Characterization of the solutions set of inconsistent least-squares problems by an extended Kaczmarz algorithm

  • Constantin Popa


We give a new characterization of the solutions set of the general (inconsistent) linear least-squares problem using the set of limit-points of an extended version of the classical Kaczmarz’s projections method. We also obtain a “ step error reduction formula” which, in some cases, can give us apriori information about the convergence properties of the algorithm. Some numerical experiments with our algorithm and comparisons between it and others existent in the literature, are made in the last section of the paper.

AMS Mathematics Subject Classification

65F10 65F20 

Key word and phrases

least-squares problems extended Kaczmarz algorithm 


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  1. 1.
    A. Bjork, T. Elfving,Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations, BIT 19(1979), 145–163.CrossRefMathSciNetGoogle Scholar
  2. 2.
    T.L. Boullion and P.L. Odell,Generalized inverse matrices, Willey-Interscience, New York, 1971.MATHGoogle Scholar
  3. 3.
    G.H. Golub and C.F. van Loan,Matrix computations, The John’s Hopkins Univ.Press, Baltimore, 1983.MATHGoogle Scholar
  4. 4.
    S. Kaczmarz,Angenaherte Auflosung von Systemen linearer Gleichungen, Bull.Acad.Polonaise Sci. et Lettres A (1937), 355–357.Google Scholar
  5. 5.
    C. Popa,An iterative method for CVBEM systems.Part I: The Kaczmarz algorithm;Part II: The unigrid method;Advances in Engineering Software, 16(1993), 61–69.Google Scholar
  6. 6.
    C. Popa,Least-Squares Solution of Overdetermined Inconsistent Linear Systems Using Kaczmarz’s Relaxation, Intern.J.Computer Math., 55(1995), 79–89.MATHCrossRefGoogle Scholar
  7. 7.
    L.D. Pyle,A generalized inverse ∈-algorithm for constructing intersection projection matrices with applications, Numer.Math., 10(1967), 86–102.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    R. Radhakrishna and S.K. Mitra,Generalized inverse of matrices and its applications, John Willey and sons Inc., New York, 1971.MATHGoogle Scholar
  9. 9.
    N. Shinozaki, M. Sibuya, and K. Tanabe,Numerical methods for the Moore- Penrose inverse of a matrix, Annals of the Institute of Statistical Mathematics, 24(1972), 193–203 and 621–629.CrossRefMathSciNetGoogle Scholar
  10. 10.
    K. Tanabe,Projection Method for Solving a Singular System of Linear Equations and its Applications, Numer.Math., 17(1971), 203–214.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    K. Tanabe,Conjugate gradient method for computing the Moore-Penrose inverse and rank of a matrix, Journal of Optimization Theory and Applications, 22(1)(1977), 1–23.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    D.M. Young,Iterative solution of large linear systems, Academic Press, New York, 1971.MATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 1999

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsOvidius UniversityConstantaRomania

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