Related papers: A new second order Taylor-like theorem with an opt…

In this paper the double-sided Taylor's approximations are studied. A short proof of a well-known theorem on the double-sided Taylor's approximations is introduced. Also, two new theorems are proved regarding the monotonicity of such…

Corrected trapezoidal rules are proved for $\int_a^b f(x)\,dx$ under the assumption that $f"\in L^p([a,b])$ for some $1\leq p\leq\infty$. Such quadrature rules involve the trapezoidal rule modified by the addition of a term…

This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…

We present a new formula for divided difference and few new schemes of divided difference tables in this paper. Through this, we derive new interpolation, numerical differentiation and numerical integration formulas with arbitrary order of…

We generalize Taylor's theorem by introducing a stochastic formulation based on an underlying Poisson point process model. We utilize this approach to propose a novel non-linear regression framework and perform statistical inference of the…

Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylor's theorem come handy for this purpose. While the former is useful in…

We consider optimization algorithms that successively minimize simple Taylor-like models of the objective function. Methods of Gauss-Newton type for minimizing the composition of a convex function and a smooth map are common examples. Our…

An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a…

In this paper a spline based integral approximation is utilized to propose a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The approximations can be improved…

In this paper, we introduce an alternative method for applying averaging theory of orders $1$ and $2$ in the plane. This is done by combining Taylor expansions of the displacement map with the integral form of the…

According to a theorem of Poincare, the solutions to differential equations are analytic functions of (and therefore have Taylor expansions in) the initial conditions and various parameters provided that the right sides of the differential…

The accuracy of the numerical solution of a fractional differential equation depends on the differentiability class of the solution. The derivatives of the solutions of fractional differential equations often have a singularity at the…

We obtain a remainder estimate for the truncated Taylor expansion for differential equations driven by weakly geometric $\Pi $-rough paths for $\Pi =\left( p_{1},\cdots ,p_{k}\right) $, $p_{i}\geq 1$. When there exists $ p\geq 1$ such that…

Fillmore Theorem says that if $A$ is a nonscalar matrix of order $n$ over a field $\mathbb{F}$ and $\gamma_1,\ldots,\gamma_n\in \mathbb{F}$ are such that $\gamma_1+\cdots+\gamma_n=\text{tr} \, A$, then there is a matrix $B$ similar to $A$…

In this paper, we introduce an algorithm that provides approximate solutions to semi-linear ordinary differential equations with highly oscillatory solutions, which, after an appropriate change of variables, can be rewritten as…

Addition theorems can be constructed by doing three-dimensional Taylor expansions according to $f (\mathbf{r} + \mathbf{r}') = \exp (\mathbf{r}' \cdot \mathbf{\nabla}) f (\mathbf{r})$. Since, however, one is normally interested in addition…

A new powerful method to calculate Feynman diagrams is proposed. It consists in setting up a Taylor series expansion in the external momenta squared (in general multivariable). The Taylor coefficients are obtained from the original diagram…

A simple version for the extension of the Taylor theorem to the operator functions was found. The expansion was done with respect to a value given by a diagonal matrix for the non-commutative case, and the coefficients are given both by…

In the present paper, we propose to give an extension to the context of Dunkl theory of the notion of translation and in connection with this a corresponding extension of Taylor's formula. More precisely, we prove some properties and…

The reconstruction approach [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] for the numerical approximation of $f'(x)$ is based on the construction of a dual function $h(x)$ whose sliding averages over the interval…