Related papers: On the efficient computation of high-order derivat…
A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms builded up from Ostrowski's method for solving systems of nonlinear equations…
Based on a class of associative algebras with zero-divisors which are called real-like algebras by us, we introduce a way of defining automatic differentiation and present different ways of doing automatic differentiation to compute the…
We describe explicit algorithms for factoring q-difference operators and solving q-difference equations. These are well known results, presented in a "concrete" form. ----- Nous decrivons des algorithmes explicites pour la factorisation…
Automatic differentiation is everywhere, but there exists only minimal documentation of how it works in complex arithmetic beyond stating "derivatives in $\mathbb{C}^d$" $\cong$ "derivatives in $\mathbb{R}^{2d}$" and, at best, shallow…
We introduce a notion of fractional (noninteger order) derivative on an arbitrary nonempty closed subset of the real numbers (on a time scale). Main properties of the new operator are proved and several illustrative examples given.
For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a…
In this work we present an algorithm to perform algorithmic differentiation in the context of quantum computing. We present two versions of the algorithm, one which is fully quantum and one which employees a classical step (hybrid…
At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly,…
Because the distant retrograde orbits dynamics inherently depends on special functions, approximate analytical solutions in the literature are commonly constrained to providing rough approximations of the qualitative behavior. We rely on…
Many algorithms for control, optimization and estimation in robotics depend on derivatives of the underlying system dynamics, e.g. to compute linearizations, sensitivities or gradient directions. However, we show that when dealing with…
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…
A heuristic formula for 5-point approximation of the first derivative of an unknown function whose values are measured with an error at unequally spaced points is proposed. The derivative at a given point is calculated using the effective…
The computation of first and second-order derivatives is a staple in many computing applications, ranging from machine learning to scientific computing. We propose an algorithm to automatically differentiate algorithms written in a subset…
A computational procedure is developed for the efficient calculation of derivatives of integrals over non-separable Gaussian-type basis functions, used for the evaluation of gradients of the total energy in quantum-mechanical simulations.…
An algorithm is proposed, analyzed, and tested experimentally for solving stochastic optimization problems in which the decision variables are constrained to satisfy equations defined by deterministic, smooth, and nonlinear functions. It is…
Many complex multi-target prediction problems that concern large target spaces are characterised by a need for efficient prediction strategies that avoid the computation of predictions for all targets explicitly. Examples of such problems…
Most of the engineering and physical systems are generally characterized by differential and difference equations based on their continuous-time and discrete-time dynamics, respectively. Moreover, these dynamical models are analyzed using…
Complex systems are composed of a large number of simple components connected to each other in the form of a network. It is shown that, for some network configurations, the equivalent dynamic behavior of the system is governed by an…
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating…
Classical neural ODEs trained with explicit methods are intrinsically limited by stability, crippling their efficiency and robustness for stiff learning problems that are common in graph learning and scientific machine learning. We present…