A Bi-Orthogonal Structure-Preserving eigensolver for large-scale linear response eigenvalue problem

Yu Li; Zijing Wang; Yong Zhang

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arXiv · 2506.08355

A Bi-Orthogonal Structure-Preserving eigensolver for large-scale linear response eigenvalue problem

Yu Li·Zijing Wang·Yong Zhang

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Abstract

The linear response eigenvalue problem, which arises from many scientific and engineering fields, is quite challenging numerically for large-scale sparse/dense system, especially when it has zero eigenvalues. Based on a direct sum decomposition of biorthogonal invariant subspaces and the minimization principles in the biorthogonal complement, using the structure of generalized nullspace, we propose a Bi-Orthogonal Structure-Preserving subspace iterative solver, which is stable, efficient, and of excellent parallel scalability. The biorthogonality is of essential importance and created by a modified Gram-Schmidt biorthogonalization (MGS-Biorth) algorithm. We naturally deflate out converged eigenvectors by computing the rest eigenpairs in the biorthogonal complementary subspace without introducing any artificial parameters. When the number of requested eigenpairs is large, we propose a moving mechanism to compute them batch by batch such that the projection matrix size is small and independent of the requested eigenpair number. For large-scale problems, one only needs to provide the matrix-vector product, thus waiving explicit matrix storage. The numerical performance is further improved when the matrix-vector product is implemented using parallel computing. Ample numerical examples are provided to demonstrate the stability, efficiency, and parallel scalability.

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Yu Li, Zijing Wang, Yong Zhang. 2025-06-10. A Bi-Orthogonal Structure-Preserving eigensolver for large-scale linear response eigenvalue problem. https://arxiv.org/abs/2506.08355

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