Journal of Numerical Analysis and Approximation Theory

Book reviews

jnaat — 2023-12-28

Book review for:

Gabriele Ciaramella, Martin J. Gander, Iterative Methods and Preconditioners for Systems of Linear Equations, SIAM, Philadelphia, 2022, X + 275 pp., ISBN 978-1-611976-89-2 (paperback), ISBN 978-1-61197-690-8 (ebook). Part of the Fundamentals of Algorithms series. Reviewed by M. Nechita.

Yutaka Yamamoto, From Vector Spaces to Function Spaces. Introduction to Functional Analysis with Applications, SIAM, Philadelphia, 2012, XIV + 268 pp., ISBN 978-1-61197-230-6 (paperback), ISBN 978-1-61197-231-3 (ebook). Part of the Other Titles in Applied Mathematics series. Reviewed by R. Precup.

Forward-backward splitting algorithm with self-adaptive method for finite family of split minimization and fixed point problems in Hilbert spaces

Hammed Anuoluwapo Abbas — 2023-12-28

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

Fuzzy Korovkin type Theorems via deferred Cesaro and deferred Euler equi-statistical convergence

Purshottam Agrawal — 2023-12-28

We establish a fuzzy Korovkin type approximation theorem by using \(eq-stat^{D}_{CE}\)(deferred Ces\'{a}ro and deferred Euler equi-statistical) convergence proposed by Saini et al. for continuous functions over \([a,b]\). Further, we determine the rate of convergence via fuzzy modulus of continuity.

Approximation of the Hilbert transform in the Lebesgue spaces

Rashid Aliev — 2023-12-28

The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. The Hilbert transform is the main part of the singular integral equations on the real line. Therefore, approximations of the Hilbert transform are of great interest. Many papers have dealt with the numerical approximation of the singular integrals in the case of bounded intervals. On the other hand, the literature concerning the numerical integration on unbounded intervals is by far poorer than the one on bounded intervals. The case of the Hilbert Transform has been considered very little. This article is devoted to the approximation of the Hilbert transform in Lebesgue spaces by operators which introduced by V.R.Kress and E.Mortensen to approximate the Hilbert transform of analytic functions in a strip. In this paper, we prove that the approximating operators are bounded maps in Lebesgue spaces and strongly converges to the Hilbert transform in these spaces.

Local convergence analysis of frozen Steffensen-type methods under generalized conditions

Ioannis K Argyros — 2023-12-28

The goal in this study is to present a unified local convergence analysis of frozen Steffensen-type methods under generalized Lipschitz-type conditions for Banach space valued operators. We also use our new idea of restricted convergence domains, where we find a more precise location, where the iterates lie leading to at least as tight majorizing functions. Consequently, the new convergence criteria are weaker than in earlier works resulting to the expansion of the applicability of these methods. The conditions do not necessarily imply the differentiability of the operator involved. This way our method is suitable for solving equations and systems of equations.

Extension of primal-dual interior point method based on a kernel function for linear fractional problem

Mousaab Bouafia — 2023-12-28

Our aim in this work is to extend the primal-dual interior point method based on a kernel function for linear fractional problem. We apply the techniques of kernel function-based interior point methods to solve a standard linear fractional program. By relying on the method of Charnes and Cooper [3], we transform the standard linear fractional problem into a linear program. This transformation will allow us to define the associated linear program and solve it efficiently using an appropriate kernel function. To show the efficiency of our approach, we apply our algorithm on the standard linear fractional programming found in numerical tests in the paper of A. Bennani et al. [4], we introduce the linear programming associated with this problem. We give three interior point conditions on this example, which depend on the dimension of the problem. We give the optimal solution for each linear program and each linear fractional program. We also obtain, using the new algorithm, the optimal solutions for the previous two problems. Moreover, some numerical results are illustrated to show the effectiveness of the method.

An extension of the Cheney-Sharma operator of the first kind

Teodora Cătinaș — 2023-12-28

We extend the Cheney-Sharma operators of the first kind using Stancu type technique and we study some approximation properties of the new operator. We calculate the moments, we study local approximation with respect to a K-functional and the preservation of the Lipschitz constant and order.

The rate of convergence of bounded linear processes on spaces of continuous functions

Heiner Gonska — 2023-12-28

Quantitative Korovkin-type theorems for approximation by bounded linear operators defined on \(C(X,d)\) are given, where \((X,d)\) is a compact metric space. Special emphasis is on positive linear operators.
As is known from previous work of Newman and Shapiro, Jimenez Pozo, Nishishiraho and the author, among others, there are two possible ways to obtain error estimates for bounded linear operator approximation: the so-called direct approach, and the smoothing technique.
We give various generalizations and refinements of earlier results which were obtained by using both techniques. Furthermore, it will be shown that, in a certain sense, none of the two methods is superior to the other one.

New sufficient conditions for the solvability of a new class of Sylvester-like absolute value matrix equation

Shubham Kumar — 2023-12-28

In this article, some new sufficient conditions for the unique solvability of a new class of Sylvester-like absolute value matrix equation \(AXB - \vert CXD \vert =F\) are given. This work is distinct from the published work by Li [Journal of Optimization Theory and Application, 195(2), 2022]. Some new conditions were also obtained, which were not covered by Li. We also provided an example in support of our result.

Quenching for discretizations of a semilinear parabolic equation with nonlinear boundary outflux

Kouakou Cyrille N'Dri — 2023-12-28

In this paper, we study numerical approximations of a semilinear parabolic problem in one-dimension, of which the nonlinearity appears both in source term and in Neumann boundary condition. By a semidiscretization using finite difference method, we obtain a system of ordinary differential equations which is an approximation of the original problem. We obtain some conditions under which the positive solution of our system quenches in a finite time and estimate its semidiscrete quenching time. Convergence of the numerical quenching time to the theoretical one is established. Next, we show that the quenching rate of the numerical scheme is different from the continuous one. Finally, we give some numerical results to illustrate our analysis.

Localization of Nash-type equilibria for systems with partial variational structure

Andrei Stan — 2023-12-28

In this paper, we aim to generalize an existing result by obtaining localized solutions within bounded convex sets, while also relaxing specific initial assumptions. To achieve this, we employ an iterative scheme that combines a fixed-point argument based on the Minty-Browder Theorem with a modified version of the Ekeland variational principle for bounded sets. An application to a system of second-order differential equations with Dirichlet boundary conditions is presented.

Nonlinear random extrapolation estimates of \(\pi\) under Dirichlet distributions

Shasha Wang — 2023-12-28

We construct optimal nonlinear extrapolation estimates of \(\pi\) based on random cyclic polygons generated from symmetric Dirichlet distributions. While the semiperimeter \( S_n \) and the area \( A_n \) of such random inscribed polygons and the semiperimeter (and area) \( S_n' \) of the corresponding random circumscribing polygons are known to converge to \( \pi \) w.p.\(1\) and their distributions are also asymptotically normal as \( n \to \infty \), we study in this paper nonlinear extrapolations of the forms \( \mathcal{W}_n = S_n^{\alpha} A_n^{\beta} S_n'^{\, \gamma} \) and \( \mathcal{W}_n (p) = ( \alpha S_n^p + \beta A_n^p + \gamma S_n'^{\, p} )^{1/p} \) where \( \alpha + \beta + \gamma = 1 \) and \( p \neq 0 \). By deriving probabilistic asymptotic expansions with carefully controlled error estimates, we show that \( \mathcal{W}_n \) and \( \mathcal{W}_n (p) \) also converge to \( \pi \) w.p.\(1\) and are asymptotically normal. Furthermore, to minimize the approximation error associated with \( \mathcal{W}_n \) and \( \mathcal{W}_n (p) \), the parameters must satisfy the optimality condition \( \alpha + 4 \beta - 2 \gamma = 0 \). Our results generalize previous work on nonlinear extrapolations of \( \pi \) which employ inscribed polygons only and the vertices are also assumed to be independently and uniformly distributed on the unit circle.