Topic
Block Lanczos algorithm
About: Block Lanczos algorithm is a research topic. Over the lifetime, 88 publications have been published within this topic receiving 3638 citations.
Papers published on a yearly basis
Papers
Book•
1 Jan 1985
TL;DR: This chapter discusses Lanczos Procedures with no Reorthogonalization for Real Symmetric Problems, and an Identification Test, 'Good' versus' spurious' Eigenvalues.
Abstract: 0 Preliminaries: Notation and Definitions.- 0.1 Notation.- 0.2 Special Types of Matrices.- 0.3 Spectral Quantities.- 0.4 Types of Matrix Transformations.- 0.5 Subspaces, Projections, and Ritz Vectors.- 0.6 Miscellaneous Definitions.- 1 Real' symmetric' Problems.- 1.1 Real Symmetric Matrices.- 1.2 Perturbation Theory.- 1.3 Residual Estimates of Errors.- 1.4 Eigenvalue Interlacing and Sturm Sequencing.- 1.5 Hermitian Matrices.- 1.6 Real Symmetric Generalized Eigenvalue Problems.- 1.7 Singular Value Problems.- 1.8 Sparse Matrices.- 1.9 Reorderings and Factorization of Matrices.- 2 Lanczos Procedures, Real Symmetric Problems.- 2.1 Definition, Basic Lanczos Procedure.- 2.2 Basic Lanczos Recursion, Exact Arithmetic.- 2.3 Basic Lanczos Recursion, Finite Precision Arithmetic.- 2.4 Types of Practical Lanczos Procedures.- 2.5 Recent Research on Lanczos Procedures.- 3 Tridiagonal Matrices.- 3.1 Introduction.- 3.2 Adjoint and Eigenvector Formulas.- 3.3 Complex Symmetric or Hermitian Tridiagonal.- 3.4 Eigenvectors, Using Inverse Iteration.- 3.5 Eigenvalues, Using Sturm Sequencing.- 4 Lanczos Procedures with no Reorthogonalization for Real Symmetric Problems.- 4.1 Introduction.- 4.2 An Equivalence, Exact Arithmetic.- 4.3 An Equivalence, Finite Precision Arithmetic.- 4.4 The Lanczos Phenomenon.- 4.5 An Identification Test, 'Good' versus' spurious' Eigenvalues.- 4.6. Example, Tracking Spurious Eigenvalues.- 4.7 Lanczos Procedures, Eigenvalues.- 4.8 Lanczos Procedures, Eigenvectors.- 4.9 Lanczos Procedure, Hermitian, Generalized Symmetric.- 5 Real Rectangular Matrices.- 5.1 Introduction.- 5.2 Relationships With Eigenvalues.- 5.3 Applications.- 5.4 Lanczos Procedure, Singular Values and Vectors.- 6 Nondefective Complex Symmetric Matrices.- 6.1 Introduction.- 6.2 Properties of Complex Symmetric Matrices.- 6.3 Lanczos Procedure, Nondefective Matrices.- 6.4 QL Algorithm, Complex Symmetric Tridiagonal Matrices.- 7 Block Lanczos Procedures, Real Symmetric Matrices.- 7.1 Introduction.- 7.2 Iterative Single-vector, Optimization Interpretation.- 7.3 Iterative Block, Optimization Interpretation.- 7.4 Iterative Block, A Practical Implementation.- 7.5 A Hybrid Lanczos Procedure.- References.- Author and Subject Indices.
1,517 citations
TL;DR: An "industrial strength" algorithm for solving sparse symmetric generalized eigenproblems that can live as a black box eigensolver inside a large applications code and is a novel combination of new techniques and extensions of old techniques.
Abstract: An "industrial strength" algorithm for solving sparse symmetric generalized eigenproblems is described. The algorithm has its foundations in known techniques in solving sparse symmetric eigenproblems, notably the spectral transformation of Ericsson and Ruhe and the block Lanczos algorithm. However, the combination of these two techniques is not trivial; there are many pitfalls awaiting the unwary implementor. The focus of this paper is on identifying those pitfalls and avoiding them, leading to a "bomb-proof" algorithm that can live as a black box eigensolver inside a large applications code. The code that results comprises a robust shift selection strategy and a block Lanczos algorithm that is a novel combination of new techniques and extensions of old techniques.
466 citations
1 Jan 1995
TL;DR: A method for the efficient computation of accurate reduced-order models of large linear circuits is described, which employs a novel block Lanczos algorithm to compute matrix Padé approximations of matrix-valued network transfer functions.
Abstract: A method for the efficient computation of accurate reduced-order models of large linear circuits is described. The method, called MPVL, employs a novel block Lanczos algorithm to compute matrix Pade approximations of matrix-valued network transfer functions. The reduced-order models, computed to the required level of accuracy, are used to speed up the analysis of circuits containing large linear blocks. The linear blocks are replaced by their reduced-order models, and the resulting smaller circuit can be analyzed with general-purpose simulators, with significant savings in simulation time and, practically, no loss of accuracy.
260 citations
1 Jan 1977
TL;DR: In this article, a Block Lanczos method for computing a few of the least or greatest eigenvalues of a sparse symmetric matrix is described, and the results of experiments conducted with this method are presented and discussed.
Abstract: In this paper, we describe a Block Lanczos method for computing a few of the least or greatest eigenvalues of a sparse symmetric matrix. A basic result of Kaniel and Paige describing the rate of convergence of Lanczos' method will be extended to the Block Lanczos method. The results of experiments conducted with this method will be presented and discussed.
234 citations
Posted Content•
TL;DR: In this article, a simple randomized block Krylov subspace method was proposed, which converges quickly for any matrix, independently of singular value gaps, and achieves a low-rank approximation within a factor of 1 + ϵ of the spectral norm error.
Abstract: Since being analyzed by Rokhlin, Szlam, and Tygert and popularized by Halko, Martinsson, and Tropp, randomized Simultaneous Power Iteration has become the method of choice for approximate singular value decomposition. It is more accurate than simpler sketching algorithms, yet still converges quickly for any matrix, independently of singular value gaps. After $\tilde{O}(1/\epsilon)$ iterations, it gives a low-rank approximation within $(1+\epsilon)$ of optimal for spectral norm error.
We give the first provable runtime improvement on Simultaneous Iteration: a simple randomized block Krylov method, closely related to the classic Block Lanczos algorithm, gives the same guarantees in just $\tilde{O}(1/\sqrt{\epsilon})$ iterations and performs substantially better experimentally. Despite their long history, our analysis is the first of a Krylov subspace method that does not depend on singular value gaps, which are unreliable in practice.
Furthermore, while it is a simple accuracy benchmark, even $(1+\epsilon)$ error for spectral norm low-rank approximation does not imply that an algorithm returns high quality principal components, a major issue for data applications. We address this problem for the first time by showing that both Block Krylov Iteration and a minor modification of Simultaneous Iteration give nearly optimal PCA for any matrix. This result further justifies their strength over non-iterative sketching methods.
Finally, we give insight beyond the worst case, justifying why both algorithms can run much faster in practice than predicted. We clarify how simple techniques can take advantage of common matrix properties to significantly improve runtime.
153 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2019 | 1 |
| 2018 | 2 |
| 2017 | 2 |
| 2016 | 6 |
| 2015 | 4 |
| 2014 | 1 |