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In this paper we consider the simpler GMRES method augmented by approximate eigenvectors for solving nonsymmetric linear systems. We modify the augmented restarted simpler GMRES proposed by Boojhawon and Bhuruth to obtain a simpler GMRES with deflated restarting. Moreover, we also propose a residual-based simpler GMRES with deflated restarting, which is numerically more stable. The main advantage over the augmented version is that the simpler GMRES with deflated restarting requires less matrix-vector products per restart cycle. Some details of implementation are also considered. Numerical experiments show that the residual-based simpler GMRES with deflated restarting is effective.

In this paper, we consider the block-GMRES method with deflated restarting for solving nonsymmetric linear systems with multiple right-hand sides. We modify slightly the restarted block-GMRES method with deflation of eigenvalues proposed by Morgan to obtain a new one. It is shown that the modified method is mathematically equivalent to the Morgan’s original method. However, the error analysis shows the modified version minimizes the numerical errors during a restart and therefore is better suited if the linear systems have to be solved with high precision. Numerical experiments report the effectiveness of the modified method.

The paper presents a convergence analysis of a modified Newton method for solving nonlinear systems of equations. The convergence results show that this method converges cubically in the nonsingular case, and linearly with the rate 3/8 under some sufficient conditions when the Jacobian is singular at the root. The convergence theory is used to analyze the convergence behavior when the modified Newton method is applied to a nonsymmetric algebraic Riccati equation arising in transport theory. Numerical experiment confirms the theoretical results.

This paper is devoted to the perturbation analysis of symmetric algebraic Riccati equations. Based on our perturbation analysis, the upper bounds for the normwise, mixed and componentwise condition numbers are presented. The results are demonstrated by our preliminary numerical experiments.

Computing Google’s PageRank via lumping the Google matrix was recently analyzed in [I.C.F. Ipsen, T.M. Selee, PageRank computation, with special attention to dangling nodes, SIAM J. Matrix Anal. Appl. 29 (2007) 1281–1296]. It was shown that all of the dangling nodes can be lumped into a single node and the PageRank could be obtained by applying the power method to the reduced matrix. Furthermore, the stochastic reduced matrix had the same nonzero eigenvalues as the full Google matrix and the power method applied to the reduced matrix had the same convergence rate as that of the power method applied to the full matrix. Therefore, a large amount of operations could be saved for computing the full PageRank vector.In this note, we show that the reduced matrix obtained by lumping the dangling nodes can be further reduced by lumping a class of nondangling nodes, called weakly nondangling nodes, to another single node, and the further reduced matrix is also stochastic with the same nonzero eigenvalues as the Google matrix.

In this paper, we consider the nonsymmetric algebraic Riccati equation arising in transport theory. An important feature of this equation is that its minimal positive solution can be obtained via computing the minimal positive solution of a vector equation. We propose a class of iterative methods to solve the vector equation. The convergence analysis shows that the sequence of vectors generated by iterative methods with two kinds of specific iterative matrices is monotonically increasing and converges to the minimal positive solution of the vector equation. Numerical experiments show that the new methods outperform the modified simple iterative method and Newton’s method.

In this paper we discuss the convergence behavior of the nonlinear inexact Uzawa algorithm for solving saddle point problems presented in a recent paper by Cao [Z.H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math. 46 (2003) 157–171]. We show that this algorithm converges under a condition weaker than that stated in this paper.

In this paper, we propose a new projection method based on global Arnoldi algorithm for solving large Sylvester matrix equations AX+XB+CD^T=0 and the large generalized Sylvester matrix equations of the form AXB+X+CD^T=0. We show how to extract low-rank approximate solutions to Sylvester matrix equations and generalized Sylvester matrix equations. Some theoretical results are given. Numerical tests report the effectiveness of these methods.

In this paper, a corrected nonlinear Uzawa algorithm for solving stabilized saddle point problems is proposed and the convergence rate is analyzed. The results of numerical experiments are presented when we apply the algorithm to Stokes equations discretized by mixed finite elements.

In this paper we consider the solution of linear systems of large nonsymmetric saddle point problems by two nonlinear iterative methods, which are similar in some respects to the Uzawa-type methods and are called corrected Uzawa methods. Their convergence rates are analyzed. The results of numerical experiments are presented when we apply them to Navier–Stokes equations discretized by mixed finite elements.

In this paper, we first introduce a block second-order Krylov subspace G m 1 ( A , B ; Q 1 ) based on a pair of square matrices A and B and an orthonormal matrix Q 1 . Then we present a block second-order Arnoldi procedure for generating an orthonormal basis of G m 1 ( A , B ; Q 1 ) and a block second-order biorthogonalization procedure for generating biorthonormal basis of G m 1 ( A , B ; Q 1 ) and G m 1 ( A T , B T ; P 1 ) . By applying the projection techniques, we derive two block second-order Krylov subspace methods for solving a large-scale quadratic eigenvalue problem (QEP). These methods are applied to the QEP directly. Hence they preserve essential structures and properties of the QEP. Some theoretical results are given. Numerical experiments report the effectiveness of these methods.

In this paper we propose minimal residual methods augmented with eigenvectors for solving large Sylvester matrix equations AX + XB = C and generalized Sylvester matrix equations AXB + X = C. The subspace, from which the approximate solution is extracted, is the Kronecker product subspace of two augmented Krylov subspaces. Numerical experiments report the effectiveness of these methods.

In this paper we propose a modified simple iterative method for nonsymmetric algebraic Riccati equations arising in transport theory. It is similar to the method proposed by Lu in [L.Z. Lu, Solution form and simple iteration of a nonsymmetric algebraic Riccati equation arising in transport theory, SIAM J. Matrix Anal. Appl. 26 (2005) 679-685]. The theoretical results and numerical experiments show that the modified version is more efficient than its counterpart.

In [J.H. Bramble, I.E. Pasciak, A.T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34 (1997) 1072–1092], a nonlinear Uzawa algorithm for solving generalized saddle point problems iteratively was considered. In [Z. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math. 46 (2003) 157–171], Cao gave another nonlinear Uzawa algorithm in order to accelerate convergence. In this paper, a new nonlinear Uzawa method, which is defined by two nonlinear approximate inverses, is proposed and its convergence result is deduced. At the same time, we analyze convergence results of three nonlinear Uzawa methods. The results of numerical experiments are presented when we apply them to Stokes equations discretized by mixed finite element.

In the paper we propose Galerkin and minimal residual methods for iteratively solving generalized Sylvester equations of the form AXB − X = C. The algorithms use Krylov subspace for which orthogonal basis are generated by the Arnoldi process and reduce the storage space required by using the structure of the matrix. We give some convergence results and present numerical experiments for large problems to show that our methods are efficient.