Paper
Iterative Refinement for a Subset of Eigenvectors of Symmetric Matrices via Matrix Multiplications
Authors
Takeshi Terao, Katsuhisa Ozaki, Toshiyuki Imamura, Takeshi Ogita
Abstract
We develop an iterative refinement method that improves the accuracy of a user-chosen subset of $k$ eigenvectors ($k\ll n$) of an $n\times n$ real symmetric matrix. Using an orthogonal matrix represented in compact WY form, the method expresses the eigenvector error through a correction matrix that can be approximated efficiently from Rayleigh quotients and residuals. Unlike refinement methods for a single eigenpair or for a full eigenbasis, the proposed method refines only the selected $k$ eigenvectors using $\mathcal{O}(nk)$ additional storage, and its dominant work can be organized as matrix--matrix multiplications. Under an eigenvalue separation condition, the refinement converges linearly; we also provide a conservative sufficient condition. Practical variants of the separation condition (e.g., via shifting) enable targeting other extremal parts of the spectrum. For tightly clustered eigenvalues, we discuss limitations and show that preprocessing can restore convergence in a representative sparse example. Numerical experiments on dense test matrices and sparse matrices from the SuiteSparse Matrix Collection illustrate attainable accuracy and problem-dependent convergence.
Metadata
Related papers
Fractal universe and quantum gravity made simple
Fabio Briscese, Gianluca Calcagni • 2026-03-25
POLY-SIM: Polyglot Speaker Identification with Missing Modality Grand Challenge 2026 Evaluation Plan
Marta Moscati, Muhammad Saad Saeed, Marina Zanoni, Mubashir Noman, Rohan Kuma... • 2026-03-25
LensWalk: Agentic Video Understanding by Planning How You See in Videos
Keliang Li, Yansong Li, Hongze Shen, Mengdi Liu, Hong Chang, Shiguang Shan • 2026-03-25
Orientation Reconstruction of Proteins using Coulomb Explosions
Tomas André, Alfredo Bellisario, Nicusor Timneanu, Carl Caleman • 2026-03-25
The role of spatial context and multitask learning in the detection of organic and conventional farming systems based on Sentinel-2 time series
Jan Hemmerling, Marcel Schwieder, Philippe Rufin, Leon-Friedrich Thomas, Mire... • 2026-03-25
Raw Data (Debug)
{
"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2602.23778v1</id>\n <title>Iterative Refinement for a Subset of Eigenvectors of Symmetric Matrices via Matrix Multiplications</title>\n <updated>2026-02-27T08:13:26Z</updated>\n <link href='https://arxiv.org/abs/2602.23778v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2602.23778v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>We develop an iterative refinement method that improves the accuracy of a user-chosen subset of $k$ eigenvectors ($k\\ll n$) of an $n\\times n$ real symmetric matrix. Using an orthogonal matrix represented in compact WY form, the method expresses the eigenvector error through a correction matrix that can be approximated efficiently from Rayleigh quotients and residuals. Unlike refinement methods for a single eigenpair or for a full eigenbasis, the proposed method refines only the selected $k$ eigenvectors using $\\mathcal{O}(nk)$ additional storage, and its dominant work can be organized as matrix--matrix multiplications. Under an eigenvalue separation condition, the refinement converges linearly; we also provide a conservative sufficient condition. Practical variants of the separation condition (e.g., via shifting) enable targeting other extremal parts of the spectrum. For tightly clustered eigenvalues, we discuss limitations and show that preprocessing can restore convergence in a representative sparse example. Numerical experiments on dense test matrices and sparse matrices from the SuiteSparse Matrix Collection illustrate attainable accuracy and problem-dependent convergence.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.NA'/>\n <published>2026-02-27T08:13:26Z</published>\n <arxiv:primary_category term='math.NA'/>\n <author>\n <name>Takeshi Terao</name>\n </author>\n <author>\n <name>Katsuhisa Ozaki</name>\n </author>\n <author>\n <name>Toshiyuki Imamura</name>\n </author>\n <author>\n <name>Takeshi Ogita</name>\n </author>\n </entry>"
}