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Least-squares solution of overdetermined inconsistent linear systems using Kaczmarz’s relaxation. (English) Zbl 0830.65027
Summary: For numerical computation of the minimal Euclidean norm (least-squares) solution of overdetermined linear systems, usually direct solvers are used. The iterative methods for such kind of problems need special assumptions about the system (consistency, full rank of the system matrix, some parameters they use or they don’t give the minimal length solution).
In the present paper, we purpose two iterative algorithms which generate sequences convergent to the minimal Euclidean length solution in the general case (inconsistent system and rank deficient matrix). The algorithms only use some combinations and properties of the well-known iterative method of S. Kaczmarz [Bull. Int. Acad. Polon. Sci. A 1937, 355-357 (1937; Zbl 0017.31703)] and need no special assumptions about the system.

MSC:
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F10 Iterative numerical methods for linear systems
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