Related papers: Nested Derivatives: A simple method for computing …

This paper provides a new approach to derive various arbitrary high order finite difference formulae for the numerical differentiation of analytic functions. In this approach, various first and second order formulae for the numerical…

The Numerov method for linear second-order differential equations is generalized to include equations containing a first derivative term. The method presented has the same degree of accuracy as the ordinary Numerov sixth-order method. A…

In various areas of applied numerics, the problem of calculating the logarithm of a matrix A emerges. Since series expansions of the logarithm usually do not converge well for matrices far away from the identity, the standard numerical…

An explicit expression of the k-th derivative of the Bessel function $J_\nu(z)$, with respect to its order $\nu$, is given. Particularizations for the cases of positive or negative $\nu$ are considered.

A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…

The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them a small imaginary step next to the real number line. The current paper proposes…

This paper derives a way to express differentiable complex-valued functions as the sum of powers of $(1-e^{\lambda x})$, where $\lambda\in\mathbb{R}$, with an explicit formula for the remainder. This formulation is then used to associate an…

Approximate computing has shown to provide new ways to improve performance and power consumption of error-resilient applications. While many of these applications can be found in image processing, data classification or machine learning, we…

Consider a sequence of real-valued functions of a real variable given by a homogeneous linear recursion with differentiable coefficients. We show that if the functions in the sequence are differentiable, then the sequence of derivatives…

We derive analytical shape derivative formulas of the system matrix representing electric field integral equation discretized with Raviart-Thomas basis functions. The arising integrals are easy to compute with similar methods as the entries…

This paper is concerned with the efficient evaluation of higher-order derivatives of functions $f$ that are composed of matrix operations. I.e., we want to compute the $D$-th derivative tensor $\nabla^D f(X) \in \mathbb R^{N^D}$, where…

An algorithm is presented to compute isolated values of the divisor summatory function in O(n^(1/3)) time and O (log n) space. The algorithm is elementary and uses a geometric approach of successive approximation combined with coordinate…

A system of nested dichotomies is a method of decomposing a multi-class problem into a collection of binary problems. Such a system recursively applies binary splits to divide the set of classes into two subsets, and trains a binary…

This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more…

We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show…

For every positive integer h, the representation function of order h associated to a subset A of the integers or, more generally, of any group or semigroup X, counts the number of ways an element of X can be written as the sum (or product,…

In this paper, we introduce directed networks called `divergence network' in order to perform graphical calculation of divergence functions. By using the divergence networks, we can easily understand the geometric meaning of calculation…

The order derivatives of the modified Bessel function of the second kind at s = .5 are obtained as finite expressions of integrals that generalize the exponential integral appearing in the first derivative (Theorem 1.) The derivatives arise…

In our recent publication [1] we presented an exponential series approximation suitable for highly accurate computation of the complex error function in a rapid algorithm. In this Short Communication we describe how a simplified…

We develop nested variational inference (NVI), a family of methods that learn proposals for nested importance samplers by minimizing an forward or reverse KL divergence at each level of nesting. NVI is applicable to many commonly-used…