JD-V: A New Variant of Jacobi–Davidson Method for Large Hermitian Eigenproblems

Q: What problem does JD-V target?

JD-V targets computing eigenpairs of large-scale Hermitian matrices, specifically eigenvalues closest to a real target τ, including interior cases near τ.

Q: How does JD-V differ from the standard Jacobi–Davidson correction equation?

JD-V is based on a newly designed correction equation that changes the orthogonal projector used in the correction equation by leveraging approximate eigenvectors associated with clustered eigenvalues near τ.

Q: Why is MINRES expected to be more efficient in JD-V?

The constructed correction equation avoids small-magnitude eigenvalues in its coefficient matrix when λ1 is close to τ, enabling MINRES to converge faster. Numerical experiments reported in the abstract support the theoretical efficiency gains over standard JD.

Jul 16, 2026

Upload by Riley

A new JD-V variant of the Jacobi–Davidson framework is presented for Hermitian eigenvalue problems, built on a newly designed correction equation. The solution of the revised correction equation is proven nearly as effective as the standard equation for subspace expansion. A rigorous convergence analysis of MINRES for these correction equations shows that JD-V inner iterations are substantially more efficient when highly clustered eigenvalues around a target are considered. A thick-restart JD-V with deflation and purgation computes multiple eigenpairs for large Hermitian matrices.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages20

Reading Time50 min

Introduction

Problem setup and eigenpairs near a target

Jacobi–Davidson framework and role of correction equations

Inexact inner solves and stopping criteria

MINRES behavior and clustered eigenvalues

arXiv :2607 . 10635v1 [math .NA] 12 Jul 2026 JD-V: A NEW VARIANT OF JACOBI–DAVIDSON METHOD FOR LARGE HERMITIAN EIGENPROBLEMS∗ JINZHI HUANG† Abstract. A novel variant of the Jacobi-Davidson (JD) type method for Hermitian eigenvalue problems, designated as JD-V, is proposed based on a newly designed correction equation, whose solution is shown to be nearly as eﬀective as that of the standard correction equation for subspace expansion. Rigorous convergence analysis of MINRES for solving these equations reveals that the inner iterations of JD-V are signiﬁcantly more eﬃcient than those of the standard JD method when highly clustered eigenvalues are of interest. A thick-restart JD-V algorithm with deﬂation and purgation is developed to compute several eigenpairs of a a large-scale Hermitian matrix. Numerical experiments conﬁrm the theoretical results and demonstrate the considerable superiority of JD-V over standard JD in overall eﬃciency. Key words. Hermitian eigenvalue problem, Jacobi-Davidson method, correction equation, MINRES method, inner iteration, convergence AMS subject classiﬁcations. 65F15, 15A18, 65F10, 65F25 1. Introduction. Let A ∈ Cn ×n be a large-scale, possibly sparse, and Hermitian matrix, i.e. , A∗ = A, where the superscript ∗ denotes the conjugate transpose. Consider its eigenvalue decomposition: (1.1) A = XΛX ∗ , where X = [x1 ,..., xn] ∈ Cn ×n is unitary, and Λ = diag{λ1 ,...,λn } ∈ Rn ×n . Such (λi , xi ) are called eigenpairs of A, with λi ∈ R the eigenvalues, and xi ∈ Cn theeigenvectors, i = 1 , . . . , n. Given a target τ ∈ R, assume that (1.2) |λ1 − τ| < ··· < |λℓ − τ| < |λℓ+1 − τ| ≤ · · · ≤ |λn − τ| . We aim to compute the ℓ eigenpairs (λ1 , x1 ) ,...,(λℓ , xℓ ) with the eigenvalues closest to τ . In particular, if τ lies within the spectrum (λmin ,λmax ) of A and is close to neither the smallest eigenvalue λmin nor the largest one λmax , then (λi , xi ) are classiﬁed as interior eigenpairs; otherwise, they are extreme, i.e., smallest or largest, ones. Several methods [1, 16 , 23] are available for solving problems of this kind. Among them, the Jacobi-Davidson (JD) method [17, 18 , 19] is a popular one. This is an inner-outer iterative method: the outer iterations compute the desired eigenpairs from the search subspace using one of the standard, harmonic, reﬁned, and reﬁned harmonic extraction approaches [11, 12 , 23], while the inner iterations employ, e.g., a Krylov subspace solver [2 , 6], to approximately solve an n-by-n Hermitian (typically indeﬁnite) linear system, called correction equation, to expand the search subspace. The inner iterations dominate the overall eﬃciency of the JD method. Fortunately, the correction equations do not need to be solved with high accuracy. Notay [15] analyzes the simpliﬁed JD method (JDCG) for Hermitian eigenvalue problems, in which the correction equations are solved by the preconditioned conjugate gradient (PCG) method [7], and the solution is directly added to the current approximate ∗ Submitted to the editors DATE. Funding: This work is supported by the Youth Fund of the National Science Foundation of China (Grant No. 12301485) and the Jiangsu Province Youth Science and Technology Talent Support Program (Grant No. JSTJ-2025-828) †School of Mathematical Sciences, Soochow University, 215006 Suzhou, China (jzhuang21@suda. [edu.cn](edu.cn)). 2 JINZHI HUANG eigenvector to form a new one. He monitors the outer convergence via the reduction of inner residual norms designs stopping criteria for the inner iterations. Hochstenbach and Notay [8] extend these results to Hermitian generalized eigenvalue problems. Stathopoulos and McCombs [22] adapt the stopping criteria from [8 , 15] to the inner iterations of their general JD methods with subspace acceleration for Hermitian eigenvalue problems. For general JD methods, Jia and Li [13 , 14] prove that solving the correction equations to low or modest accuracy with the relative error of appro

FAQs

What problem does JD-V target?

JD-V targets computing eigenpairs of large-scale Hermitian matrices, specifically eigenvalues closest to a real target τ, including interior cases near τ.

How does JD-V differ from the standard Jacobi–Davidson correction equation?

JD-V is based on a newly designed correction equation that changes the orthogonal projector used in the correction equation by leveraging approximate eigenvectors associated with clustered eigenvalues near τ.

Why is MINRES expected to be more efficient in JD-V?

The constructed correction equation avoids small-magnitude eigenvalues in its coefficient matrix when λ1 is close to τ, enabling MINRES to converge faster. Numerical experiments reported in the abstract support the theoretical efficiency gains over standard JD.

Jul 16, 2026

Upload by Riley