Symmetric successive overrelaxation methods for solving the rank deficient linear least squares problem
Introduction
The linear systemwhere A is a complex m × n matrix and b a complex m-vector, is studied by various people. Numerous iterative methods have been obtained for the case in which A is a nonsingular matrix, such as the well-known ones of Jacobi, Gauss-Seidel, SOR, which can be derived from the AOR method, and SSOR. With splitting the coefficient matrix A and extending the above results, methods for finding the least squares solution of minimal norm to (1.1) have been suggested by Berman and Plemmons [1], Tanabe [2], Berman and Neumann [3], [4], Neumann [5], Elfving [6] and others.
For the case where A is rank deficient, Miller and Neumann [7] give an approach based on subproper splittings as developed by Berman and Plemmons [1] and establish the theory of SOR iterations for the augmented and consistent linear system that Chen provides in [8] to solve the least squares problem. Recently, Tian [9] presents the AOR methods for the system (1.1) in the same manner.
In this paper, we apply subproper splittings on SSOR iterations for rank deficient linear system and give another subproper splitting differing from Miller and Neumann’s for the coefficient matrix of the augmented system. As in their paper, we first augment (1.1) to a block 4 × 4 consistent system, and then develop subproper techniques to determine a solution to the augmented system. This requires us to overcome the complicating feature of having to analyze and bound the subdominant or controlling eigenvalue of the iteration matrix which is singular. Finally, we give the least squares solution of minimal norm, and a numerical example is given to illustrate our theory.
Throughout the paper, for , A∗, N(A), R(A), rank(A), σ(A) and ρ(A) denotes the conjugate transpose, the null space, the range space, the rank, the spectrum and the spectral radius of A, respectively. ∥·∥2 denotes the 2-norm over , i.e., . PV,W denotes the projector on V along W ().
Section snippets
Preliminaries
In this section, we recall the notions of generalized inverses and subproper splittings, and present the augmented and consistent system for (1.1).
Let , then a matrix is called Moore–Penrose generalized inverse of A, denoted by X = A†, if it satisfiesA† is unique and always exists. It is well known that A†b is the least square solution of minimal 2-norm to the system (1.1).
Assume that m = n and index(A) ⩽ 1, or equivalently, , then
Subproper SSOR splittings
We now concentrate our efforts on the augmented system (2.12). For the augmented systemwe first split asObviously, D is nonsingular. Let , then the SSOR splitting of isSince L is strictly lower triangular and I − ωU is nonsingular (ω ≠ 1), the matrix D(I − ωL)(I − ωU)/ω(2 − ω
Computations of least squares solution of minimal norm
As mentioned earlier in Section 2, beginning with x0, the iterations (3.3), (3.15) converge to a solution to (2.12), given byand
The following theorem shows that the least square solution of minimal norm can be obtained by multiplying the solution of the augmented system by (I − Hω)†(I − Hω) or . Lemma 4.1 Let . Then PV,WA = A if and only if R(A) ⊆ V. Let . Then APV,W = A if and only if N(A) ⊇ W.[10]
Theorem 4.2
Numerical example
We use the following example to illustrate the procedures by Matlab. Example Consider The least squares solution of minimal norm is , and the coefficient matrixis of rank 2. Let be a block form like (2.8), where , and . In this case, and . Simple calculation yields Therefore, we can choose the proper parameter ω in SSOR scheme (3.3)
Conclusion
In this paper, we investigate the SSOR methods for the rank deficient equation Ax = b, and the necessary and sufficient conditions for the semiconvergence of the methods are explicitly given out. The numerical example shows that the presented methods are applicable and efficient to solve such rank deficient equations.
Acknowledgment
The first author was supported by the start-up fund and 985 specific program of Lanzhou University, PR China.
References (12)
Subproper splitting for rectangular matrices
Linear Algebra Appl.
(1976)- et al.
Successive overrelaxation methods for solving the rank deficient linear least squares problem
Linear Algebra Appl.
(1987) Accelerate overrelaxation methods for rank deficient linear systems
Appl. Math. Comput.
(2003)- et al.
Cones and iterative methods for best least squares solutions of linear systems
SIAM J. Numer. Anal.
(1974) Characterization of linear stationary iterative processes for solving a singular system of linear equations
Numer. Math.
(1974)- et al.
Proper splittings of rectangular matrices
SIAM J. Appl. Math.
(1976)
Cited by (10)
Symmetric block-SOR methods for rank-deficient least squares problems
2008, Journal of Computational and Applied MathematicsComments on "Symmetric successive overrelaxation methods for rank deficient linear systems". [Applied Mathematics and Computation 173 (2006) 404-420]
2006, Applied Mathematics and ComputationA necessary and sufficient condition for semiconvergence and optimal parameter of the SSOR method for solving the rank deficient linear least squares problem
2006, Applied Mathematics and Computation