Randomized double and triple Kaczmarz for solving extended normal equations
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The randomized Kaczmarz algorithm has received considerable attention recently because of its simplicity, speed, and the ability to approximately solve large-scale linear systems of equations. In this paper we propose randomized double and triple Kaczmarz algorithms to solve extended normal equations of the form A^⊤ Ax=A^⊤ b-c. The proposed algorithms avoid forming A^⊤ A explicitly and work for arbitrary A∈^m× n (full rank or rank deficient, m≥ n or m<n). Tight upper bounds showing exponential convergence in the mean square sense of the proposed algorithms are presented and numerical experiments are given to illustrate the theoretical results.
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