The weighted moore-Penrose inverse of a partitioned matrix A=(U V) is discussed. Representations for the weighted Moore-Penrose inverse of the matrix A are derived, which extend some early results.

In this paper, we shall give a new method for computing the weighted generalized inversion of partitioned matrices, i.e. solve the weighted problems with the aid of the unweighted ones, which is quite efficient and convenient for either dealing with the inconsistent equations or computing the generalized inverses, now we discuss some problems about the latter. The theorem I below is an extension of the main result in [2], especially, the proof is quite succinct here.

Algebraic perturbation methods were first proposed for the solution of nonsingular linear systems by R. E. Lynch and T. J. Aird [2]. Since then, the algebraic perturbation methods for generalized inverses have been discussed by many scholars [3]-[6]. In [4], a singular square matrix was perturbed algebraically to obtain a nonsingular matrix, resulting in the algebraic perturbation method for the Moore-Penrose generalized inverse. In [5], some results on the relations between nonsingular perturbations and generalized inverses of $m\times n$ matrices were obtained, which generalized the results in [4]. For the Drazin generalized inverse, the author has derived an algebraic perturbation method in [6].
In this paper, we will discuss the algebraic perturbation method for generalized inverses with prescribed range and null space, which generalizes the results in [5] and [6].
We remark that the algebraic perturbation methods for generalized inverses are quite useful. The applications can be found in [5] and [8].
In this paper, we use the same terms and notations as in [1].  

The solution of certain Toeplitz linear systems is considered in this paper. This kind of systems are encountered when we solve certain partial differential equations by finite difference techniques and approximate functions using higher order splines. The methods presented here are more efficient than the Cholesky decomposition method and are based on the circulant factorization of the symmetric "banded circulant" matrix, the Woodbury formula and the algebraic perturbation method.

A numerical method for solving the ill-posed Ramm integral equation is presented in this paper. It is found that the method is stable and more accurate. Particularly, when the given data is contaminated by noise, satisfactory results are obtained by using the algorithm of this paper.

In this paper, we mainly discuss the evolution of initial small disturbance in discrete computation of the contour dynamics method. For one class of smooth contour, we prove the stability of evolution of initial small disturbance based on the analysis of the convergence of the contour dynamics method with Euler's explicit method in time. Namely, at terminal time T, the evolving disturbance is going to zero as initial small disturbance goes to zero. The numerical experiment on the stability of contour dynamics has been given in [5,6].

In this paper, the generalized inversion theory and its application in inverse scattering problems are discussed. An iterative solution of joint inversion of parameters describing the earth structures and sources is given and a numerical example is also shown.

The Galerkin methods are studied for two-point boundary value problems and the related one-dimensional parabolic and hyperbolic problems. The boundary value problem considered here is of non-adjoint from and with mixed boundary conditions. The optimal order error estimate in the max-norm is first derived for the boundary problem for the finite element subspace. This result then gives optimal order max-norm error estimates for the continuous and discrete time approximations for the evolution problems described above.

Let $A$, $\tilde{A}\in C^{m\times n}$, rank (A)=rank ($\tilde{A}$)=$n$. Suppose that $A=QH$ and $\tilde{A}=\tilde{Q}\tilde{H}$ are the polar decompositions of $A$ and $\tilde{A}$, respectively. It is proved that $$\|\tilde{Q}-Q\|_F\leq 2\|A^+\|_2\|\tilde{A}-A\|_F$$ and $$\|\tilde{H}-H\|_F\leq \sqrt{2}\|\tilde{A}-A\|_F$$ hold, where $A^+$ is the Moore-Penrose inverse of $A$, and $\| \|_2$ and $\| \|_F$ denote the spectral norm and the Frobenius norm, respectively.

In this paper, we derive the error expansion for finite element method under natural assumption and discuss the superconvergence as a special case of error expansion.

Jacobi algorithm has been developed for the eigenproblem of real symmetric matrices, singular value decomposition of matrices and least squares of the overdetermined system on a parallel computer. In this paper, the parallel schemes and fast algorithm are discussed, and the error analysis and a new bound are presented.