Pith. sign in

REVIEW 4 major objections 5 minor 35 references

An Integral-Based Framework for Preconditioning $f(A)b$

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper shows that f(A)b can be preconditioned through a Cauchy contour integral by building one Krylov basis from a shifted operator, with a rational closed form and a matrix-free polynomial branch.

desk verdict A coherent preconditioning framework for f(A)b with real algebraic novelty, but the headline LQCD speedup rests on an unvalidated stopping proxy and an omitted eigensolve cost — worth serious refereeing, not desk rejection. read the letter →

arxiv 2608.01003 v1 pith:XUSAX6SD submitted 2026-08-02 math.NA cs.NAhep-lat

Gustavo Ramirez-Hidalgo This is my paper · ORCID
classification math.NAcs.NAhep-lat MSC 65F6065F1065N22
keywords matrixfunctionspolynomialpreconditioningrationalKrylovsubspacemethodsCauchyintegralrepresentationshift-invarianceLR-deflationinversesquareroot
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that the expensive action $f(A)b$ can be preconditioned without leaving the Cauchy integral representation: a single shift-invariance condition lets one generate one Krylov basis for a well-conditioned shifted operator and then recover $f(A)b$ through a mapped contour integral. In the rational branch this produces an exact closed-form extraction, $f_m = \mu(\mu I_N + A)^{-1} V_m (I_m - \tilde H_m)^{-1} f(\hat H_m)(\beta e_1)$, so no numerical contour quadrature is needed, and double reorthogonalization suppresses the phantom poles that small shifts otherwise create. In the polynomial branch the same identity yields a matrix-free method in which a Schur decomposition reduces each contour integrand to triangular back-substitution, a Newton basis keeps high-degree coefficients stable, and LR-deflation protects the quadrature from near-singular low modes. The payoff, if correct, is that ill-conditioned evaluations—stiff diffusion and the Wilson-Dirac operator from lattice QCD—can be computed with far fewer iterations and a much smaller Krylov basis than unpreconditioned Arnoldi.

What carries the argument

The load-bearing mechanism is shift-invariance of Krylov subspaces fused with the Cauchy integral representation of $f(A)b$. For every contour node $z$, the paper asks for a scalar $\eta_z$ and a shifted preconditioner $M_z^{-1}$ satisfying $\eta_z I_N - A M_0^{-1} = (z I_N - A) M_z^{-1}$; then $\mathcal{K}_m(A M_0^{-1}, b) = \mathcal{K}_m((z I_N - A)M_z^{-1}, b)$, so one orthonormal basis $V_m$ and its projected Hessenberg matrix $\tilde H_m$ represent every shifted solve. The rational specialization of this identity factors all $z$-dependence into a small matrix $\hat H_m = \mu(I_m - \tilde H_m)^{-1}\tilde H_m$ and delivers the closed-form extraction $f_m = \mu(\mu I_N + A)^{-1} V_m (I_m -

What would settle it

Compute the polynomial branch on a non-Hermitian matrix with explicitly known spectrum and compare its reported $\varepsilon_m$ with the true relative error $\|f_m - f_{\mathrm{exact}}\|/\|f_{\mathrm{exact}}\|$; if $\varepsilon_m$ reaches the stopping tolerance while the true error remains orders of magnitude larger, the matrix-free stopping criterion is falsified and the reported LQCD convergence is not certified.

Watch

Extended reading notes

Core claim

The central claim is that preconditioning $f(A)b$ reduces to solving, for each contour node $z$, the operator equivalence $\eta_z I_N - A M_0^{-1} = (z I_N - A) M_z^{-1}$; when it holds, the Krylov subspace generated by $A M_0^{-1}$ is identical to the one generated by every shifted operator $(z I_N - A) M_z^{-1}$, so one basis serves the whole integral. With the rational choice $M_0^{-1} = (\mu I_N + A)^{-1}$, the shifted systems collapse to the closed form $f_m = \mu(\mu I_N + A)^{-1} V_m (I_m - \tilde H_m)^{-1} f(\hat H_m)(\beta e_1)$, where $\hat H_m = \mu(I_m - \tilde H_m)^{-1}\tilde H_m$; the paper proves a lower bound on $\mu$ keeps $(I_m - \tilde H_m)$ nonsingular and uses double mod

Load-bearing premise

The stopping guarantee rests on the assumption that the measured lack of change in the base polynomial component inside the small projected space, $\varepsilon_m$, faithfully tracks the true error of the full-space solution; this was validated on one 64×64 Laplace example and then used to stop the LQCD runs without further support.

Editorial extensions

If this is right

  • On the 2D Laplacian on a 512×512 grid with $f(A)=\exp(-cA)$, the polynomial branch at degree $d=5$ reaches the precision floor in about 88 Krylov iterations (about 527 SpMVs), while standard Arnoldi needs 600 iterations and a memory-heavy basis of dimension 600; increasing stiffness is absorbed by tuning the rational shift $\mu$.
  • For the 32768-dimensional twisted-mass Wilson-Dirac operator with $f(A)b = A^{-1/2}b$, a degree-40 harmonic-Ritz polynomial with 400 deflated modes reaches a $10^{-10}$ tolerance in about 15 Krylov iterations (fewer than 600 SpMVs), whereas the unpreconditioned baseline requires a Krylov dimension around 1300.
  • The rational branch's closed form $f_m = \mu(\mu I_N+A)^{-1}V_m(I_m-\tilde H_m)^{-1}f(\hat H_m)(\beta e_1)$ removes contour quadrature entirely, and the lower bound $\mu > \lambda_{\max}\tau$ is the tunable safety margin protecting the extraction from singularity.
  • The matrix-free stopping criterion $\varepsilon_m$ allows the iteration to be halted from small-space coefficient vectors only, with zero additional SpMVs; the paper demonstrates on the Laplacian that it shadows the true error's descent.
  • Because the polynomial branch never algebraically isolates the function from the squared polynomial, it avoids the sign ambiguity that earlier polynomial preconditioners for $A^{-1/2}$ had to manage.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An extension the paper leaves implicit: with many right-hand sides $b_i$, the Krylov basis, Schur factorization, and deflated subspace are constructed once and reused, so the per-vector cost of the polynomial branch should approach the cost of the polynomial assembly loops alone; a timed benchmark varying the number of right-hand sides would quantify this.
  • A testable extension of the same framework is to apply the polynomial branch to other analytic functions with branch cuts, such as $f(A)=\log(A)$ or a different contour for $A^{-1/2}$, on the same twisted-mass operator; the machinery does not depend on the sign function's specific structure.
  • Since the paper's equivalence condition is sufficient rather than necessary, an approximate version of $M_z^{-1}$ may still give an acceptable single-basis approximation; measuring how the contour quadrature degrades as the equivalence is relaxed would map the boundary of the framework's applicability.
  • A natural strengthening would be a formal bound linking the cheap stagnation measure $\varepsilon_m$ to the full-space error; such a bound would certify the LQCD results without relying on the proxy's observed behavior.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes an integral-based framework for preconditioning f(A)b. Starting from the Cauchy integral representation, the author uses shift-invariance to construct a single Krylov subspace for a preconditioned operator and evaluates f(A)b through a mapped integral. Two branches are developed: (i) rational shift-and-invert preconditioning, yielding the closed-form extraction in Eq. (5.7) with Double MGS reorthogonalization to suppress phantom poles; and (ii) polynomial preconditioning, where Schur decomposition and dual-regime scaling stabilize contour quadrature, and LR-deflation removes critical low modes. Numerical experiments on the 2D Laplacian with f=exp and on a coarse Wilson-Dirac operator from lattice QCD with f=A^{-1/2} are presented. The algebraic derivations in Sections 4-6 are internally consistent, and the shift-invariance operator equivalence is a useful unification of existing techniques.

Significance. If the experimental claims are fully established, the paper would be a valuable contribution: it extends shift-invariance ideas from multi-shift linear systems to the continuous contour-integral setting, provides a closed-form extraction that avoids explicit contour quadrature in the rational case, and proposes a polynomial branch with a matrix-free stopping criterion. The derivations in Sections 4-6 are a strength, as is the availability of Octave code on GitHub. However, the headline LQCD efficiency result is conditional on an unvalidated stagnation proxy and an incomplete cost accounting, so the significance of the numerical claims is currently limited.

major comments (4)
  1. [Section 6.7, Eq. (6.19)] The matrix-free stagnation measure epsilon_m is not a bound or reliable estimate of the true error ||f_m - f_exact||. It tracks only the relative change of the base polynomial component in the small Schur space; no theorem connects it to the full-space error. The only validation, Figure 4, is a single small Laplacian case, and the caption contradicts the text on the grid size (32x32 vs 64x64). In Figure 13, the y-axis is 'relative stagnation' for both methods, so the claimed 1e-10 convergence for the LQCD run is not a true-error statement. This is load-bearing for the LQCD speedup claim. The paper should either prove a bound relating epsilon_m to ||f_m - f_exact||, or present reference-error comparisons for the LQCD cases, and clearly label what is plotted.
  2. [Section 7.3.1, Figure 13] The SpMV comparison for POLY-DEF with p=400 omits the cost of computing the 400 deflated eigenmodes of the preconditioned operator. The text states these are computed to 1e-10 relative tolerance, but no cost is included. Since m is only about 15 in this run, the eigensolve likely dominates the total cost. The comparison is fair only if the eigensolve cost is amortized over many right-hand sides, as suggested in the introduction. The paper should state this assumption explicitly and provide either the eigensolve cost or an amortized break-even analysis.
  3. [Sections 6.4 and 7.3.1] The SpMV-based complexity metric ignores the cost of the contour quadrature itself. For the non-deflated LQCD case (p=0), the paper reports 1e6 quadrature nodes; each node requires solving an m x m triangular system (O(m^2) work). With m=47, this is about 2e9 flops, which is not negligible relative to the reported ~1900 SpMVs. The paper should report wall-clock time or total flops, or at least quantify the quadrature cost, before claiming that the p=0 POLY-DEF run is advantageous in overall efficiency.
  4. [Section 5.2, Eqs. (5.8)-(5.9), Figure 7] The theoretical bound in Section 5.2 concerns only eigenvalues: it requires the largest eigenvalue of W to be bounded away from 1.0. The matrix (I_m - \tilde H_m) whose inverse appears in the closed-form extraction (5.7) is non-normal (a Hessenberg matrix), so eigenvalue bounds do not control ||(I_m - \tilde H_m)^{-1}||. The claim that the extraction is shielded from singularity is therefore not fully supported by the asymptotic analysis. The paper should either prove a norm bound, or explicitly state that the empirical conditioning history in Figure 7 is the primary evidence for stability.
minor comments (5)
  1. [Figure 4] The text in Section 7.2.1 states that validation of the stopping criterion uses a 64x64 grid (N=4096), but the Figure 4 caption says '32x32 grid'. Correct the inconsistency.
  2. [Figure 13] The legend describes 'STD-ARN (True Error Proxy)' while the y-axis is 'Relative Stagnation'. Clarify what proxy is used for the baseline and whether it is the true error or a residual estimate.
  3. [Section 6.7] After Eq. (6.19), the text says the criterion demands 'absolutely zero SpMVs'. This is true per iteration, but the Arnoldi process itself still generates the basis; clarify that the saving is relative to reconstructing f_m at each iteration, not relative to the overall algorithm.
  4. [Section 7.3] The operator D is called non-Hermitian, but the matrix actually used, A=DD^dagger, is Hermitian positive definite. Distinguish D from A to avoid confusion.
  5. [Figure 4 legend] The legend references 'Eq. 31', which does not match the paper's equation numbering; it should refer to Eq. (6.19).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained and benchmarked against independent Arnoldi baselines.

full rationale

The paper's central derivation chain is self-contained. The rational closed form (5.7) follows from the Cauchy integral representation (3.1), the preconditioned Arnoldi relation (4.4), and the algebraic factorization (5.6); the identity is verified directly and does not assume the result. The polynomial branch (6.13) is obtained by substituting the projected shifted resolvent into the exact integral and factoring out basis operators; it is coefficient matching, not an input-output loop. The matrix-free stagnation criterion (6.19) is derived from the isometry of the orthonormal Arnoldi basis and tracks one small-space component; it is not a fitted parameter relabeled as a prediction, and Figure 4 explicitly compares it against the true full-space error on one Laplacian example. The LQCD convergence claim relies on this proxy without a theorem linking it to ||f_m - f_exact||; that is a validation/correctness risk, not a circularity. Self-citations [12,20] are used as sources of techniques (polynomial preconditioning, deflation) rather than as the basis for the derived formulas; the load-bearing shift-invariance relation is attributed to external work [1,10] and is re-derived in the framework. No equation reduces to its own input by construction, and no fitted value is renamed as a prediction. Score 0.

Assumptions & free parameters 7 free parameters · 9 assumptions · 0 invented entities

The framework rests on standard complex analysis and Krylov-space facts plus several domain assumptions and hand-tuned parameters. The most consequential free parameters are the shift μ/tolerance τ, the polynomial degree d, the deflation size p, and the contour quadrature node counts; each is chosen empirically per benchmark. The paper introduces no new physical or mathematical entities.

free parameters (7)
  • SAI shift parameter μ = μ = λ_max · τ; e.g., λ_max/10^k for k=3,4,5 in experiments
    Controls spectral compression vs. finite-precision stability in shift-and-invert preconditioner; chosen via tunable tolerance τ (eqs 5.8-5.9).
  • Stabilization tolerance τ = not given as a number; inferred from μ choices
    Tunable parameter balancing compression against conditioning of (I - H_m); no formula for optimal τ.
  • Polynomial degree d = d=5 (Laplacian, Chebyshev); d=40 (LQCD, Harmonic Ritz)
    Empirically tuned based on spectral analysis; higher d improves compression but increases cost.
  • Deflation subspace dimension p = p=400 (LQCD)
    Number of smallest-real eigenmodes extracted; choice is heuristic and its extraction cost is not included in SpMV counts.
  • Quadrature nodes on contour = 10^6 (p=0), 10^5 (p=400) for LQCD
    Chosen to resolve near-singular poles; cost grows with m and node count.
  • Contour Γ (type and parameters) = parabolic for exp, elliptical for inverse square root
    Standard choices from cited literature; shape parameters not reported.
  • Convergence tolerance tol = ~1e-10
    Used to stop iteration; the matrix-free proxy's relationship to true error is not rigorously established.
assumptions (9)
  • standard math Cauchy integral representation: f(A)b = (1/2πi)∫Γ f(z)(zI-A)^{-1}b dz for f analytic on Γ enclosing σ(A)
    Eq 3.1; standard complex analysis.
  • standard math Shift-invariance of Krylov subspaces: K_m(zI-A,b) = K_m(A,b)
    Section 3, eq 3.2 and ref [10]; standard property.
  • domain assumption Operator equivalence condition (4.3): η_z I - A M0^{-1} = (zI-A) M_z^{-1}
    The design condition the framework is built on; for SAI it is solved explicitly (5.2-5.3), for polynomials by coefficient matching (6.4-6.5). Not proven for all preconditioners.
  • domain assumption Assumption 5.1: Re(λ)>0 for all λ∈σ(A)
    Stated in Assumption 5.1; restricts applicability to positive-real-spectrum matrices.
  • standard math FOM projection: the solution of the shifted system in the Krylov subspace is V_m(η_z I-H_m)^{-1}V_m^H b
    Standard Galerkin condition; used in eqs 4.5, 6.13.
  • ad hoc to paper Double Modified Gram-Schmidt reorthogonalization fully removes phantom poles
    Numerically demonstrated for the Laplacian (Fig 7) but not proven; used to justify the stability of the SAI closed form.
  • domain assumption Ritz values from the active Krylov subspace provide good roots for the polynomial preconditioner
    Standard practice [19,21]; harmonic Ritz variant targets low modes.
  • standard math LR-deflation of the preconditioned operator yields exact evaluation of the deflated component via dense projection H_p = L_p^H A R_p
    Section 6.6.2; relies on invariant subspace theory.
  • ad hoc to paper The matrix-free stagnation proxy ϵ_m tracks the true full-space error
    Eq 6.19 computes a cheap relative change of the base polynomial component; only one numerical validation (Fig 4) is provided, and no theorem links it to ||f_m - f_exact||.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An Integral-Based Framework for Preconditioning $f(A)b$." pith.science (2026). https://pith.science/paper/XUSAX6SD

@misc{pith2026260801003,
  author       = {Pith},
  title        = {Pith review of: An Integral-Based Framework for Preconditioning $f(A)b$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XUSAX6SD}},
  note         = {Machine review of arXiv:2608.01003}
}
abstract

The computation of the action of a matrix function on a vector, $f(A)b$, is a major computational bottleneck for large, sparse matrices, particularly when unfavorable spectral distributions cause standard Krylov subspace methods to stagnate. In this work, we propose a unified framework for preconditioning $f(A)b$ based on the Cauchy integral representation of the matrix function. By exploiting shift-invariance properties, we decouple the preconditioner evaluation from the Krylov subspace generation. We develop this framework in two distinct directions. First, for rational shift-and-invert preconditioning, we resolve a fundamental trade-off between optimal spectral compression and finite-precision instability. We achieve this by formulating a closed-form extraction stabilized via Double Modified Gram-Schmidt reorthogonalization, which eliminates the formation of spurious phantom poles. Second, we present a matrix-free polynomial approach. To ensure numerical stability, we isolate the continuous numerical quadrature step using a Schur decomposition of the projected Hessenberg matrix. To further stabilize the integration near contour singularities and accelerate overall convergence, we incorporate an exact LR-deflation scheme targeting the critical low modes of the preconditioned operator. We analyze the asymptotic stability and proximity to singularity of these methods, and present numerical experiments demonstrating their efficiency on the 2D Laplacian with $f=\textrm{exp}$, and a highly ill-conditioned Wilson-Dirac operator from lattice quantum chromodynamics with $f=\textrm{sign}$, although the framework can be in principle used with any $f$ and it is particularly beneficial when applying $f(A)b_{i}$ with many different vectors $b_{i}$.

Figures

Figures reproduced from arXiv: 2608.01003 by the authors.

Figure 1
Figure 1. The 100 smallest eigenvalues of W, with A the 2D Laplace matrix for a 512×512 grid, for which λmin = 19.74 and λmax = 2.1 · 106 . When µ is forced to be relatively large (left side along the horizontal axis), the low modes are severely clustered towards zero. Shrinking µ (toward the right) cures this compression but forces the highest modes of Hem toward 1.0. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Eigenvalue λW 1D… view at source ↗
Figure 2
Figure 2. Distribution of the 100 smallest eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Original spectrum of A (top) and the compressed spectra of the preconditioned [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Comparison of the true full-space relative error (solid black line) and the [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Convergence comparison on the highly stiff 512 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Convergence of SAI-MG vs. STD-ARN for increasingly stiff diffusion param [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Conditioning history of H˜m and (Im − H˜m) corresponding to the runs in [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Eigenvalue spectrum of the coarsest-level non-Hermitian twisted mass operator [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Spectral compression achieved by a standard Chebyshev polynomial for degrees [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Comparison of spectral compression using non-harmonic (left) vs. harmonic [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Zoomed-in view of the 30 smallest-real mapped eigenvalues for the Har [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Mapping of the original elliptical contour Γ (dashed black) to [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Algorithmic convergence on the non-Hermitian twisted mass operator ( [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 30 canonical work pages

  1. [1]

    Analysis of some

    Saad, Yousef , journal=. Analysis of some. 1992 , publisher=

    work page 1992

  2. [2]

    USSR Computational Mathematics and Mathematical Physics , volume=

    Two polynomial methods of calculating functions of symmetric matrices , author=. USSR Computational Mathematics and Mathematical Physics , volume=. 1989 , publisher=

    work page 1989

  3. [3]

    2013 , publisher=

    Matrix computations , author=. 2013 , publisher=

    2013

  4. [4]

    2003 , publisher=

    Iterative methods for sparse linear systems , author=. 2003 , publisher=

    2003

  5. [5]

    Journal of scientific computing , volume=

    Preconditioning the matrix exponential operator with applications , author=. Journal of scientific computing , volume=. 1998 , publisher=

    work page 1998

  6. [6]

    SIAM Journal on Matrix Analysis and Applications , volume=

    Acceleration techniques for approximating the matrix exponential operator , author=. SIAM Journal on Matrix Analysis and Applications , volume=. 2008 , publisher=

    work page 2008

  7. [7]

    Toward efficient polynomial preconditioning for

    Loe, Jennifer A and Morgan, Ronald B , journal=. Toward efficient polynomial preconditioning for. 2022 , publisher=

    work page 2022

  8. [8]

    Preconditioned multishift

    Ahmad, Mian Ilyas and Szyld, Daniel B and Van Gijzen, Martin B , journal=. Preconditioned multishift. 2017 , publisher=

    work page 2017

Show all 35 references
  1. [9]

    An iterative method to compute the sign function of a non-

    Bloch, Jacques and Frommer, Andreas and Lang, Bruno and Wettig, Tilo , journal=. An iterative method to compute the sign function of a non-. 2007 , publisher=

    2007

  2. [10]

    PoS , pages=

    Deflation and polynomial preconditioning in the application of the overlap operator at nonzero chemical potential , author=. PoS , pages=

  3. [11]

    Gonnet, Pedro and Guttel, Stefan and Trefethen, Lloyd N , journal=. Robust. 2013 , publisher=

    2013

  4. [12]

    2008 , publisher=

    Functions of matrices: theory and computation , author=. 2008 , publisher=

    2008

  5. [13]

    Linear Algebra and its applications , volume=

    Computing real square roots of a real matrix , author=. Linear Algebra and its applications , volume=. 1987 , publisher=

    1987

  6. [14]

    Bj. A. Linear algebra and its applications , volume=. 1983 , publisher=

    1983

  7. [15]

    Parabolic and hyperbolic contours for computing the

    Weideman, J and Trefethen, Lloyd , year=. Parabolic and hyperbolic contours for computing the

  8. [16]

    Evaluating matrix functions for exponential integrators via

    Trefethen, LLOYD N , journal=. Evaluating matrix functions for exponential integrators via

  9. [17]

    2009 , publisher=

    Quantum chromodynamics on the lattice: an introductory presentation , author=. 2009 , publisher=

    2009

  10. [18]

    Adaptive aggregation based domain decomposition multigrid for the lattice

    Frommer, Andreas and Kahl, Karsten and Krieg, Stefan and Leder, Bj. Adaptive aggregation based domain decomposition multigrid for the lattice. arXiv preprint arXiv:1303.1377 , year=

    arXiv

  11. [19]

    2019 , school=

    Simulating maximally twisted fermions at the physical point with multigrid methods , author=. 2019 , school=

    2019

  12. [20]

    The application of Leja points to

    Reichel, Lothar , journal=. The application of Leja points to. 1991 , publisher=

    1991

  13. [22]

    Coarsest-level improvements in multigrid for lattice

    Espinoza-Valverde, Jesus and Frommer, Andreas and Ramirez-Hidalgo, Gustavo and Rottmann, Matthias , journal=. Coarsest-level improvements in multigrid for lattice. 2023 , publisher=

    2023

  14. [23]

    arXiv preprint arXiv:2209.14163 , year=

    Krylov subspace recycling for matrix functions , author=. arXiv preprint arXiv:2209.14163 , year=

    arXiv

  15. [24]

    Hochbruck, Marlis and Lubich, Christian , journal=. On. 1997 , publisher=

    1997

  16. [25]

    Davies, Philip I and Higham, Nicholas J , journal=. A. 2003 , publisher=

    2003

  17. [26]

    The loss of orthogonality in the

    Giraud, Luc and Langou, Julien and Rozloznik, Miroslav , journal=. The loss of orthogonality in the. 2005 , publisher=

    2005

  18. [27]

    2003 , publisher=

    Frommer, Andreas , journal=. 2003 , publisher=

    2003

  19. [28]

    G. H. Golub and C. F. Van Loan , Matrix Computations, JHU Press, 2013

    2013

  20. [29]

    Saad , Iterative Methods for Sparse Linear Systems, SIAM, 2003

    Y. Saad , Iterative Methods for Sparse Linear Systems, SIAM, 2003

    2003

  21. [30]

    Castillo and Y

    P. Castillo and Y. Saad , Preconditioning the matrix exp. operator, UMSI, 1997

    1997

  22. [31]

    Simoncini , Acceleration techniques for approximating the matrix exponential operator, SIAM J

    V. Simoncini , Acceleration techniques for approximating the matrix exponential operator, SIAM J. Sci. Comput., 28 (2006)

    2006

  23. [32]

    J. A. Loe and R. B. Morgan , Toward efficient polynomial preconditioning for GMRES, Numer. Linear Algebra Appl., 29 (2022)

    2022

  24. [33]

    M. I. Ahmad, D. B. Szyld, and M. B. Van Gijzen , Preconditioned multishift BiCG, SIAM J. Matrix Anal. Appl., 38 (2017)

    2017

  25. [34]

    Bloch, A

    J. Bloch, A. Frommer, B. Lang, and T. Wettig , An iterative method to compute the sign function of a non-Hermitian matrix, Comput. Phys. Commun., 177 (2007)

    2007

  26. [35]

    [AUTHOR(S)] , [Exact Title], https://pos.sissa.it/466/284

  27. [36]

    [AUTHOR(S)] , [Exact Title], arXiv preprint https://arxiv.org/abs/2401.06684

    arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.