A Structure-Preserving LOBPCG Method for Computing Several of the Largest Generalized Singular Triplets
Abstract
We develop a structure-preserving locally optimal block preconditioned conjugate gradient(LOBPCG) method for computing several of the largest positive generalized singular values of a matrix pair (A, B), where B has full column rank, together with the associated generalized singular vectors. The method is derived from the Hermitian-definite Jordan--Wielandt pencil and therefore avoids explicitly forming the normal-equation matrices. By exploiting the block structure and spectral symmetry of this pencil, we construct separate search subspaces for the left and right generalized singular vectors. The resulting Rayleigh--Ritz procedure requires only a singular value decomposition of a small projected cross matrix. We adapt the improved Hetmaniuk--Lehoucq (IHL) trick to construct stable conjugate search directions in the two component spaces. We also establish an explicit correspondence between the coefficient matrices produced by the componentwise and augmented Rayleigh--Ritz procedures. This correspondence permits the componentwise IHL construction to be used in an augmented, structure-preserving LOBPCG formulation without explicitly assembling the augmented basis. Numerical experiments demonstrate the computational advantages of the proposed method over unstructured and Gram-matrix-based LOBPCG implementations.
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BibTeX
@misc{shan2026structurepreserving,
title = {{A Structure-Preserving LOBPCG Method for Computing Several of the Largest Generalized Singular Triplets}},
author = {Xinyu Shan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32406},
eprint = {2609.32406},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32406},
}
RIS
TY - UNPB
TI - A Structure-Preserving LOBPCG Method for Computing Several of the Largest Generalized Singular Triplets
AU - Xinyu Shan
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32406
UR - https://arxiv.org/abs/2609.32406
AB - We develop a structure-preserving locally optimal block preconditioned conjugate gradient(LOBPCG) method for computing several of the largest positive generalized singular values of a matrix pair (A, B), where B has full column rank, together with the associated generalized singular vectors. The method is derived from the Hermitian-definite Jordan--Wielandt pencil and therefore avoids explicitly forming the normal-equation matrices. By exploiting the block structure and spectral symmetry of this pencil, we construct separate search subspaces for the left and right generalized singular vectors. The resulting Rayleigh--Ritz procedure requires only a singular value decomposition of a small projected cross matrix. We adapt the improved Hetmaniuk--Lehoucq (IHL) trick to construct stable conjugate search directions in the two component spaces. We also establish an explicit correspondence between the coefficient matrices produced by the componentwise and augmented Rayleigh--Ritz procedures. This correspondence permits the componentwise IHL construction to be used in an augmented, structure-preserving LOBPCG formulation without explicitly assembling the augmented basis. Numerical experiments demonstrate the computational advantages of the proposed method over unstructured and Gram-matrix-based LOBPCG implementations.
ER -