Related papers: On averaging block Kaczmarz methods for solving no…
This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging…
Solving large-scale systems of nonlinear equations/inequalities is a fundamental problem in computing and optimization. In this paper, we propose a generic successive projection (SP) framework for this problem. The SP sequentially projects…
A three-point iterative method for solving scalar non-linear equations was selected and then adapted to solve systems of non-linear equations. Subsequently, by applying Taylor's theorem to functions of $\R^{n}$ in $\R^{n}$, it is shown that…
This paper presents a methodology for constructing iterative schemes of any order of convergence for solving nonlinear systems of equations. It also provides formulas for the order of convergence of any iterative schemes constructed using…
The sketch-and-project, as a general archetypal algorithm for solving linear systems, unifies a variety of randomized iterative methods such as the randomized Kaczmarz and randomized coordinate descent. However, since it aims to find a…
In this paper we make a theoretical analysis of the convergence rates of Kaczmarz and Extended Kaczmarz projection algorithms for some of the most practically used control sequences. We first prove an at least linear convergence rate for…
The article mainly introduces preprocessing algorithms for solving linear equation systems. This algorithm uses three algorithms as inner iterations, namely RPCG algorithm, ADI algorithm, and Kaczmarz algorithm. Then, it uses BA-GMRES as an…
We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the…
We present a randomized iterative algorithm that exponentially converges in expectation to the minimum Euclidean norm least squares solution of a given linear system of equations. The expected number of arithmetic operations required to…
The classical Kaczmarz iteration and its randomized variants are popular tools for fast inversion of linear overdetermined systems. This method extends naturally to the setting of the phase retrieval problem via substituting at each…
The distributed Kaczmarz algorithm is an adaptation of the standard Kaczmarz algorithm to the situation in which data is distributed throughout a network represented by a tree. We isolate substructures of the network and study convergence…
The randomized Kaczmarz (RK) method is a well-known approach for solving linear least-squares problems with a large number of rows. RK accesses and processes just one row at a time, leading to exponentially fast convergence for consistent…
Randomized Kaczmarz (RK), Motzkin Method (MM) and Sampling Kaczmarz Motzkin (SKM) algorithms are commonly used iterative techniques for solving a system of linear inequalities (i.e., $Ax \leq b$). As linear systems of equations represent a…
The object of the present paper is to extend the third-order iterative method for solving nonlinear equations into systems of nonlinear equations. Since our motive is to develop the method which improve the order of convergence of Newton's…
This paper investigates the randomized version of the Kaczmarz method to solve linear systems in the case where the adjoint of the system matrix is not exact---a situation we refer to as "mismatched adjoint". We show that the method may…
We propose iterative projection methods for solving square or rectangular consistent linear systems Ax = b. Existing projection methods use sketching matrices (possibly randomized) to generate a sequence of small projected subproblems, but…
With a quite different way to determine the working rows, we propose a novel greedy Kaczmarz method for solving consistent linear systems. Convergence analysis of the new method is provided. Numerical experiments show that, for the same…
Randomized Kaczmarz is a simple iterative method for finding solutions of linear systems $Ax = b$. We point out that the arising sequence $(x_k)_{k=1}^{\infty}$ tends to converge to the solution $x$ in an interesting way: generically, as $k…
We present a Projection onto Convex Sets (POCS) type algorithm for solving systems of linear equations. POCS methods have found many applications ranging from computer tomography to digital signal and image processing. The Kaczmarz method…
Solving a large-scale system of linear equations is a key step at the heart of many algorithms in machine learning, scientific computing, and beyond. When the problem dimension is large, computational and/or memory constraints make it…