We construct a modified Bernoulli iteration method for solving the quadratic matrix equation $AX^{2} + BX + C = 0$, where $A$, $B$ and $C$ are square matrices. This method is motivated from the Gauss-Seidel iteration for solving linear systems and the Sherman-Morrison-Woodbury formula for updating matrices. Under suitable conditions, we prove the local linear convergence of the new method. An algorithm is presented to find the solution of the quadratic matrix equation and some numerical results are given to show the feasibility and the effectiveness of the algorithm. In addition, we also describe and analyze the block version of the modified Bernoulli iteration method.

In this paper, the theoretical analysis for the Rayleigh quotient matrix is studied, and some results of the Rayleigh quotient (matrix) of Hermitian matrices are extended to those for arbitrary matrix on one hand. On the other hand, some unitarily invariant norm bounds for singular values are presented for Rayleigh quotient matrices. Our results improve the existing bounds.

We are interested in the numerical solution of the large nonsymmetric shifted linear system, $(A + \alpha I) x = b$, for many different values of the shift $\alpha$ in a wide range. We apply the Saad's flexible preconditioning technique [14] to the solution of the shifted systems. Such flexible preconditioning with a few parameters could probably cover all the shifted systems with the shift in a wide range. Numerical experiments report the effectiveness of our approach on some problems.

Based on the generalized minimal residual (GMRES) principle, Hu and Reichel proposed a minimal residual algorithm for the Sylvester equation. The algorithm requires the solution of a structured least squares problem. They form the normal equations of the least squares problem and then solve it by a direct solver, so it is susceptible to instability. In this paper, by exploiting the special structure of the least squares problem and working on the problem directly, a numerically stable QR decomposition based algorithm is presented for the problem. The new algorithm is more stable than the normal equations algorithm of Hu and Reichel. Numerical experiments are reported to confirm the superior stability of the new algorithm.

Let $S_E$ denote the least-squares symmetric solution set of the matrix equation $AXB=C$, where $A$, $B$ and $C$ are given matrices of suitable size. To find the optimal approximate solution in the set $S_E$ to a given matrix, we give a new feasible method based on the projection theorem, the generalized SVD and the canonical correction decomposition.

In this paper, we present a useful result on the structures of circulant inverse M-matrices. It is shown that if the $n\times n$ nonnegative circulant matrix $A=Circ[c_0, c_1, \cdots, c_{n-1}]$ is not a positive matrix and not equal to $c_0 I$, then $A$ is an inverse M-matrix if and only if there exists a positive integer $k$, which is a proper factor of $n$, such that $c_{jk}>0$ for $j=0, 1,\cdots, [\frac{n-k}{k}]$, the other $c_i$ are zero and $Circ[c_0, c_k, \cdots, c_{n-k}]$ is an inverse M-matrix. The result is then extended to the so-called generalized circulant inverse M-matrices.

In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem: $\min_x \|Ax-b\|_2$, where $A$ is an $m$-by-$n$ ($m \ge n$) rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm $\|\left[AT, \beta b\right] \|_F$ of the data $A$ and $b$, where $T$ is a positive diagonal matrix and $\beta$ is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.

Based on the finite difference discretization of partial differential equations, we propose a kind of semi-implicit $\theta$-schemes of incremental unknowns type for the heat equation with time-dependent coefficients. The stability of the new schemes is carefully studied. Some new types of conditions give better stability when $\theta$ is closed to $1/2$ even if we have variable coefficients.

We propose a quadratically convergent algorithm for computing the invariant subspaces of a Hermitian matrix. Each iteration of the algorithm consists of one matrix-matrix multiplication and one QR decomposition. We present an accurate convergence analysis of the algorithm without using the big $O$ notation. We also propose a general framework based on implicit rational transformations which allows us to make connections with several existing algorithms and to derive classes of extensions to our basic algorithm with faster convergence rates. Several numerical examples are given which compare some aspects of the existing algorithms and the new algorithms.

Recently, Wei in [18] proved that perturbed stiff weighted pseudoinverses and stiff weighted least squares problems are stable, if and only if the original and perturbed coefficient matrices $A$ and $\overline A$  satisfy several row rank preservation conditions. According to these conditions, in this paper we show that in general, ordinary  modified Gram-Schmidt with column pivoting is not numerically stable for solving the stiff weighted least squares problem. We then propose a row block modified Gram-Schmidt algorithm with column pivoting, and show that with appropriately chosen tolerance, this algorithm can correctly determine the numerical ranks of these row partitioned sub-matrices, and the computed QR factor $\overline R$ contains small roundoff error which  is row stable. Several numerical experiments are also provided to compare the results of the ordinary Modified Gram-Schmidt algorithm with column pivoting and the row block Modified Gram-Schmidt algorithm with column pivoting.

In this paper, we discuss an inverse eigenvalue problem for constructing a $2n\times 2n$ Jacobi matrix $T$ such that its $2n$ eigenvalues are given distinct real values and its leading principal submatrix of order $n$ is a given Jacobi matrix. A new sufficient and necessary condition for the solvability of the above problem is given in this paper. Furthermore, we present a new algorithm and give some numerical results.