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The extensions of convergence rates of Kaczmarz-type methods. (English) Zbl 1452.65062
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.
##### MSC:
 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
##### Keywords:
Kaczmarz-type methods; convergence rate; error estimates
##### Software:
Regularization tools
Full Text:
##### References:
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