On the Spectral Decomposition of Hermitian Matrices Modified by Low Rank Perturbations with Applications

Arbenz, Peter; Golub, Gene H.

doi:10.1137/0609004

Cited by 68 publications

(50 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Arbenz and coauthors [3,4,5] investigated a generalized D&C approach for computing the eigenpairs of symmetric band matrices. The authors provide many important theoretical results concerning the eigenanalysis of a low rank modification of a diagonal matrix.…”

Section: Previous Workmentioning

confidence: 99%

Toward a High Performance Tile Divide and Conquer Algorithm for the Dense Symmetric Eigenvalue Problem

Haidar¹,

Ltaief²,

Dongarra³

2012

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. Classical solvers for the dense symmetric eigenvalue problem suffer from the first step, which involves a reduction to tridiagonal form that is dominated by the cost of accessing memory during the panel factorization. The solution is to reduce the matrix to a banded form, which then requires the eigenvalues of the banded matrix to be computed. The standard divide and conquer algorithm can be modified for this purpose. The paper combines this insight with tile algorithms that can be scheduled via a dynamic runtime system to multicore architectures. A detailed analysis of performance and accuracy is included. Performance improvements of 14-fold and 4-fold speedups are reported relative to LAPACK and Intel's Math Kernel Library.Key words. divide and conquer, symmetric eigenvalue solver, tile algorithms, dynamic scheduling AMS subject classifications. 15A18, 65F15, 65F18, 65Y05, 65Y20, 68W10 DOI. 10.1137/1108236991. Introduction. The objective of this paper is to introduce a new high performance tile divide and conquer (TD&C) eigenvalue solver for dense symmetric matrices on homogeneous multicore architectures. The necessity of calculating eigenvalues emerges from various computational science disciplines, e.g., in quantum physics [33], chemistry [37], and mechanics [25], as well as in statistics when computing the principal component analysis of the symmetric covariance matrix. As multicore systems continue to gain ground in the high performance computing community, linear algebra algorithms have to be redesigned or new algorithms have to be developed in order to take advantage of the architectural features brought by these processing units.In particular, tile algorithms have recently shown very promising performance results for solving linear systems of equations on multicore architectures using Cholesky, QR/LQ, and LU factorizations available in the PLASMA [34] library and other similar projects like FLAME [44]. The PLASMA concepts consist of splitting the input matrix into square tiles and reorganizing the data within each tile to be contiguous in memory (block data layout) for efficient cache reuse. The whole dataflow execution can then be represented as a directed acyclic graph (DAG) where nodes are tasks operating on tiles, and edges represent dependencies between them. An efficient and lightweight runtime system environment named QUARK [30] (internally

show abstract

Section: Previous Workmentioning

confidence: 99%

Toward a High Performance Tile Divide and Conquer Algorithm for the Dense Symmetric Eigenvalue Problem

Haidar¹,

Ltaief²,

Dongarra³

2012

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…Upper bound estimates for the gap are well studied in the perturbation theory [4,11]. By considering A + V as a low rank perturbation of A, the interlace structure of eigenvalues of A + V and of A is studied by [1,3]. In the second result of this paper, we give a nontrivial lower bound for …”

Section: Introductionmentioning

confidence: 89%

Eigenvalue Estimates Using the Kolmogorov-Sinai Entropy

Shieh

2011

Entropy

View full text Add to dashboard Cite

show abstract

“…Among these are, to name just a few, updating the singular value decomposition of matrices (e.g., [4]) and divide and conquer methods, based on the important paper by J.J.M. Cuppen ([6]), for the singular value decomposition of matrices ( [1]) and for eigenproblems for symmetric matrices (e.g., [6,7,13,16,17,18]). In these applications, a so-called secular equation appears ([14]).…”

Section: Introductionmentioning

confidence: 99%

“…In §2 preliminary results are presented, which are used in §3 to obtain our main results concerning equation (1). In §4 we consider equation (2).…”

Section: Introductionmentioning

confidence: 99%

A unifying convergence analysis of second-order methods for secular equations

Melman

1997

Math. Comp.

View full text Add to dashboard Cite

Abstract. Existing numerical methods of second-order are considered for a so-called secular equation. We give a brief description of the most important of these methods and show that all of them can be interpreted as improvements of Newton's method for an equivalent problem for which Newton's method exhibits convergence from any point in a given interval. This interpretation unifies the convergence analysis of these methods, provides convergence proofs where they were lacking and furnishes ways to construct improved methods. In addition, we show that some of these methods are, in fact, equivalent. A second secular equation is also briefly considered.

show abstract

On the Spectral Decomposition of Hermitian Matrices Modified by Low Rank Perturbations with Applications

Cited by 68 publications

References 18 publications

Toward a High Performance Tile Divide and Conquer Algorithm for the Dense Symmetric Eigenvalue Problem

Toward a High Performance Tile Divide and Conquer Algorithm for the Dense Symmetric Eigenvalue Problem

Eigenvalue Estimates Using the Kolmogorov-Sinai Entropy

A unifying convergence analysis of second-order methods for secular equations

Contact Info

Product

Resources

About