Xiezhang Li

A Perturbation Bound of the Drazin Inverse of a Matrix by Separation of Simple Invariant Subspaces

A constructive perturbation bound of the Drazin inverse of a square matrix is derived using a technique proposed by G. Stewart and based on perturbation theory for invariant subspaces. This is an improvement of the result published by the authors Wei and Li [Numer. Linear Algebra Appl., 10 (2003), pp. 563--575]. It is a totally new approach to developing perturbation bounds for the Drazin inverse of a matrix. A numerical example which indicates the sharpness of the perturbation bound is presented.

An improvement on the perturbation of the group inverse and oblique projection

The perturbations of the group inverse A# and oblique projection AA# of a square matrix A have been previously studied. Under certain assumptions on the matrix A and a perturbation matrix E, upper bounds for ∥B#∥,∥BB#∥,∥B#−A#∥∥A#∥and∥BB#−AA#∥∥AA#∥, where B=A+E, are given in the literature. Recently, upper bounds for the general case have been published by Y. Wei [Appl. Math. Comp. 98 (1999) 29]. However, the special cases in the literature and the continuity of the group inverse do not follow from the general upper bounds. In this paper, we derive new general upper bounds which not only cover all the special cases but also are sharper than Weis results such that the continuity of the group inverse directly follows. A numerical example is given to illustrate the sharpness of the new general upper bounds.

Iterative methods for the Drazin inverse of a matrix with a complex spectrum

Iterative methods for the Drazin inverse of a square matrix with a real spectrum have been developed recently. These methods are generalized in the case of matrices with complex spectra. Semiiterative method for the Drazin inverse is also discussed. Numerical examples are given to illustrate the theoretic results.

A Representation for the Drazin Inverse of Block Matrices with a Singular Generalized Schur Complement

Abstract Consider a 2 × 2 block complex square matrix M = A B C D , where A and D are square matrices. Suppose that ( I - AA D ) B = O and C ( I - AA D ) = O , where A D is the Drazin inverse of A. The representations of the Drazin inverse M D have been studied in the case where the generalized Schur complement, S = A - CA D B , is either zero or nonsingular. In this paper, we develop a representation, under certain conditions, for M D when S is singular and group invertible. Moreover, this formula includes the case where S = O or nonsingular. A numerical example is given to illustrate the result.

A note on the perturbation bound of the Drazin inverse

A perturbation bound for the Drazin inverse A^D with Ind(A+E)=1 has recently been developed. However, those upper bounds are not satisfied since it is not tight enough. In this paper, a sharper upper bounds for ||(A+E)^#-A^D|| with weaker conditions is derived. That new bound is also a generalization of a new general upper bound of the group inverse. We also derive a new expression of the Drazin inverse (A+E)^D with Ind(A+E)>1 and the corresponding upper bound of ||(A+E)^D-A^D|| in a special case. Numerical examples are given to illustrate the sharpness of the new bounds.

An expression of the Drazin inverse of a perturbed matrix

Given a square matrix A and its perturbation matrix E, a new expression for the Drazin inverse B^D of B=A+E is derived if AA^DB^2=(AA^DB)^2 or B^2AA^D=(BAA^D)^2. Based on the new expression, a bound of the relative error of B^D is developed. Some known results in the literature on the Drazin inverse and the perturbation bound are included by this new formula as special cases. A numerical example is given to compare the upper bounds.

The convergence of the block cyclic projection with an overrelaxation parameter for compressed sensing based tomography

The convergence of the block cyclic projection for compressed sensing based tomography (BCPCS) algorithm had been proven recently in the case of underrelaxation parameter λ ? ( 0 , 1 ] . In this paper, we prove its convergence with overrelaxation parameter λ ? ( 1 , 2 ) . As a result, the convergence of the other two algorithms (BCAVCS and BDROPCS) with overrelaxation parameter λ ? ( 1 , 2 ) in a special case is derived. Experiments are given to demonstrate the convergence behavior of the BCPCS algorithm with different values of λ .

The convergence rate of the Chebyshev sim under a perturbation of a complex line-segment spectrum

Abstract The Chebyshev semiiterative method ( chsim ) is probably the best known and most often used method for the iterative solution of linear system x = T x + c, where the spectrum of T is located in a complex line segment [α, β] excluding 1. The asymptotic convergence factor (ACF) of the chsim , under a perturbation of [α, β], is considered. Several formulae for the approximation to the ACFs, up to the second order of a perturbation, are derived. This generalizes the results about the sensitivity of the asymptotic rate of convergence to the estimated eigenvalues by Hageman and Young in the case that both α and β are real. Two numerical examples are given to illustrate the theoretical results.

A UNIFORM ERROR BOUND FOR THE OVERRELAXATION METHODS

Abstract Let A x = b be a system of linear equations where A is symmetric and positive definite. Suppose that the associated block Jacobi matrix B is consistently ordered, weekly cyclic of index 2, and convergent [i.e., μ 1 ≔ ϱ ( B ) n + 1 = T ω x n + c ω for n ⩾ 0, to solve the system. We derive a uniform error bound for the overrelaxation methods, ∥x−x n ∥ 2 ⩽ 1 [1+s(μ 1 2 ) + t(μ 1 2 )] 2 x (t 0 + |t 1 |μ 1 2 ) 2 ∥δ n ∥ 2 − 2t 0 〈δ n ,δ n+1 〉 +|t 1 |μ 1 2 ∥δ n ∥∥δ n+1 ∥+∥δ n+1 ∥ 2 where ∥ · ∥ = ∥ · ∥ 2 , δ n = x n − x n − 1 , and s ( μ 2 ) and t ( μ 2 ) ≔ t 0 + t 1 μ 2 are two coefficients of the corresponding functional equation connecting the eigenvalues λ of T ω to the eigenvalues μ of B . As special cases of the uniform error bound, we will give two error bounds for the SSOR and USSOR methods.

Construction of a full row-rank matrix system for multiple scanning directions in discrete tomography

A full row-rank system matrix generated by scans along two directions in discrete tomography was recently studied. In this paper, we generalize the result to multiple directions. Let A x = h be a reduced binary linear system generated by scans along three directions. Using geometry, it is shown in this paper that the linearly dependent rows of the system matrix A can be explicitly identified and a full row-rank matrix can be obtained after the removal of those rows. The results could be extended to any number of multiple directions. Therefore, certain software packages requiring a full row-rank system matrix can be adopted to reconstruct an image. Meanwhile, the cost of computation is reduced by using a full row-rank matrix.