Topic
Lanczos approximation
About: Lanczos approximation is a research topic. Over the lifetime, 506 publications have been published within this topic receiving 14976 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
23 Sep 1994
TL;DR: PVL, an algorithm for computing the Pad6 approximation of Laplace-domain transfer functions of large linear networks via a Lanczos process, has significantly superior numerical stability and renders unnecessary many of the heuristics that AWE and its derivatives had to employ.
Abstract: In this paper, we introduce PVL, an algorithm for computing the Pad6 approximation of Laplace-domain transfer functions of large linear networks via a Lanczos process. The PVL algorithm has significantly superior numerical stability, while retaining the same efficiency as algorithms that compute the Pad6 approximation directly through moment matching, such as AWE (l), (2) and its derivatives. As a consequence, it produces more accurate and higher-order approximations, and it renders unnecessary many of the heuristics that AWE and its derivatives had to employ. The algorithm also computes an error bound that permits to identify the true poles and zeros of the original network. We present results of numerical experiments with the PVL algorithm for several large examples.
816 citations
Book•
1 Jan 1964
Abstract: 2 Approximation of functions 11 2.1 The least squares method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 The projection (or Galerkin) method . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Example: linear approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Implementation of the least squares method . . . . . . . . . . . . . . . . . . . . . 13 2.5 Perfect approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Ill-conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.7 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.8 Orthogonal basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.9 Numerical computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.10 The interpolation (or collocation) method . . . . . . . . . . . . . . . . . . . . . . 22 2.11 The regression method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.12 Lagrange polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
566 citations
TL;DR: It is shown how a modification called selective orthogonalization stifles the formation of duplicate eigenvectors without increasing the cost of a Lanczos step signifi'cantly.
Abstract: The simple Lanczos process is very effective for finding a few extreme eigenvalues of a large symmetric matrix along with the associated eigenvectors. Unfortunately, the process computes redundant copies of the outermost eigen- vectors and has to be used with some skill. In this paper it is shown how a modification called selective orthogonalization stifles the formation of duplicate eigenvectors without increasing the cost of a Lanczos step signifi'cantly. The degree of linear independence among the Lanczos vectors is controlled without
455 citations
422 citations
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Performance Metrics
| Year | Papers |
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| 2021 | 1 |
| 2020 | 1 |
| 2019 | 1 |
| 2018 | 1 |
| 2017 | 10 |
| 2016 | 14 |