Convergence of integration-based methods for the solution of standard and generalized Hermitian eigenvalue problems Article Swipe

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YOU? · · 2018 · Open Access · · DOI: https://doi.org/10.1553/etna_vol48s183

Recently, methods based on spectral projection and numerical integration have been proposed in the literature as candidates for reliable high-performance eigenvalue solvers. The key ingredients of this type of eigenvalue solver are a Rayleigh-Ritz process and a routine to compute an approximation to the desired eigenspace. The latter computation can be performed by numerical integration of the resolvent. In this article we investigate the progress of the Rayleigh-Ritz process and the achievable quality of the computed eigenpairs for the case that an upper bound for the normwise difference between the currently used subspace and the desired eigenspace is available. Then, such bounds are derived for the Gauß-Legendre rule and the trapezoidal rule.

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Mathematics

Barbara Kingsolver

Mathematical Analysis

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Mathematics Eigenvalues and eigenvectors Hermitian matrix Applied mathematics Solver Subspace topology Convergence (economics) Krylov subspace Inverse iteration Resolvent Projection (relational algebra) Mathematical analysis Algorithm Mathematical optimization Linear system Pure mathematics Quantum mechanics Economic growth Physics Economics

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Type: article
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Landing Page: https://doi.org/10.1553/etna_vol48s183
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DOI: https://doi.org/10.1553/etna_vol48s183

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Title: Convergence of integration-based methods for the solution of standard and generalized Hermitian eigenvalue problems

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Language: en

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Publication year: 2018

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Authors: Lukas Krämer, Bruno Lang

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Landing page: https://doi.org/10.1553/etna_vol48s183

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Concepts: Mathematics, Eigenvalues and eigenvectors, Hermitian matrix, Applied mathematics, Solver, Subspace topology, Convergence (economics), Krylov subspace, Inverse iteration, Resolvent, Projection (relational algebra), Mathematical analysis, Algorithm, Mathematical optimization, Linear system, Pure mathematics, Quantum mechanics, Economic growth, Physics, Economics

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Cited by: 2

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abstract_inverted_index.eigenvalue	20, 29
abstract_inverted_index.literature	14
abstract_inverted_index.projection	5
abstract_inverted_index.resolvent.	57
abstract_inverted_index.computation	48
abstract_inverted_index.eigenspace.	45
abstract_inverted_index.ingredients	24
abstract_inverted_index.integration	8, 54
abstract_inverted_index.investigate	62
abstract_inverted_index.trapezoidal	110
abstract_inverted_index.Rayleigh-Ritz	33, 67
abstract_inverted_index.approximation	41
abstract_inverted_index.Gauß-Legendre	106
abstract_inverted_index.high-performance	19
cited_by_percentile_year.max	94
cited_by_percentile_year.min	89
corresponding_author_ids	https://openalex.org/A5070913387
countries_distinct_count	1
institutions_distinct_count	2
corresponding_institution_ids	https://openalex.org/I167360494
citation_normalized_percentile.value	0.54147915
citation_normalized_percentile.is_in_top_1_percent	False
citation_normalized_percentile.is_in_top_10_percent	False