Related papers: Nested Derivatives: A simple method for computing …
We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power…
A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of extensor fields is present using algebraic and analytical tools developed in previous papers. Several important formulas are derived.
In this paper we propose a method for proving some exponential inequalities based on power series expansion and analysis of derivations of the corresponding functions. Our approach provides a simple proof and generates a new class of…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
We introduce a new model of linear regression for random functional inputs taking into account the first order derivative of the data. We propose an estimation method which comes down to solving a special linear inverse problem. Our…
The derivatives with respect to order {\nu} for the Bessel functions of argument x (real or complex) are studied. Representations are derived in terms of integrals that involve the products pairs of Bessel functions, and in turn series…
In a previous work, we developed an algorithm for the computation of incomplete Bessel functions, which pose as a numerical challenge, based on the $G_{n}^{(1)}$ transformation and Slevinsky-Safouhi formula for differentiation. In the…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
We compute the nth derivative of a function given parametrically, and of one given implicitly, and some history for both problems. I am posting this version of the paper at the request of Shaul Zemel, whose forthcoming paper The…
We give estimates for the convolution product of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power…
The recursive Neville algorithm allows one to calculate interpolating functions recursively. Upon a judicious choice of the abscissas used for the interpolation (and extrapolation), this algorithm leads to a method for convergence…
This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
In distributed optimization and distributed numerical linear algebra, we often encounter an inversion bias: if we want to compute a quantity that depends on the inverse of a sum of distributed matrices, then the sum of the inverses does not…
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
Derivative-based algorithms are ubiquitous in statistics, machine learning, and applied mathematics. Automatic differentiation offers an algorithmic way to efficiently evaluate these derivatives from computer programs that execute relevant…
We introduce $p$-derivations and give a few basic ways in which they act like derivatives by numbers.
There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of…
The derivative expansion approach to the calculation of the interaction between two surfaces, is a generalization of the proximity force approximation, a technique of widespread use in different areas of physics. The derivative expansion…