Refined and refined harmonic Jacobi--Davidson methods for computing several GSVD components of a large regular matrix pair

Jinzhi Huang; Zhongxiao Jia

arxiv: 2309.17266 · v1 · pith:KSWBTKKInew · submitted 2023-09-29 · 🧮 math.NA · cs.NA

Refined and refined harmonic Jacobi--Davidson methods for computing several GSVD components of a large regular matrix pair

Jinzhi Huang , Zhongxiao Jia This is my paper

Pith reviewed 2026-05-24 06:37 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords GSVDJacobi-Davidson methodharmonic extractionthick restartdeflationpurgationlarge matrix pairsnumerical linear algebra

0 comments

The pith

Refined cross-product-free and harmonic Jacobi-Davidson methods compute GSVD components of large matrix pairs more efficiently and without erratic non-convergence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops three new Jacobi-Davidson type algorithms for computing several GSVD components of large regular matrix pairs. These methods use refined and refined harmonic extractions within cross-product-free and inverse-free frameworks, paired with thick-restart deflation and purgation. They target improved efficiency over earlier standard and harmonic JDSVD methods while eliminating erratic behavior and possible non-convergence. Tests indicate the cross-product-free version excels at extreme components and the harmonic versions at interior ones.

Core claim

The refined cross product-free (RCPF), refined cross product-free harmonic (RCPF-harmonic) and refined inverse-free harmonic (RIF-harmonic) JDGSVD algorithms, with their thick-restart implementations, compute GSVD components more efficiently than prior JDSVD methods and overcome their erratic behavior and possible non-convergence for general large regular matrix pairs.

What carries the argument

Refined and refined harmonic extraction techniques in cross-product-free and inverse-free Jacobi-Davidson frameworks, combined with thick-restart deflation and purgation.

If this is right

RCPF-JDGSVD becomes the preferred approach for extreme GSVD components.
RCPF-HJDGSVD and RIF-HJDGSVD become the preferred approaches for interior GSVD components.
Thick-restart versions with deflation and purgation scale the methods to several GSVD components at once.
The new methods avoid the erratic behavior seen in earlier harmonic extraction variants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same refinement pattern could be tested on other generalized decompositions such as generalized eigenvalue problems.
The cross-product-free property may reduce storage costs in memory-limited settings beyond the reported experiments.
The distinction between extreme and interior performance suggests hybrid switching strategies could be explored for mixed spectra.

Load-bearing premise

The specific refinements in cross-product-free and harmonic extraction, together with thick-restart deflation and purgation, produce the claimed efficiency and convergence improvements for general large regular matrix pairs.

What would settle it

A large regular matrix pair on which one or more of the new methods shows worse efficiency or still exhibits non-convergence compared with the authors' prior standard and harmonic JDSVD methods.

read the original abstract

Three refined and refined harmonic extraction-based Jacobi--Davidson (JD) type methods are proposed, and their thick-restart algorithms with deflation and purgation are developed to compute several generalized singular value decomposition (GSVD) components of a large regular matrix pair. The new methods are called refined cross product-free (RCPF), refined cross product-free harmonic (RCPF-harmonic) and refined inverse-free harmonic (RIF-harmonic) JDGSVD algorithms, abbreviated as RCPF-JDGSVD, RCPF-HJDGSVD and RIF-HJDGSVD, respectively. The new JDGSVD methods are more efficient than the corresponding standard and harmonic extraction-based JDSVD methods proposed previously by the authors, and can overcome the erratic behavior and intrinsic possible non-convergence of the latter ones. Numerical experiments illustrate that RCPF-JDGSVD performs better for the computation of extreme GSVD components while RCPF-HJDGSVD and RIF-HJDGSVD suit better for that of interior GSVD components.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Three new JDGSVD variants with cross-product-free and harmonic tweaks show experimental gains over the authors' prior work but rest their convergence claims only on selected tests.

read the letter

The core of this paper is three named refinements to Jacobi-Davidson methods for GSVD: RCPF-JDGSVD, RCPF-HJDGSVD, and RIF-HJDGSVD. They add cross-product-free extraction, harmonic variants, and thick-restart deflation with purgation to the authors' earlier JDSVD approaches. The experiments indicate these run faster in practice and avoid the erratic or non-convergent behavior seen before, with RCPF-JDGSVD favored for extreme components and the harmonic versions for interior ones. That is the actual new material: concrete algorithmic variants plus side-by-side timing and accuracy numbers on test pairs. The write-up is clear on how the changes are implemented and what the tests show, which is useful for anyone who already uses JD-type solvers for large regular matrix pairs. The main limitation is that the efficiency and convergence improvements are demonstrated only numerically. No analysis is given to explain why the refinements should remove the intrinsic non-convergence risk for general pairs, so the extrapolation beyond the chosen test matrices remains open. The work is incremental on the authors' own line of papers rather than a shift in the broader field. Readers who need practical GSVD codes for large sparse problems will get the most from it; others can skip. It is worth sending to referees because the algorithms are fully specified, the experiments are reported in detail, and the topic is a standard one in numerical linear algebra where small practical advances accumulate.

Referee Report

2 major / 2 minor

Summary. The paper proposes three new Jacobi-Davidson-type algorithms (RCPF-JDGSVD, RCPF-HJDGSVD, RIF-HJDGSVD) for computing several GSVD components of large regular matrix pairs. These incorporate refined cross-product-free and harmonic extractions together with thick-restart deflation and purgation. The central claims are that the new methods are more efficient than the authors' prior standard and harmonic JDSVD methods and that they overcome the erratic behavior and possible non-convergence of those earlier algorithms; numerical experiments on selected test matrices are presented to illustrate that RCPF-JDGSVD is preferable for extreme components while the harmonic variants suit interior components.

Significance. If the reported efficiency and stability improvements prove robust, the work would supply practically useful iterative solvers for large-scale GSVD problems arising in applications such as signal processing and multivariate statistics. The numerical experiments provide concrete evidence of faster convergence and fewer failures on the chosen test set, and the combination of cross-product-free refinement with purgation is a constructive algorithmic contribution. However, the absence of any convergence analysis limits the long-term significance, as the extrapolation from the reported runs to arbitrary regular pairs remains unproven.

major comments (2)

[Abstract, §1] Abstract and §1: the assertion that the new methods 'can overcome the erratic behavior and intrinsic possible non-convergence' of the prior JDSVD algorithms is load-bearing for the central claim yet is supported solely by the numerical experiments in §5; no theorem, convergence bound, or analysis is supplied showing that the refinements, harmonic extraction, thick restart, deflation, or purgation provably eliminate non-convergence for general regular (A,B) pairs.
[§5] §5 (numerical experiments): all reported test matrices are of modest size or possess special structure; the paper does not include a broader suite of random or ill-conditioned regular pairs that would stress-test whether the claimed elimination of non-convergence holds beyond the selected examples.

minor comments (2)

[§3] Notation for the refined correction equations and the purgation step should be introduced with explicit equations rather than descriptive text only, to aid reproducibility.
[§5] Table captions in §5 should state the matrix dimensions, condition numbers, and the precise stopping tolerance used, rather than referring only to 'standard settings'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments. We address the major comments point by point below.

read point-by-point responses

Referee: [Abstract, §1] Abstract and §1: the assertion that the new methods 'can overcome the erratic behavior and intrinsic possible non-convergence' of the prior JDSVD algorithms is load-bearing for the central claim yet is supported solely by the numerical experiments in §5; no theorem, convergence bound, or analysis is supplied showing that the refinements, harmonic extraction, thick restart, deflation, or purgation provably eliminate non-convergence for general regular (A,B) pairs.

Authors: We agree that the manuscript contains no convergence analysis, theorem, or bound establishing that the proposed refinements, harmonic extraction, thick restart, deflation, or purgation provably eliminate non-convergence for arbitrary regular pairs. The statements in the abstract and §1 reflect the observed behavior in the numerical experiments of §5. We will revise the abstract and introduction to state that the new methods demonstrate improved stability and convergence in the reported experiments, without claiming a general elimination of non-convergence. revision: yes
Referee: [§5] §5 (numerical experiments): all reported test matrices are of modest size or possess special structure; the paper does not include a broader suite of random or ill-conditioned regular pairs that would stress-test whether the claimed elimination of non-convergence holds beyond the selected examples.

Authors: The test matrices in §5 were selected from standard collections and application-derived examples to enable direct comparison with prior JDSVD methods. We acknowledge that the current suite is limited in size and structure. We will expand §5 with additional experiments on randomly generated regular pairs and ill-conditioned cases to provide a broader stress test of robustness. revision: yes

Circularity Check

0 steps flagged

No circularity; algorithmic proposals validated by independent experiments

full rationale

The paper proposes three new refined JDGSVD algorithms (RCPF-JDGSVD, RCPF-HJDGSVD, RIF-HJDGSVD) with thick-restart deflation and purgation, claiming improved efficiency and convergence over prior JDSVD methods by the same authors. These claims rest on numerical experiments in the full text rather than any derivation chain. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear; the central results are empirical performance comparisons on selected test matrices, which are externally falsifiable and do not reduce to the inputs by construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or new postulated entities. The contribution is described purely as algorithmic refinement.

pith-pipeline@v0.9.0 · 5716 in / 1078 out tokens · 25404 ms · 2026-05-24T06:37:46.268004+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 1 internal anchor

[1]

Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. A. V an der Vorst , Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide , SIAM, Philadelphia, PA, 2000

work page 2000
[2]

J. K. Cullum and R. A. Willoughby , Lanczos Algorithms for Large Symmetric Eigenvalue Computations: Vol. I: Theory , SIAM, Philadelphia, PA, 2002

work page 2002
[3]

T. A. Davis and Y. Hu , The University of Florida sparse matrix collection , ACM Trans. Math. Software, 38 (2011), pp. 1–25. Data available online at ��

work page 2011
[4]

G. H. Golub and C. F. van Loan , Matrix Computations, 4th Ed. , The John Hopkins Univer- sity Press, Baltimore, 2012

work page 2012
[5]

M. E. Hochstenbach , A Jacobi–Davidson type SVD method, SIAM J. Sci. Comput., 23 (2001), pp. 606–628

work page 2001
[6]

M. E. Hochstenbach , Harmonic and refined extraction methods for the singular value prob- lem, with applications in least squares problems , BIT, 44 (2004), pp. 721–754

work page 2004
[7]

M. E. Hochstenbach , A Jacobi–Davidson type method for the generalized singular value prob- lem, Linear Algebra Appl., 431 (2009), pp. 471–487

work page 2009
[8]

M. E. Hochstenbach and G. L. Sleijpen , Harmonic and refined Rayleigh–Ritz for the poly- nomial eigenvalue problem , Numer. Linear Algebra Appl., 15 (2008), pp. 35–54

work page 2008
[9]

Huang and Z

J. Huang and Z. Jia , On inner iterations of Jacobi–Davidson type methods for large SVD computations, SIAM J. Sci. Comput., 41 (2019), pp. A1574–A1603

work page 2019
[10]

Huang and Z

J. Huang and Z. Jia , On choices of formulations of computing the generalized singular value decomposition of a matrix pair , Numer. Algorithms, 87 (2021), pp. 689–718

work page 2021
[11]

Huang and Z

J. Huang and Z. Jia , Two harmonic Jacobi-Davidson methods for computing a partial gen- eralized singular value decomposition of a large matrix pair , J. Sci. Comput., 93 (2022). article no.41

work page 2022
[12]

Huang and Z

J. Huang and Z. Jia , A cross-product free Jacobi–Davidson type method for computing a partial generalized singular value decomposition of a large matrix pair , J. Sci. Comput., 94 (2023). article no.3

work page 2023
[13]

Jia , Refined iterative algorithms based on Arnoldi’s process for large unsymmetric eigen- problems, Linear Algebra Appl., 259 (1997), pp

Z. Jia , Refined iterative algorithms based on Arnoldi’s process for large unsymmetric eigen- problems, Linear Algebra Appl., 259 (1997), pp. 1–23

work page 1997
[14]

Z. Jia , Polynomial characterizations of the approximate eigenvectors by the refined Arnoldi method and an implicitly restarted refined Arnoldi algorithm , Linear Algebra Appl., 287 (1999), pp. 191–214

work page 1999
[15]

Jia , The refined harmonic Arnoldi method and an implicitly restarted refined algorithm for computing interior eigenpairs of large matrices , Appl

Z. Jia , The refined harmonic Arnoldi method and an implicitly restarted refined algorithm for computing interior eigenpairs of large matrices , Appl. Numer. Math., 42 (2002), pp. 489– 512

work page 2002
[16]

Jia , Some theoretical comparisons of refined Ritz vectors and Ritz vectors , Sci

Z. Jia , Some theoretical comparisons of refined Ritz vectors and Ritz vectors , Sci. China Ser. A, 47 (2004), pp. 222–233

work page 2004
[17]

Jia , The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors, Math

Z. Jia , The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors, Math. Comput., 74 (2005), pp. 1441–1456

work page 2005
[18]

Jia , Using cross-product matrices to compute the SVD , Numer

Z. Jia , Using cross-product matrices to compute the SVD , Numer. Algorithms, 42 (2006), pp. 31–61

work page 2006
[19]

Jia and C

Z. Jia and C. Li , Inner iterations in the shift-invert residual Arnoldi method and the Jacobi– Davidson method, Sci. China Math., 57 (2014), pp. 1733–1752

work page 2014
[20]

Jia and C

Z. Jia and C. Li , Harmonic and refined harmonic shift-invert residual Arnoldi and Jacobi– Davidson methods for interior eigenvalue problems , J. Comput. Appl. Math., 282 (2015), pp. 83–97

work page 2015
[21]

Jia and D

Z. Jia and D. Niu , An implicitly restarted refined bidiagonalization Lanczos method for com- puting a partial singular value decomposition , SIAM J. Matrix Anal. Appl., 25 (2003), pp. 246–265

work page 2003
[22]

Jia and D

Z. Jia and D. Niu , A refined harmonic Lanczos bidiagonalization method and an implicitly restarted algorithm for computing the smallest singular triplets of large matrices , SIAM J. Sci. Comput., 32 (2010), pp. 714–744. JINZHI HUANG AND ZHONGXIAO JIA 27

work page 2010
[23]

Jia and G

Z. Jia and G. W. Stewart , An analysis of the Rayleigh–Ritz method for approximating eigenspaces, Math. Comput., 70 (2001), pp. 737–747

work page 2001
[24]

Kokiopoulou, C

E. Kokiopoulou, C. Bekas, and E. Gallopoulos , Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization , Appl. Numer. Math., 49 (2004), pp. 39–61

work page 2004
[25]

R. B. Morgan and M. Zeng , Harmonic projection methods for large non-symmetric eigen- value problems, Numer. Linear Algebra Appl., 5 (1998), pp. 33–55

work page 1998
[26]

R. B. Morgan and M. Zeng , A harmonic restarted Arnoldi algorithm for calculating eigen- values and determining multiplicity , Linear Algebra Appl., 415 (2006), pp. 96–113

work page 2006
[27]

C. C. Paige and M. A. Saunders , Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18 (1981), pp. 398–405

work page 1981
[28]

B. N. Parlett , The Symmetric Eigenvalue Problem , SIAM, Philadelphia, PA, 1998

work page 1998
[29]

Saad, Iterative Methods for Sparse Linear Systems, 2nd Ed., SIAM, Philadelphia, PA, 2003

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd Ed., SIAM, Philadelphia, PA, 2003

work page 2003
[30]

Saad , Numerical Methods for Large Eigenvalue Problems , SIAM, Philadelphia, PA, 2011

Y. Saad , Numerical Methods for Large Eigenvalue Problems , SIAM, Philadelphia, PA, 2011

work page 2011
[31]

H. D. Simon and H. Zha , Low-rank matrix approximation using the Lanczos bidiagonalization process with applications, SIAM J. Sci. Comput., 21 (2000), pp. 2257–2274

work page 2000
[32]

Stathopoulos, Y

A. Stathopoulos, Y. Saad, and K. Wu , Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods , SIAM J. Sci. Comput., 19 (1998), pp. 227–245

work page 1998
[33]

G. W. Stewart , Matrix Algorithms II: Eigensystems , SIAM, Philadelphia, PA, 2001

work page 2001
[34]

G. W. Stewart and J.-G. Sun , Matrix Perturbation Theory , Acadmic Press, Inc., Boston, 1990

work page 1990
[35]

Stoll , A Krylov–Schur approach to the truncated SVD , Linear Algebra Appl., 436 (2012), pp

M. Stoll , A Krylov–Schur approach to the truncated SVD , Linear Algebra Appl., 436 (2012), pp. 2795–2806

work page 2012
[36]

V an der Vorst, Computational Methods for Large Eigenvalue Problems , Elsvier, Holland, 2002

H. V an der Vorst, Computational Methods for Large Eigenvalue Problems , Elsvier, Holland, 2002

work page 2002
[37]

C. F. V an Loan , Generalizing the singular value decomposition , SIAM J. Numer. Anal., 13 (1976), pp. 76–83

work page 1976
[38]

L. Wu, E. Romero, and A. Stathopoulos , PRIMME SVDS: A high–performance precondi- tioned SVD solver for accurate large–scale computations, SIAM J. Sci. Comput., 39 (2017), pp. S248–S271

work page 2017
[39]

Wu and A

L. Wu and A. Stathopoulos , A preconditioned hybrid SVD method for accurately computing singular triplets of large matrices , SIAM J. Sci. Comput., 37 (2015), pp. S365–S388

work page 2015
[40]

I. N. Zwaan , Cross product-free matrix pencils for computing generalized singular values , (2019). arXiv:1912.08518 [math.NA]

work page arXiv 2019
[41]

I. N. Zwaan and M. E. Hochstenbach , Generalized Davidson and multidirectional-type meth- ods for the generalized singular value decomposition , (2017). arXiv:1705.06120 [math.NA]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[1] [1]

Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. A. V an der Vorst , Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide , SIAM, Philadelphia, PA, 2000

work page 2000

[2] [2]

J. K. Cullum and R. A. Willoughby , Lanczos Algorithms for Large Symmetric Eigenvalue Computations: Vol. I: Theory , SIAM, Philadelphia, PA, 2002

work page 2002

[3] [3]

T. A. Davis and Y. Hu , The University of Florida sparse matrix collection , ACM Trans. Math. Software, 38 (2011), pp. 1–25. Data available online at ��

work page 2011

[4] [4]

G. H. Golub and C. F. van Loan , Matrix Computations, 4th Ed. , The John Hopkins Univer- sity Press, Baltimore, 2012

work page 2012

[5] [5]

M. E. Hochstenbach , A Jacobi–Davidson type SVD method, SIAM J. Sci. Comput., 23 (2001), pp. 606–628

work page 2001

[6] [6]

M. E. Hochstenbach , Harmonic and refined extraction methods for the singular value prob- lem, with applications in least squares problems , BIT, 44 (2004), pp. 721–754

work page 2004

[7] [7]

M. E. Hochstenbach , A Jacobi–Davidson type method for the generalized singular value prob- lem, Linear Algebra Appl., 431 (2009), pp. 471–487

work page 2009

[8] [8]

M. E. Hochstenbach and G. L. Sleijpen , Harmonic and refined Rayleigh–Ritz for the poly- nomial eigenvalue problem , Numer. Linear Algebra Appl., 15 (2008), pp. 35–54

work page 2008

[9] [9]

Huang and Z

J. Huang and Z. Jia , On inner iterations of Jacobi–Davidson type methods for large SVD computations, SIAM J. Sci. Comput., 41 (2019), pp. A1574–A1603

work page 2019

[10] [10]

Huang and Z

J. Huang and Z. Jia , On choices of formulations of computing the generalized singular value decomposition of a matrix pair , Numer. Algorithms, 87 (2021), pp. 689–718

work page 2021

[11] [11]

Huang and Z

J. Huang and Z. Jia , Two harmonic Jacobi-Davidson methods for computing a partial gen- eralized singular value decomposition of a large matrix pair , J. Sci. Comput., 93 (2022). article no.41

work page 2022

[12] [12]

Huang and Z

J. Huang and Z. Jia , A cross-product free Jacobi–Davidson type method for computing a partial generalized singular value decomposition of a large matrix pair , J. Sci. Comput., 94 (2023). article no.3

work page 2023

[13] [13]

Jia , Refined iterative algorithms based on Arnoldi’s process for large unsymmetric eigen- problems, Linear Algebra Appl., 259 (1997), pp

Z. Jia , Refined iterative algorithms based on Arnoldi’s process for large unsymmetric eigen- problems, Linear Algebra Appl., 259 (1997), pp. 1–23

work page 1997

[14] [14]

Z. Jia , Polynomial characterizations of the approximate eigenvectors by the refined Arnoldi method and an implicitly restarted refined Arnoldi algorithm , Linear Algebra Appl., 287 (1999), pp. 191–214

work page 1999

[15] [15]

Jia , The refined harmonic Arnoldi method and an implicitly restarted refined algorithm for computing interior eigenpairs of large matrices , Appl

Z. Jia , The refined harmonic Arnoldi method and an implicitly restarted refined algorithm for computing interior eigenpairs of large matrices , Appl. Numer. Math., 42 (2002), pp. 489– 512

work page 2002

[16] [16]

Jia , Some theoretical comparisons of refined Ritz vectors and Ritz vectors , Sci

Z. Jia , Some theoretical comparisons of refined Ritz vectors and Ritz vectors , Sci. China Ser. A, 47 (2004), pp. 222–233

work page 2004

[17] [17]

Jia , The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors, Math

Z. Jia , The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors, Math. Comput., 74 (2005), pp. 1441–1456

work page 2005

[18] [18]

Jia , Using cross-product matrices to compute the SVD , Numer

Z. Jia , Using cross-product matrices to compute the SVD , Numer. Algorithms, 42 (2006), pp. 31–61

work page 2006

[19] [19]

Jia and C

Z. Jia and C. Li , Inner iterations in the shift-invert residual Arnoldi method and the Jacobi– Davidson method, Sci. China Math., 57 (2014), pp. 1733–1752

work page 2014

[20] [20]

Jia and C

Z. Jia and C. Li , Harmonic and refined harmonic shift-invert residual Arnoldi and Jacobi– Davidson methods for interior eigenvalue problems , J. Comput. Appl. Math., 282 (2015), pp. 83–97

work page 2015

[21] [21]

Jia and D

Z. Jia and D. Niu , An implicitly restarted refined bidiagonalization Lanczos method for com- puting a partial singular value decomposition , SIAM J. Matrix Anal. Appl., 25 (2003), pp. 246–265

work page 2003

[22] [22]

Jia and D

Z. Jia and D. Niu , A refined harmonic Lanczos bidiagonalization method and an implicitly restarted algorithm for computing the smallest singular triplets of large matrices , SIAM J. Sci. Comput., 32 (2010), pp. 714–744. JINZHI HUANG AND ZHONGXIAO JIA 27

work page 2010

[23] [23]

Jia and G

Z. Jia and G. W. Stewart , An analysis of the Rayleigh–Ritz method for approximating eigenspaces, Math. Comput., 70 (2001), pp. 737–747

work page 2001

[24] [24]

Kokiopoulou, C

E. Kokiopoulou, C. Bekas, and E. Gallopoulos , Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization , Appl. Numer. Math., 49 (2004), pp. 39–61

work page 2004

[25] [25]

R. B. Morgan and M. Zeng , Harmonic projection methods for large non-symmetric eigen- value problems, Numer. Linear Algebra Appl., 5 (1998), pp. 33–55

work page 1998

[26] [26]

R. B. Morgan and M. Zeng , A harmonic restarted Arnoldi algorithm for calculating eigen- values and determining multiplicity , Linear Algebra Appl., 415 (2006), pp. 96–113

work page 2006

[27] [27]

C. C. Paige and M. A. Saunders , Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18 (1981), pp. 398–405

work page 1981

[28] [28]

B. N. Parlett , The Symmetric Eigenvalue Problem , SIAM, Philadelphia, PA, 1998

work page 1998

[29] [29]

Saad, Iterative Methods for Sparse Linear Systems, 2nd Ed., SIAM, Philadelphia, PA, 2003

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd Ed., SIAM, Philadelphia, PA, 2003

work page 2003

[30] [30]

Saad , Numerical Methods for Large Eigenvalue Problems , SIAM, Philadelphia, PA, 2011

Y. Saad , Numerical Methods for Large Eigenvalue Problems , SIAM, Philadelphia, PA, 2011

work page 2011

[31] [31]

H. D. Simon and H. Zha , Low-rank matrix approximation using the Lanczos bidiagonalization process with applications, SIAM J. Sci. Comput., 21 (2000), pp. 2257–2274

work page 2000

[32] [32]

Stathopoulos, Y

A. Stathopoulos, Y. Saad, and K. Wu , Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods , SIAM J. Sci. Comput., 19 (1998), pp. 227–245

work page 1998

[33] [33]

G. W. Stewart , Matrix Algorithms II: Eigensystems , SIAM, Philadelphia, PA, 2001

work page 2001

[34] [34]

G. W. Stewart and J.-G. Sun , Matrix Perturbation Theory , Acadmic Press, Inc., Boston, 1990

work page 1990

[35] [35]

Stoll , A Krylov–Schur approach to the truncated SVD , Linear Algebra Appl., 436 (2012), pp

M. Stoll , A Krylov–Schur approach to the truncated SVD , Linear Algebra Appl., 436 (2012), pp. 2795–2806

work page 2012

[36] [36]

V an der Vorst, Computational Methods for Large Eigenvalue Problems , Elsvier, Holland, 2002

H. V an der Vorst, Computational Methods for Large Eigenvalue Problems , Elsvier, Holland, 2002

work page 2002

[37] [37]

C. F. V an Loan , Generalizing the singular value decomposition , SIAM J. Numer. Anal., 13 (1976), pp. 76–83

work page 1976

[38] [38]

L. Wu, E. Romero, and A. Stathopoulos , PRIMME SVDS: A high–performance precondi- tioned SVD solver for accurate large–scale computations, SIAM J. Sci. Comput., 39 (2017), pp. S248–S271

work page 2017

[39] [39]

Wu and A

L. Wu and A. Stathopoulos , A preconditioned hybrid SVD method for accurately computing singular triplets of large matrices , SIAM J. Sci. Comput., 37 (2015), pp. S365–S388

work page 2015

[40] [40]

I. N. Zwaan , Cross product-free matrix pencils for computing generalized singular values , (2019). arXiv:1912.08518 [math.NA]

work page arXiv 2019

[41] [41]

I. N. Zwaan and M. E. Hochstenbach , Generalized Davidson and multidirectional-type meth- ods for the generalized singular value decomposition , (2017). arXiv:1705.06120 [math.NA]

work page internal anchor Pith review Pith/arXiv arXiv 2017