an:07241425
Zbl 1452.65062
Kang, Chuan-gang; Zhou, Heng
The extensions of convergence rates of Kaczmarz-type methods
EN
J. Comput. Appl. Math. 382, Article ID 113099, 11 p. (2021).
00452504
2021
j
65F10 65F08 65N22 65J20
Kaczmarz-type methods; convergence rate; error estimates
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.