Successive overrelaxation methods for solving the rank deficient linear squares problem

Abstract We develop successive overrelaxation (SOR) methods for finding the least squares solution of minimal norm to the system Ax = b , (∗) where A is an m × n matrix of rank r . The methods are obtained by first augmenting the system (∗) to a block 4 × 4 consistent system of linear equations. The augmented coefficient matrix is then split by a subproper SOR splitting. An interval for the relaxation parameter in which the subproper SOR iteration matrix is semiconvergent is determined along with the optimal relaxation parameter which minimizes the modulus of the controlling eigenvalue of the SOR matrix. Since the scheme computes at first only a solution to the augmented system, it is subsequently shown how to transform such a solution to the unique solution of minimal 2-norm. Analysis of the practical implementation of the algorithms developed here will be given elsewhere.

[1]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[2]  C. D. Meyer,et al.  Generalized inverses of linear transformations , 1979 .

[3]  M. Neumann,et al.  Subproper splitting for rectangular matrices , 1976 .

[4]  K. Tanabe.  Projection method for solving a singular system of linear equations and its applications , 1971 .

[5]  Ben Noble.  A Method for Computing the Generalized Inverse of a Matrix , 1966 .

[6]  T. Markham,et al.  Convergence of a direct-iterative method for large-scale least-squares problems , 1985 .

[7]  C. W. Groetsch,et al.  Generalized inverses of linear operators : representation and approximation , 1977 .

[8]  A. Berman,et al.  Proper Splittings of Rectangular Matrices , 1976 .

[9]  T. Greville.  Note on the Generalized Inverse of a Matrix Product , 1966 .

[10]  A. Berman,et al.  Cones and Iterative Methods for Best Least Squares Solutions of Linear Systems , 1974 .

[11]  V. N. Joshi.  A Note on the Solution of Rectangular Linear Systems by Iteration , 1970 .

[12]  Abraham Berman,et al.  Consistency and Splittings , 1976 .

[13]  R. Oldenburger,et al.  Infinite powers of matrices and characteristic roots , 1940 .

[14]  Richard S. Varga,et al.  Convergence of block iterative methods applied to sparse least-squares problems , 1984 .

[15]  T. Elfving.  Block-iterative methods for consistent and inconsistent linear equations , 1980 .

[16]  K. Tanabe.  Characterization of linear stationary iterative processes for solving a singular system of linear equations , 1974 .

[17]  Åke Bjürck.  METHODS FOR SPARSE LINEAR LEAST SQUARES PROBLEMS , 1976 .

[18]  L. M. Lawson.  Computational methods for generalized inverse matrices arising from proper splittings , 1975 .

[19]  Randall E. Cline,et al.  Inverses of Rank Invariant Powers of a Matrix , 1968 .

[20]  Ying-Tung Chen.  Iterative methods for linear least squares problems. , 1974 .

[21]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[22]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .