zbMATH — the first resource for mathematics

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.

65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F10 Iterative numerical methods for linear systems
Full Text: DOI
[1] Boullion T. L., Generalized inverse matrices (1971)
[2] DOI: 10.1016/0024-3795(88)90226-1 · Zbl 0649.65023 · doi:10.1016/0024-3795(88)90226-1
[3] DOI: 10.1016/0024-3795(87)90110-8 · Zbl 0621.65060 · doi:10.1016/0024-3795(87)90110-8
[4] Van Loan C. F., Matrix computations (1983) · Zbl 0559.65011
[5] DOI: 10.1016/0024-3795(90)90207-S · Zbl 0708.65033 · doi:10.1016/0024-3795(90)90207-S
[6] Kelley, J. L. 1955. ”General topology”. New York: Springer. · Zbl 0066.16604
[7] Knopp, K. 1956. ”Infinite sequences and series”. New York: Dover Publ. · Zbl 0070.05807
[8] DOI: 10.1016/0024-3795(85)90073-4 · Zbl 0576.65026 · doi:10.1016/0024-3795(85)90073-4
[9] DOI: 10.1090/S0025-5718-1983-0701623-0 · doi:10.1090/S0025-5718-1983-0701623-0
[10] DOI: 10.1016/0024-3795(84)90218-0 · Zbl 0565.65019 · doi:10.1016/0024-3795(84)90218-0
[11] DOI: 10.1016/0965-9978(93)90031-N · doi:10.1016/0965-9978(93)90031-N
[12] Radhakrishna Rao, C. and K. Mitra, Sujit. 1971. ”Generalized inverse of matrices and its applications”. New York: John Willey and sons Inc. · Zbl 0236.15004
[13] DOI: 10.1007/BF01436376 · Zbl 0228.65032 · doi:10.1007/BF01436376
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.