A deterministic Kaczmarz algorithm for solving linear systems

05/17/2021

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by   Changpeng Shao, et al.

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We propose a deterministic Kaczmarz method for solving linear systems A=$̱ withAnonsingular. Instead of using orthogonal projections, we use reflections in the original Kaczmarz iterative method. This generates a series of points on ann-spherecentered at the solution_*=A^-1$̱. We show that these points are nicely distributed on . 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.