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We propose a deterministic Kaczmarz method for solving linear systems $Ax=b$ with $A$ nonsingular. Instead of using orthogonal projections, we use reflections in the original Kaczmarz iterative method. This generates a series of points on an $n$-sphere $S$ centered at the solution $x_*=A^{-1}b$. We show that these points are nicely distributed on $S$. Taking the average of several points will lead to an effective approximation to the solution. We will show how to choose these points efficiently. The numerical tests show that in practice this deterministic scheme converges much faster than we expected and can beat the (block) randomized Kaczmarz methods.
The famous greedy randomized Kaczmarz (GRK) method uses the greedy selection rule on maximum distance to determine a subset of the indices of working rows. In this paper, with the greedy selection rule on maximum residual, we propose the greedy rando
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 accuracy, our
In this paper, combining count sketch and maximal weighted residual Kaczmarz method, we propose a fast randomized algorithm for large overdetermined linear systems. Convergence analysis of the new algorithm is provided. Numerical experiments show tha
Based on the geometric {it Triangle Algorithm} for testing membership of a point in a convex set, we present a novel iterative algorithm for testing the solvability of a real linear system $Ax=b$, where $A$ is an $m times n$ matrix of arbitrary rank.
The sampling Kaczmarz-Motzkin (SKM) method is a generalization of the randomized Kaczmarz and Motzkin methods. It first samples some rows of coefficient matrix randomly to build a set and then makes use of the maximum violation criterion within this