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The extensions of convergence rates of Kaczmarz-type methods. (English) Zbl 07241425
Summary: Kaczmarz-type methods, such as the randomized Kaczmarz method, the block Kaczmarz method and the Cimmino method, can be derived from the Kaczmarz method. In this paper, we introduce a new error term \(\|x_k-P_{N(A)}x_0-x^\dagger\|_2\) for Kaczmarz-type methods, where \(x^\dagger\) is the generalized solution of \(Ax=b\) and \(P_{N(A)}x_0\) is the orthogonal projection of a given initial value \(x_0\) onto the null space \(N(A)\). It includes the well-known error term \(\|x_k-x_\ast\|_2\) as a special case when \(x_0=0\) and \(x^\dagger=x^\ast\), where \(x^\ast\) is a true solution of \(Ax=b\). We investigate the behavior of the new error term and establish the corresponding convergence rates for Kaczmarz-type methods. Especially, from the estimate of new error term for the Kaczmarz method, we can get a more simple proof for the convergence of the Kaczmarz method.
65F10 Iterative numerical methods for linear systems
65F08 Preconditioners for iterative methods
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
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