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In this paper, we developed a new type iterative scheme to approximate a common solution of split monotone variational inclusion, variational inequality and fixed point problems for an infinite family of nonexpansive mappings in the framework of Hilbert spaces. Further, we proved that the sequence generated by the proposed iterative method converges strongly to a common solution of split monotone variational inclusion, variational inequality and fixed point problems. Furthermore, we give some consequences of the main result. Finally, we discuss a numerical example to demonstrate the applicability of the iterative algorithm. The result presented in this paper unifies and extends some known results in this area. [ABSTRACT FROM AUTHOR]

Large-scale linear systems, Ax = b, frequently arise in practice and demand effective iterative solvers. Often, these systems are noisy due to operational errors or faulty data-collection processes. In the past decade, the randomized Kaczmarz algorithm (RK) has been studied extensively as an efficient iterative solver for such systems. However, the convergence study of RK in the noisy regime is limited and considers measurement noise in the right-hand side vector, b. Unfortunately, in practice, that is not always the case; the coefficient matrix A can also be noisy. In this paper, we analyze the convergence of RK for doubly noisy linear systems, i.e., when the coefficient matrix, A, has additive or multiplicative noise, and b is also noisy. In our analyses, the quantity R = ||Æ ||²||Ã||²F influences the convergence of RK, where A represents a noisy version of A. We claim that our analysis is robust and realistically applicable, as we do not require information about the noiseless coefficient matrix, A, and by considering different conditions on noise, we can control the convergence of RK. We perform numerical experiments to substantiate our theoretical findings. [ABSTRACT FROM AUTHOR]

The paper concerns a new iterative method for approximating a solution of an equilibrium problem involving a monotone and Lipschitz-type bifunction in a Hilbert space. Our method combines the proximal mapping with a regularization technique. When compared with known extragradient methods, the proposed method is seen to have a simpler and more elegant structure. A strong convergence theorem is established under some appropriate conditions imposed on the control parameters. Several experiments are performed in order to illustrate the numerical effectiveness of our new algorithm in comparison with that of existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a computing algorithm to obtain optimal checking times which minimize the expected cost rate of sequential inspection models. This algorithm is applied to two modified inspection models with finite intervals such that the unit is checked at times T k (k = 1 , 2 , ... , N) for given N and given S ≡ T N . The modified algorithms are proposed and optimal checking times to minimize the expected cost rates are obtained. Numerical examples of each model are given to show how to use the algorithm. This method would be applied to other sequential inspection models by its suitable modifications. [ABSTRACT FROM AUTHOR]

Solving systems of nonlinear equations has evolved into an active research field, with numerous iterative methods being proposed. Notably, iterative methods characterized by fast convergence remain of interest. In this paper, based on the modified line search scheme by Ou and Li, we introduce a derivative-free algorithm with a relaxed-inertial technique for approximating solutions of nonlinear systems involving pseudo-monotone mappings in Euclidean space. The global convergence of the proposed algorithm is established without Lipschitz continuity of the underlying mapping. Moreover, our approach allows flexibility in selecting the inertial extrapolation step length within the interval [0, 1]. To show the efficiency of the proposed method, we embed a derivative-free search direction into the scheme. Numerical experiments are given to illustrate the efficiency of the proposed algorithm for large-scale systems and sparse signal reconstruction. [ABSTRACT FROM AUTHOR]

In this paper, using the concept of inertial extrapolation, we introduce a globally convergent inertial extrapolation method for solving nonlinear equations with convex constraints for which the underlying mapping is monotone and Lipschitz continuous. The method can be viewed as a combination of the efficient three-term derivative-free method of Gao and He [Calcolo. 55(4), 1-17, 2018] with the inertial extrapolation step. Moreover, the algorithm is designed such that at every iteration, the method is free from derivative evaluations. Under standard assumptions, we establish the global convergence results for the proposed method. Numerical implementations illustrate the performance and advantage of this new method. Moreover, we also extend this method to solve the LASSO problems to decode a sparse signal in compressive sensing. Performance comparisons illustrate the effectiveness and competitiveness of our algorithm. [ABSTRACT FROM AUTHOR]

The aim of this paper is to approximate fixed points of nonexpansive type mappings in Banach spaces when the set of fixed points is nonempty. We study the general Picard–Mann (GPM) algorithm, obtaining the weak and strong convergence theorems. We provide an example to illustrate the convergence behaviour of the GPM algorithm. We compare the GPM algorithm with other existing (well known) algorithms numerically (under different parameters and initial guesses). [ABSTRACT FROM AUTHOR]

In this paper, in order to solve systems of nonlinear equations, a new class of frozen Jacobian multi-step iterative methods is presented. Our proposed algorithms are characterized by a highly convergent order and an excellent efficiency index. The theoretical analysis is presented in detail. Finally, numerical experiments are presented for showing the performance of the proposed methods, when compared with known algorithms taken from the literature. [ABSTRACT FROM AUTHOR]

The split feasibility problem (SFP) has many practical applications, which has attracted the attention of many authors. In this paper, we propose a different method to solve the SFP and the fixed-point problem involving quasi-nonexpansive mappings. We relax the conditions of the operator as well as consider the inertial iteration and the adaptive step size. For example, the convergence generated by our new method is better than that of other algorithms, and the convergence rate of our algorithm greatly improves that of previous algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we study and introduce a self adaptive method together with a Halpern iterative algorithm for approximating solutions of multiple-sets split monotone variational inclusion problem which includes the multiple-sets split feasibility problem, split feasibility problem, split monotone variational inclusion problem and split variational inclusion problem, to mention a few. Using our iterative algorithm, we prove a strong convergence result for approximating the solution of the aforementioned problems. Numerical examples on finite-dimensional and infinite-dimensional spaces are displayed to illustrate the performance of our iterative method. The result discussed in this article extends and complements many related results in literature. [ABSTRACT FROM AUTHOR]

In this study, our examination centers around the numerical resolution of non-coercive issues using a multi-grid approach. Our particular emphasis is directed towards employing multi-grid methodologies to tackle non-linear variational inequalities. Our primary goal involves confirming the consistent convergence of the multi-grid algorithm. To attain this objective, we make use of fundamental sub-differential calculus and glean insights from the convergence principles of non-linear multi-grid techniques. [ABSTRACT FROM AUTHOR]

We propose an algorithm that estimates the real roots of differentiable functions on closed intervals. Then, we extend this algorithm to real differentiable functions that are dominated by a polynomial. For each starting point, our method converges to the nearest root to the right or left hand side of that point. Our algorithm can look for missed roots as well and theoretically it misses no root. Furthermore, we do not find the roots by randomly chosen initial guesses. The iterated sequences in our algorithms converge linearly. Therefore, the rate of convergence can be accelerated considerably to make it comparable to Newton-Raphson and other high-speed methods. We have illustrated our algorithms with some concrete examples. Finally, the pseudo-codes of the related algorithms are presented at the end of this manuscript. [ABSTRACT FROM AUTHOR]

This paper presents a third order iterative method for obtaining the Moore–Penrose and Drazin inverses with a computational cost of O (n 3) , where n ∈ N . The performance of the new approach is compared with other methods discussed in the literature. The results show that the algorithm is remarkably efficient and accurate. Furthermore, sufficient criteria in the fractional sense are presented, both for smooth and non-smooth solutions. The fractional elliptic Poisson and fractional sub-diffusion equations in the Caputo sense are considered as prototype examples. The results can be extended to other scientific areas involving numerical linear algebra. [ABSTRACT FROM AUTHOR]

In this work we focus on the convex feasibility problem (CFP) in Hilbert space. A specific method in this area that has gained a lot of interest in recent years is the Douglas-Rachford (DR) algorithm. This algorithm was originally introduced in 1956 for solving stationary and non-stationary heat equations. Then in 1979, Lions and Mercier adjusted and extended the algorithm with the aim of solving CFPs and even more general problems, such as finding zeros of the sum of two maximally monotone operators. Many developments which implement various concepts concerning this algorithm have occurred during the last decade. We introduce an unrestricted DR algorithm, which provides a general framework for such concepts. Using unrestricted products of a finite number of strongly nonexpansive operators, we apply this framework to provide new iterative methods, where, inter alia, such operators may be interlaced between the operators used in the scheme of our unrestricted DR algorithm. [ABSTRACT FROM AUTHOR]

Consider a linear system Ax = b where the coefficient matrix A is rectangular and of full-column rank. We propose an iterative algorithm for solving this linear system, based on gradientdescent optimization technique, aiming to produce a sequence of well-approximate least-squares solutions. Here, we consider least-squares solutions in a full generality, that is, we measure any related error through an arbitrary vector norm induced from weighted positive definite matrices W. It turns out that when the system has a unique solution, the proposed algorithm produces approximated solutions converging to the unique solution. When the system is inconsistent, the sequence of residual norms converges to the weighted least-squares error. Our work includes the usual least-squares solution when W = I. Numerical experiments are performed to validate the capability of the algorithm. Moreover, the performance of this algorithm is better than that of recent gradient-based iterative algorithms in both iteration numbers and computational time. [ABSTRACT FROM AUTHOR]

We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schrödinger perturbation theory and termed iterative perturbative theory (IPT). Contrary to standard eigenvalue algorithms, which are either "direct" (to compute all eigenpairs) or "iterative" (to compute just a few), IPT computes any number of eigenpairs with the same basic iterative procedure. Thanks to this perfect parallelism, IPT proves more efficient than classical methods (LAPACK or CUSOLVER for the full-spectrum problem, preconditioned Davidson solvers for extremal eigenvalues). We give sufficient conditions for linear convergence and demonstrate performance on dense and sparse test matrices, including one from quantum chemistry. The code is available at http://github.com/msmerlak/IterativePerturbationTheory.jl. [ABSTRACT FROM AUTHOR]

This paper proposes an algorithm based on the Shultz iterative method and divided differences to find the roots of nonlinear equations. Using the new method, we present a rapid and powerful algorithm to compute an approximate inverse of an invertible tensor. Analysis of the convergence error shows that the convergence order of the method is a linear combination of the Fibonacci sequence and also is rapid and powerful in finding and keeping sparsity of the obtained approximate inverse of the sparse tensors. The algorithm is extended for computing the Moore-Penrose inverse of a tensor. As an application, we use the iterates obtained by the algorithm as a preconditioner for the tensorized Krylov subspace method, e.g., LSQR based tensor form to solve the multilinear system A ★ N X = B. Several examples are also provided to show the efficiency of the proposed method. Finally, some concluding remarks are given. [ABSTRACT FROM AUTHOR]

In this paper, combining count sketch and maximal weighted residual Kaczmarz method, we propose a fast randomized algorithm for highly overdetermined linear systems. Convergence analysis of the new algorithm is provided. Numerical experiments show that, for the same accuracy, our method behaves better in computing time compared with the maximal weighted residual Kaczmarz algorithm. [ABSTRACT FROM AUTHOR]

In recent times, various algorithms have been incorporated with the inertial extrapolation step to speed up the convergence of the sequence generated by these algorithms. As far as we know, very few results exist regarding algorithms of the inertial derivative-free projection method for solving convex constrained monotone nonlinear equations. In this article, the convergence analysis of a derivative-free iterative algorithm (Liu and Feng in Numer. Algorithms 82(1):245–262, 2019) with an inertial extrapolation step for solving large scale convex constrained monotone nonlinear equations is studied. The proposed method generates a sufficient descent direction at each iteration. Under some mild assumptions, the global convergence of the sequence generated by the proposed method is established. Furthermore, some experimental results are presented to support the theoretical analysis of the proposed method. [ABSTRACT FROM AUTHOR]