Perturbation analysis for the symplectic QR factorization

In this paper, the perturbation analysis for the symplectic QR factorization is considered. Some first-order and rigorous normwise perturbation bounds with normwise or componentwise perturbations in the given matrix are presented.     

Accurate solutions of diagonally dominant tridiagonal linear systems

In this paper, we settle Higham’s conjecture for the LU factorization of diagonally dominant tridiagonal matrices. We establish a strong componentwise perturbation bound for the solution of a diagonally dominant tridiagonal linear system, independent of the traditional condition number of the coefficient matrix. We then accurately and efficiently solve the linear system by the GTH-like algorithm without pivoting, as suggested by the perturbation result.     

Stability and Sensitivity of Tridiagonal LU Factorization without Pivoting

In this paper the accuracy of LU factorization of tridiagonal matrices without pivoting is considered. Two types of componentwise condition numbers for the L and U factors of tridiadonal matrices are presented and compared. One type is a condition number with respect to small relative perturbations of each entry of the matrix. The other type is a condition number with respect to small componentwise perturbations of the kind appearing in the backward error analysis of the usual algorithm for the LU factorization. We show that both condition numbers are of similar magnitude. This means that the algorithm is componentwise forward stable, i.e., the forward errors are of similar magnitude to those produced by a componentwise backward stable method. Moreover the presented condition numbers can be computed in O(n) flops, which allows to estimate with low cost the forward errors. AMS subject classification (2000) 65F35, 65F50, 15A12, 15A23, 65G50.Received October 2003. Accepted August 2004. Communicated by Per Christian Hansen.Froilán M. Dopico: This research has been partially supported by the Ministerio de Ciencia y Tecnología of Spain through grants BFM2003-06335-C03-02 (M. I. Bueno) and BFM2000-0008 (F. M. Dopico).     

On the perturbation of the Q‐factor of the QR factorization

This paper gives normwise and componentwise perturbation analyses for the Q‐factor of the QR factorization of the matrix A with full column rank when A suffers from an additive perturbation. Rigorous perturbation bounds are derived on the projections of the perturbation of the Q‐factor in the range of A and its orthogonal complement. These bounds overcome a serious shortcoming of the first‐order perturbation bounds in the literature and can be used safely. From these bounds, identical or equivalent first‐order perturbation bounds in the literature can easily be derived. When A is square and nonsingular, tighter and simpler rigorous perturbation bounds on the perturbation of the Q‐factor are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Improved componentwise verified error bounds for least squares problems and underdetermined linear systems

Recently Miyajima presented algorithms to compute componentwise verified error bounds for the solution of full-rank least squares problems and underdetermined linear systems. In this paper we derive simpler and improved componentwise error bounds which are based on equalities for the error of a given approximate solution. Equalities are not improvable, and the expressions are formulated in a way that direct evaluation yields componentwise and rigorous estimates of good quality. The computed bounds are correct in a mathematical sense covering all sources of errors, in particular rounding errors. Numerical results show a gain in accuracy compared to previous results.     

Some Features of Gaussian Elimination with Rook Pivoting

Rook pivoting is a relatively new pivoting strategy used in Gaussian elimination (GE). It can be as computationally cheap as partial pivoting and as stable as complete pivoting. This paper shows some new attractive features of rook pivoting. We first derive error bounds for the LU factors computed by GE and show rook pivoting usually gives a highly accurate U factor. Then we show accuracy of the computed solution of a linear system by rook pivoting is essentially independent of row scaling of the coefficient matrix. Thus if the matrix is ill-conditioned due to bad row scaling a highly accurate solution can usually be obtained. Finally for a typical inversion method involving the LU factorization we show rook pivoting usually makes both left and right residuals for the computed inverse of a matrix small.     

Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

We derive new perturbation bounds for eigenvalues of Hermitian matrices with block tridiagonal structure. The main message of this paper is that an eigenvalue is insensitive to blockwise perturbation, if it is well-separated from the spectrum of the diagonal blocks nearby the perturbed blocks. Our bound is particularly effective when the matrix is block-diagonally dominant and graded. Our approach is to obtain eigenvalue bounds via bounding eigenvector components, which is based on the observation that an eigenvalue is insensitive to componentwise perturbation if the corresponding eigenvector components are small. We use the same idea to explain two well-known phenomena, one concerning aggressive early deflation used in the symmetric tridiagonal QR algorithm and the other concerning the extremal eigenvalues of Wilkinson matrices.     

A condition analysis of the weighted linear least squares problem using dual norms

In this paper, based on the theory of adjoint operators and dual norms, we define condition numbers for a linear solution function of the weighted linear least squares problem. The explicit expressions of the normwise and componentwise condition numbers derived in this paper can be computed at low cost when the dimension of the linear function is low due to dual operator theory. Moreover, we use the augmented system to perform a componentwise perturbation analysis of the solution and residual of the weighted linear least squares problems. We also propose two efficient condition number estimators. Our numerical experiments demonstrate that our condition numbers give accurate perturbation bounds and can reveal the conditioning of individual components of the solution. Our condition number estimators are accurate as well as efficient.     

Componentwise perturbation analyses for the QR factorization

Summary. This paper gives componentwise perturbation analyses for Q and R in the QR factorization A=QR, , R upper triangular, for a given real $m\times n$ matrix A of rank n. Such specific analyses are important for example when the columns of A are badly scaled. First order perturbation bounds are given for both Q and R. The analyses more accurately reflect the sensitivity of the problem than previous such results. The condition number for R is bounded for a fixed n when the standard column pivoting strategy is used. This strategy also tends to improve the condition of Q, so usually the computed Q and R will both have higher accuracy when we use the standard column pivoting strategy. Practical condition estimators are derived. The assumptions on the form of the perturbation are explained and extended. Weaker rigorous bounds are also given. Received April 11, 1999 / Published online October 16, 2000     

SR分解的乘法扰动界

In this paper, we present the first order perturbation bounds for the SR factorization with respect to left multiplicative perturbation, and the first order and rigorous perturbation bounds for this factorization with respect to right multiplicative perturbation.Moreover, taking the properties of SR factors into consideration, we also provide some refined perturbation bounds.     

Mixed,componentwise condition numbers and small sample statistical condition estimation of Sylvester equations

We present a componentwise perturbation analysis for the continuous‐time Sylvester equations. Componentwise, mixed condition numbers and new perturbation bounds are derived for the matrix equations. The small sample statistical method can also be applied for the condition estimation. These condition numbers and perturbation bounds are tested on numerical examples and compared with the normwise condition number. The numerical examples illustrate that the mixed condition number gives sharper bounds than the normwise one. Copyright © 2011 John Wiley & Sons, Ltd.     

Iterative refinement enhances the stability ofQR factorization methods for solving linear equations

Iterative refinement is a well-known technique for improving the quality of an approximate solution to a linear system. In the traditional usage residuals are computed in extended precision, but more recent work has shown that fixed precision is sufficient to yield benefits for stability. We extend existing results to show that fixed precision iterative refinement renders anarbitrary linear equations solver backward stable in a strong, componentwise sense, under suitable assumptions. Two particular applications involving theQR factorization are discussed in detail: solution of square linear systems and solution of least squares problems. In the former case we show that one step of iterative refinement suffices to produce a small componentwise relative backward error. Our results are weaker for the least squares problem, but again we find that iterative refinement improves a componentwise measure of backward stability. In particular, iterative refinement mitigates the effect of poor row scaling of the coefficient matrix, and so provides an alternative to the use of row interchanges in the HouseholderQR factorization. A further application of the results is described to fast methods for solving Vandermonde-like systems.     

Remark on convergence of algebraic multigrid in the form of matrix decomposition

We introduce the convergence of algebraic multigrid in the form of matrix decomposition. The convergence is proved in block versions of the multi-elimination incomplete LU (BILUM) factorization technique and the approximation of their inverses to preserve sparsity. The convergence theorem can be applied to general interpolation operator. Furthermore, we discuss the error caused by the error matrix.     

HR分解的精化扰动界（英文）

给出了HR分解的分量型和范数型的一阶扰动界.对于范数型,新的精化扰动界至少优于已有结果,特别的,新的关于R因子的扰动界远远优于已有的扰动界.     

New perturbation bounds and condition numbers for the hyperbolic QR factorization

Using the modified matrix-vector equation approach, the technique of Lyapunov majorant function and the Banach fixed point theorem, we obtain some new rigorous perturbation bounds for R factor of the hyperbolic QR factorization under normwise perturbation. These bounds are always tighter than the one given in the literature. Moreover, the optimal first-order perturbation bounds and the normwise condition numbers for the hyperbolic QR factorization are also presented.     

Normwise, mixed and componentwise condition numbers of nonsymmetric algebraic Riccati equations

This paper is devoted to the perturbation analysis for nonsymmetric algebraic Riccati equations. The upper bounds for the normwise, mixed and componentwise condition numbers are presented. The results are illustrated by numerical examples.     

Block LU factorizations of M–matrices

Summary. It is well known that any nonsingular M–matrix admits an LU factorization into M–matrices (with L and U lower and upper triangular respectively) and any singular M–matrix is permutation similar to an M–matrix which admits an LU factorization into M–matrices. Varga and Cai establish necessary and sufficient conditions for a singular M–matrix (without permutation) to allow an LU factorization with L nonsingular. We generalize these results in two directions. First, we find necessary and sufficient conditions for the existence of an LU factorization of a singular M-matrix where L and U are both permitted to be singular. Second, we establish the minimal block structure that a block LU factorization of a singular M–matrix can have when L and U are M–matrices. Received November 21, 1994 / Revised version received August 4, 1997     

Non‐recursive solution of sparse block Hessenberg systems

A double‐phase algorithm, based on the block recursive LU decomposition, has been recently proposed to solve block Hessenberg systems with sparsity properties. In the first phase the information related to the factorization of A and required to solve the system, is computed and stored. The solution of the system is then computed in the second phase. In the present paper the algorithm is modified: the two phases are merged into a one‐phase version having the same computational cost and allowing a saving of storage. Moreover, the corresponding non‐recursive version of the new algorithm is presented, which is especially suitable to solve infinite systems where the coefficient matrix dimension is not a priori fixed and a subsequent size enlargement technique is used. A special version of the algorithm, well suited to deal with block Hessenberg matrices having also a block band structure, is presented. Theoretical asymptotic bounds on the computational costs are proved. They indicate that, under suitable sparsity conditions, the proposed algorithms outperform Gaussian elimination. Numerical experiments have been carried out, showing the effectiveness of the algorithms when the size of the system is of practical interest. Copyright © 2004 John Wiley & Sons, Ltd.     

LU分解的增长因子

In this article, we derive upper bounds of different growth factors for the LU factorization, which are dominated by A11(k)-1A12(k),A21(k)A11(k)-1, where A11(k), A12(k), A21(k), A22(k) are sub-matrices of A. We also derive upper bounds of growth factors for the Cholesky factorization. Numerical examples are presented to verify our findings.     

ON GROWTH FACTORS OF THE LU FACTORIZATION

In this article, we derive upper bounds of different growth factors for the LU factorization, which are dominated by A11(k)-1A12(k),A21(k)A11(k)-1, where A11(k), A12(k), A21(k), A22(k) are sub-matrices of A. We also derive upper bounds of growth factors for the Cholesky factorization. Numerical examples are presented to verify our findings.