Optimal Backward Perturbation Research Articles

On the Kahan-Parlett-Jiang theorem—A globally optimal backward perturbation error for two-sided invariant subspaces

A crucial question in large non-Hermitian eigenproblems is whether or not any residual bounds can be very useful in practical situations. For this problem, the famous Kahan-Parlett-Jiang theorem Kahan et al. (1982) [15] seems to be the best compromise between generality and sharpness. In essence, it presents an a posteriori error bound on approximate solutions to eigenproblems, which provides a powerful tool to evaluate quality of the computed two-sided invariant subspaces of large-scale non-Hermitian matrices. However, the perturbation error determined by this theorem is only locally optimal rather than globally optimal. In this work, we revisit this problem and derive a globally optimal backward perturbation error for given two-sided approximate invariant subspaces. Our theorem enhances the Kahan-Parlett-Jiang theorem, and numerical experiments demonstrate the effectiveness of our theoretical results.

Analysis of the structured perturbation for the BSCCB linear system

In this paper, based on block style spectral decomposition of the block skew circulant with circulant blocks (BSCCB) matrix, the structure perturbation is discussed, which includes the condition number and relative error of the BSCCB linear system. Then the optimal backward perturbation bound of the BSCCB linear system is analyzed. Simultaneously, the algorithm for the optimal backward perturbation bound is presented. At the end of the paper, a numerical example is provided to verify the effectiveness of the algorithm.

Analysis of the Structured Perturbation for the BCSCB Linear System

Circulant and block circulant type matrices are important tools in solving networked systems. In this paper, based on the style spectral decomposition of the basic circulant matrix and the basic skew circulant matrix, the block style spectral decomposition of the BCSCB matrix is obtained. And then, the structure perturbation is analysed, which includes the condition number and relative error of the BCSCB linear system. Then the optimal backward perturbation bound of the BCSCB linear system is discussed. Simultaneously, the algorithm for the optimal backward perturbation bound is given. Finally, a numerical example is provided to verify the effectiveness of the algorithm.

Backward perturbation analysis and residual-based error bounds for the linear response eigenvalue problem

The numerical solution of a large scale linear response eigenvalue problem is often accomplished by computing a pair of deflating subspaces associated with the interesting part of the spectrum. This paper is concerned with the backward perturbation analysis for a given pair of approximate deflating subspaces or an approximate eigenquadruple. Various optimal backward perturbation bounds are obtained, as well as bounds for approximate eigenvalues computed through the pair of approximate deflating subspaces or approximate eigenquadruple. These results are reminiscent of many existing classical ones for the standard eigenvalue problem.

Optimal Backward Perturbation Analysis for the Block Skew Circulant Linear Systems with Skew Circulant Blocks

We first give the block style spectral decomposition of arbitrary block skew circulant matrix with skew circulant blocks. Secondly, we obtain the singular value of block skew circulant matrix with skew circulant blocks as well. Finally, based on the block style spectral decomposition, we deal with the optimal backward perturbation analysis for the block skew circulant linear system with skew circulant blocks.

On Optimal Backward Perturbation Analysis for the Linear System with Skew Circulant Coefficient Matrix

We first give the style spectral decomposition of a special skew circulant matrix C and then get the style decomposition of arbitrary skew circulant matrix by making use of the Kronecker products between the elements of first row in skew circulant and the special skew circulant C. Besides that, we obtain the singular value of skew circulant matrix as well. Finally, we deal with the optimal backward perturbation analysis for the linear system with skew circulant coefficient matrix on the base of its style spectral decomposition.

Condition numbers and backward perturbation bound for linear matrix equations

This paper deals with the normwise perturbation theory for linear (Hermitian) matrix equations. The definition of condition number for the linear (Hermitian) matrix equations is presented. The lower and upper bounds for the condition number are derived. The estimation for the optimal backward perturbation bound for the Hermitian matrix equations is obtained. Copyright © 2010 John Wiley & Sons, Ltd.

Estimation of optimal backward perturbation bounds for the linear least squares problem

In this paper a method of estimating the optimal backward perturbation bound for the linear least squares problem is presented. In contrast with the optimal bound, which requires a singular value decomposition, this method is better suited for practical use on large problems since it requiresO(mn) operations. The method presented involves the computation of a strict lower bound for the spectral norm and a strict upper bound for the Frobenius norm which gives a gap in which the optimal bounds for the spectral and the Frobenius norm must be. Numerical tests are performed showing that this method produces an efficient estimate of the optimal backward perturbation bound.

On optimal backward perturbation bounds for the linear least squares problem

Consider the linear least squares problem min x ‖b−Ax‖2 whereA is anm×n (m<n) matrix, andb is anm-dimensional vector. Lety be ann-dimensional vector, and let ηls(y) be the optimal backward perturbation bound defined by $$\eta _{LS} (y) = \inf \{ ||F||_F :y is a solution to \mathop {min}\limits_x ||b - (A + F)x||_2 \} .$$ . An explicit expression of ηls(y) (y≠0) has been given in [8]. However, if we define the optimal backward perturbation bounds ηmls(y) by $$\eta _{MLS} (y) = \inf \{ ||F||_F :y is the minimum 2 - norm solution to \mathop {min}\limits_x ||b - (A + F)x||_2 \} ,$$ , it may well be asked: How to derive an explicit expression of ηmls(y)? This note gives an answer. The main result is: Ifb≠0 andy≠0, then ηmls(y)=ηls (y).

Optimal backward perturbation bounds for the linear least-squares problem with multiple right-hand sides

Let A be an m x n matrix, B be an m x r matrix, and (X) over tilde be an approximate solution to the problem of minimizing \\AX-B\\(F). In this note we consider the following open problem: find an explicit expression of the optimal backward perturbation bound eta(F)((X) over tilde) defined by eta(F)((X) over tilde)=min{\\(E,theta F)\\(F):(X) over tilde minimizes \\(A+E)X-(B+F)\\(F)} where theta is a positive number. This problem is solved when (X) over tilde is of full column rank.

Optimal backward perturbation bounds for the linear least squares problem

AbstractLet A be an m × n matrix, b be an m‐vector, and x̃ be a purported solution to the problem of minimizing ‖b — Ax‖2. We consider the following open problem: find the smallest perturbation E of A such that the vector x̃ exactly minimizes ‖b — (A+E)x‖2. This problem is completely solved when E is measured in the Frobenius norm. When using the spectral norm of E, upper and lower bounds are given, and the optimum is found under certain conditions.

Backward perturbation analysis of certain characteristic subspaces

This paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemAx=?x, the generalized eigenvalue problem βAx=?Bx, and the singular value decomposition of a matrixA. This paper also gives residual bounds for certain eigenvalues, generalized eigenvalues and singular values.