function f (x) in order to construct the code for the respective derivatives. It is well-known that the generation of the code by Adi reqiures additional execution time, which increases with the complexity of the function to be differentiated (see, e.g., [21]). Consecutive evaluations of the derivatives depend on the complexity of the function f (x) and its code.…”

“…For this reason, the execution times of the algorithms are not provided here. To give an idea of the performance of the used framework of the Infinity Computer, the papers [21,41,50] can be studied where the computational performance of the Infinity Computer is analyzed and compared with its competitors. Moreover, since the Infinity Computer with finite powers of ① is equivalent to Levi-Civita field from the implementational and computational points of view, then the paper [15] can be also consulted for possible execution times and performance.…”

“…Finally, the number X = c① 0 , i.e., having only the power of ① equal to zero, is called purely finite. A more detailed description of the Infinity Computer including its technical implementation information can be found, e.g., in [14,21,45].…”

“…However, it has been constructed for univariate derivatives only and has not been implemented for the gradient or Hessian. Multidimensional differentiation techniques have been also proposed in [20,21], but they are applicable only for Lie derivatives embedded in initial value problems. In [52], there has been proposed the method for higherorder mixed differentiation.…”

“…There exist different approaches for working with infinite and infinitesimal quantities: see, e.g., [48] for the Infinity Computer, [52] for the Levi-Civita fields, [44] for the Non-Standard Analysis, [5] for labeled sets and numerosities, etc. In this paper, the Infinity Computer (see, e.g., [46,48] for its complete description and the patents [45] for technical implementation details) is used for this purpose, since it has already been successfully used in different applications, where a high precision computation of the derivatives is involved (see, e.g., [13,20,21,47,51]). Moreover, it has been also successfully used in many other different applied fields, e.g., in optimization [7,8,11,12], in ordinary differential equations [2,50], in handling ill-conditioning and high precision computations [1,17,49], numerical simulation and modeling [13,14], etc.…”