Relative Perturbation Theory for Diagonally Dominant Matrices

Dailey, Megan; Dopico, Froilán M.; Ye, Qiang

doi:10.1137/130943613

Cited by 17 publications

(12 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result combined with the perturbation results for the D and U factors presented in [14] show that the column diagonal dominance pivoting strategy for row diagonally dominant matrices leads, simultaneously, to LDU factorizations that are guaranteed to be rank-revealing decompositions, i.e., the factors L and U are guaranteed to have small condition numbers, and that always undergo small relative perturbations under the structured perturbations considered in this work. The perturbation results presented in this paper are fundamental to prove in [10] that essentially all interesting magnitudes corresponding to row diagonally dominant matrices undergo small relative variations under small relative perturbations in the diagonally dominant parts and off-diagonal entries and, therefore, that these magnitudes can be computed with high accuracy by algorithms based on rank-revealing decompositions [5,11,15,17,39].…”

Section: Then We Havementioning

confidence: 96%

A New Perturbation Bound for the LDU Factorization of Diagonally Dominant Matrices

Dailey¹,

Dopico²,

Ye³

2014

SIAM J. Matrix Anal. & Appl.

Self Cite

View full text Add to dashboard Cite

Abstract. This work introduces a new perturbation bound for the L factor of the LDU factorization of (row) diagonally dominant matrices computed via the column diagonal dominance pivoting strategy. This strategy yields L and U factors which are always well-conditioned and, so, the LDU factorization is guaranteed to be a rank-revealing decomposition. The new bound together with those for the D and U factors in [F. M. Dopico and P. Koev, Numer. Math., 119 (2011), pp. 337-371] establish that if diagonally dominant matrices are parameterized via their diagonally dominant parts and off-diagonal entries, then tiny relative componentwise perturbations of these parameters produce tiny relative normwise variations of L and U and tiny relative entrywise variations of D when column diagonal dominance pivoting is used. These results will allow us to prove in a follow-up work that such perturbations also lead to strong perturbation bounds for many other problems involving diagonally dominant matrices.Key words. accurate computations, column diagonal dominance pivoting, diagonally dominant matrices, diagonally dominant parts, LDU factorization, rank-revealing decomposition, relative perturbation theory AMS subject classifications. 65F05, 65F15, 15A18, 15A23, 15B99 DOI. 10.1137/13093858X1. Introduction. Perturbation analysis is a classical topic in matrix theory and numerical linear algebra [23,24,35] which still attracts a lot of attention. In recent years, considerable effort has been devoted to deriving sharper perturbation bounds when structured perturbations of important classes of structured matrices are considered (see, as a sample, [1,3,4,7,12,14,18,21,22,26,27,28,29,30,31,33,34,37,38,40]). In this paper, we present a new perturbation bound for the L factor of the LDU factorization of diagonally dominant matrices under a class of componentwise structure-preserving perturbations which are important in numerical computations [14,39,40]. Here, A = LDU is an LDU factorization of A if L is a unit lower triangular matrix, D is a diagonal matrix, and U is a unit upper triangular matrix.Solution of this problem is motivated by several facts. First, apart from its classical applications [20], the LDU factorization has been applied recently to computing accurate rank-revealing decompositions (RRD) [11] of many classes of structure matrices, which are used to perform matrix computations with high relative accuracy [5,11,13,15,17]. In this context, an LDU factorization is an RRD if L and U are well-conditioned. A key point in computing an LDU factorization as an RRD is that the standard partial pivoting strategy does not produce, in general, well-conditioned factors L and U , and that neither complete nor rook pivoting guarantees that L and

show abstract

Section: Then We Havementioning

confidence: 96%

A New Perturbation Bound for the LDU Factorization of Diagonally Dominant Matrices

Dailey¹,

Dopico²,

Ye³

2014

SIAM J. Matrix Anal. & Appl.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We compute λ 1,h by applying inverse iteration 2 to B h and we use the termination criterion B −1 h x k −µ k x k / x k ≤ nu|µ k |. In applying B −1 h , we compare the methods of using the Cholesky factorization of B h and using the accurate LDL T factorization of T N + h 2 D and T N in (18). We denote the computed eigenvalues by µ chol 1 and µ aldu 1 respectively and present the results in Table 2.…”

Section: Biharmonic Operator With the Natural Boundary Conditionmentioning

confidence: 99%

“…Many special classes of matrices have been identified for which the singular values (or eigenvalues) are determined and can be computed to high relative accuracy (i.e. removing κ 2 (A) in (1)); see [4,18,19,21,22,24,25,26,28,33,43] and the recent survey [30] for more details.…”

Section: Introductionmentioning

confidence: 99%

Accurate inverses for computing eigenvalues of extremely ill-conditioned matrices and differential operators

2017

Math. Comp.

Self Cite

View full text Add to dashboard Cite

This paper is concerned with computations of a few smallest eigenvalues (in absolute value) of a large extremely ill-conditioned matrix. It is shown that a few smallest eigenvalues can be accurately computed for a diagonally dominant matrix or a product of diagonally dominant matrices by combining a standard iterative method with the accurate inversion algorithms that have been developed for such matrices. Applications to the finite difference discretization of differential operators are discussed. In particular, a new discretization is derived for the 1-dimensional biharmonic operator that can be written as a product of diagonally dominant matrices. Numerical examples are presented to demonstrate the accuracy achieved by the new algorithms. * 2010 Mathematics Subject Classification: 65F15, 65F35, 65N06, 65N25.

show abstract

“…However, the accuracy is still expected to depend on 1/c i . See [9,Theorem 6.3] for some some related discussions for diagonally dominant matrices. Finally, we discuss an application to discretizations of differential operators, which is a large source of ill-conditioned problems.…”

Section: Application To Eigenvalue Problemsmentioning

confidence: 99%

Preconditioning for Accurate Solutions of Linear Systems and Eigenvalue Problems

Ye¹

2017

Preprint

Self Cite

View full text Add to dashboard Cite

This paper develops the preconditioning technique as a method to address the accuracy issue caused by ill-conditioning. Given a preconditioner M for an ill-conditioned linear system Ax = b, we show that, if the inverse of the preconditioner M −1 can be applied to vectors accurately, then the linear system can be solved accurately. A stability concept called inverse-equivalent accuracy is introduced to describe higher accuracy that is achieved and an error analysis will be presented. As an application, we use the preconditioning approach to accurately compute a few smallest eigenvalues of certain ill-conditioned matrices. Numerical examples are presented to illustrate the error analysis and the performance of the methods.

show abstract

Relative Perturbation Theory for Diagonally Dominant Matrices

Cited by 17 publications

References 41 publications

A New Perturbation Bound for the LDU Factorization of Diagonally Dominant Matrices

A New Perturbation Bound for the LDU Factorization of Diagonally Dominant Matrices

Accurate inverses for computing eigenvalues of extremely ill-conditioned matrices and differential operators

Preconditioning for Accurate Solutions of Linear Systems and Eigenvalue Problems

Contact Info

Product

Resources

About