arXiv subjects

Eleonora Denich

Publications and source records attributed to Eleonora Denich.

9 recordsLinked to original sources

arXiv2026

A simple algorithm for the summation of alternating series

This paper deals with the computation of the sum of alternating series, whose general terms can be expressed by means of analytic functions. After rewriting the series as a weighted integral with the Abel weight, we employ the sinc (trapezoidal) rule and analyze the remainder term with respect to the number of quadrature points. We provide some numerical experiments to show the reliability of the derived error estimate and to test the algorithm for automatic summation with prescribed accuracy. All the Matlab codes used in the present paper can be found as open-source software at the authors' homepage.

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arXiv2026

On modified anti-Gaussian rules for Jacobi weight functions

Anti-Gaussian formulas represent an efficient tool for a dynamical estimation of the error of the underlying Gaussian rule. When applied to the Jacobi weight function it is known that such formulas are not always internal. In this work we show how to overcome this problem by using the so called modified anti-Gaussian rule with suitable parameter {\theta} = {\theta}(n), that depends on the number n of quadrature points of the Gaussian formula. Next we study theoretically the asymptotic rate of convergence of the corresponding modified averaged Gaussian formulas. We conclude by showing the benefits of this approach via numerical experiments. All the Matlab codes used in this work are available as open-source software.

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arXiv2023

Some notes on the trapezoidal rule for Fourier type integrals

This paper deals with the error analysis of the trapezoidal rule for the computation of Fourier type integrals, based on two double exponential transformations. The theory allows to construct algorithms in which the steplength and the number of nodes can be a priori selected. The analysis is also used to design an automatic integrator that can be employed without any knowledge of the function involved in the problem. Several numerical examples, which confirm the reliability of this strategy, are reported.

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arXiv2023

A fast and simple algorithm for the computation of the Lerch transcendent

This paper deals with the computation of the Lerch transcendent by means of the Gauss-Laguerre formula. An a priori estimate of the quadrature error, that allows to compute the number of quadrature nodes necessary to achieve an arbitrary precision, is derived. Exploiting the properties of the Gauss-Laguerre rule and the error estimate, a truncated approach is also considered. The algorithm used and its Matlab implementation are reported. The numerical examples confirm the reliability of this approach.

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arXiv2023

Error estimates for a Gaussian rule involving Bessel functions

This paper deals with the estimation of the quadrature error of a Gaussian formula for weight functions involving fractional powers, exponentials and Bessel functions of the first kind. For this purpose, in this work the averaged and generalized averaged Gaussian rules are employed, together with a tentative a priori approximation of the error. The numerical examples confirm the reliability of these approaches.

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arXiv2022

A fast and accurate numerical approach for electromagnetic inversion

This paper deals with the solution of Maxwell's equations to model the electromagnetic fields in the case of a layered earth. The integrals involved in the solution are approximated by means of a novel approach based on the splitting of the reflection term. The inverse problem, consisting in the computation of the unknown underground conductivity distribution from a set of modeled magnetic field components, is also considered. Two optimization algorithms are applied, based on line- and global-search methods, and a new minimization approach is presented. Several EM surveys from the ground surface are simulated, considering the horizontal coplanar (HCP) and perpendicular (PRP) magnetic dipolar configurations. The numerical experiments, carried out for the study of river-levees integrity, allowed to estimate the errors associated to these kind of investigations, and confirm the reliability of the technique.

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arXiv2022

A Gauss Laguerre approach for the resolvent of fractional powers

This paper introduces a very fast method for the computation of the resolvent of fractional powers of operators. The analysis is kept in the continuous setting of (potentially unbounded) self adjoint positive operators in Hilbert spaces. The method is based on the Gauss-Laguerre rule, exploiting a particular integral representation of the resolvent. We provide sharp error estimates that can be used to a priori select the number of nodes to achieve a prescribed tolerance.

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arXiv2022

A Gaussian method for the operator square root

We consider the approximation of the inverse square root of regularly accretive operators in Hilbert spaces. The approximation is of rational type and comes from the use of the Gauss-Legendre rule applied to a special integral formulation of the problem. We derive sharp error estimates, based on the use of the numerical range, and provide some numerical experiments. For practical purposes, the finite dimensional case is also considered. In this setting, the convergence is shown to be of exponential type.

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arXiv2021

Gaussian rule for integrals involving Bessel functions

In this work we develop the Gaussian quadrature rule for weight functions involving fractional powers, exponentials and Bessel functions of the first kind. Besides the computation based on the use of the standard and the modified Chebyshev algorithm, here we present a very stable algorithm based on the preconditioning of the moment matrix. Numerical experiments are provided and a geophysical application is considered.

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