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Rational Krylov for Stieltjes matrix functions: convergence and pole selection

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Pith's one-line read Rational Krylov poles chosen from the Zolotarev problem make the action of Laplace-Stieltjes and Cauchy-Stieltjes matrix functions converge exponentially, with rates expressed purely in terms of the spectral interval and with analogous…

desk verdict Solid rational Krylov pole-selection paper with explicit bounds for Laplace- and Cauchy-Stieltjes functions; the math is credible, and the main caveats are experimental documentation and an unproven sharp constant that does not affect the rates. read the letter →

arxiv 1908.02032 v4 pith:S6HOYCVZ submitted 2019-08-06 math.NA cs.NA

Stefano Massei , Leonardo Robol This is my paper · ORCID
classification math.NAcs.NA MSC 65F6065F3041A20
keywords rationalKrylovmethodsStieltjesmatrixfunctionsLaplace-StieltjesCauchy-StieltjesZolotarevpolespoleselectionKroneckerstructurelow-rankapproximation
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

This paper proves that rational Krylov methods with carefully chosen Zolotarev poles evaluate Stieltjes matrix functions on vectors with exponential convergence, where the rate is determined only by the matrix spectrum. For Laplace-Stieltjes functions, such as matrix exponentials and completely monotonic functions, the error after $\ell$ steps is $O(\rho_{[a,b]}^{\ell/2})$; for Cauchy-Stieltjes functions, such as inverse fractional powers, it is $O(\rho_{[a,4b]}^{\ell})$, with $\rho_{[\alpha,\beta]}=\exp(-\pi^2/\log(4\beta/\alpha))$. The same analysis extends to Kronecker-sum matrices $M=I\otimes A-B^T\otimes I$ acting on vectorized low-rank matrices, where tensorized rational Krylov subspaces preserve structure and the error rates become $O(\rho_{[a,b]}^{\ell/2})$ and $O(\rho_{[a,2b]}^{\ell})$. This matters because fractional diffusion solvers, exponential integrators, and Sylvester-type matrix equations are precisely the settings where these functions and structures arise.

What carries the argument

The engine of the argument is the reduction of the projection error to a rational approximation problem. For the resolvent kernel $(tI+A)^{-1}$, the skeleton approximation yields the explicit error formula $1-(t+\lambda)f_{\mathrm{skel}}(t,\lambda)=r(\lambda)/r(-t)$, and for the exponential kernel the Laplace transform converts the same skeleton error into a bound involving the inverse Laplace transform of $(1/s)p(s)/p(-s)$. The optimal rational functions solving the third Zolotarev problem, that is, the poles of $p(z)/p(-z)$ that minimize $\max_{[a,b]}|r(z)|/\min_{[a,b]}|r(-z)|$, provide the pole sets; Möbius transformations map the relevant intervals $[-\infty,0]\cup[a,b]$ onto the symmetric Zolotarev configuration. In the Kronecker case the machinery is the tensorized subspace $U\otimes V$, where the projected matrix retains the same Kronecker-sum form, and the Cauchy-Stieltjes analysis passes through uniformly accurate solution of parameter-dependent Sylvester equations $(tI+A)X_t-X_tB=F$ using rational Galerkin residual bounds.

What would settle it

Pick the 1D Laplacian $A$ of size $10^5$ with known spectrum in $[a,b]\approx[1/n,4]$, set $f(z)=e^{-z}$, and run rational Krylov with the paper's Zolotarev poles for $\ell=5,10,15,20$. The log-error should fall with slope at least $-\pi^2/(2\log(4b/a))$; a markedly flatter slope, or failure of the error to decay once past the constant regime, would falsify the bound.

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Extended reading notes

Core claim

For Hermitian positive definite $A$ with spectrum in $[a,b]$, the rational Krylov iterate with Zolotarev poles satisfies $\|f(A)v-x_\ell\|_2\le C\rho_{[a,b]}^{\ell/2}$ when $f$ is Laplace-Stieltjes and $\|f(A)v-x_\ell\|_2\le C\rho_{[a,4b]}^{\ell}$ when $f$ is Cauchy-Stieltjes. For $M=I\otimes A-B^T\otimes I$ with $A,-B$ Hermitian positive definite and $v=\mathrm{vec}(F)$ for low-rank $F$, tensorized rational Krylov spaces give $\|X-X_\ell\|_2\le C\rho_{[a,b]}^{\ell/2}$ for Laplace-Stieltjes functions and $\|X-X_\ell\|_2\le C\rho_{[a,2b]}^{\ell}$ for Cauchy-Stieltjes functions. The constants depend only on $f(0+)$ or $f(2a)$, the spectral ratio $\kappa=b/a$, and a mild logarithmic factor $\gamma_{\ell,\kappa}$, but not on the density defining $f$. These are explicit, parameter-free convergence rates, and the paper also derives matching singular-value decay bounds for the low-rank solution $X$.

Load-bearing premise

The convergence rates assume $A$ and $-B$ are symmetric positive definite with spectra exactly contained in a known interval $[a,b]$, and the poles are built from those endpoints; if the interval endpoints are inaccurate, the predicted rates no longer apply.

Editorial extensions

If this is right

  • Ill-conditioned arguments are where the gain appears: the rate depends on the spectral ratio only through $\log(4b/a)$, so a widening spectrum costs little extra.
  • Nested sequences of poles with the same asymptotic rate can be generated by equidistributed sequences, so the method can be run adaptively without fixing $\ell$ in advance.
  • For Laplace-Stieltjes functions and rank-$k$ right-hand sides, the solution $X$ of $\mathrm{vec}(X)=f(I\otimes A-B^T\otimes I)\mathrm{vec}(F)$ has singular values bounded by $O(\rho_{[a,b]}^{\ell/2})$ after the $k\ell$-th, justifying low-rank compression.
  • For Cauchy-Stieltjes functions of Kronecker-sum matrices, the analogous singular-value decay $O(\rho_{[a,2b]}^{\ell})$ holds without the condition-number factor $(1+\kappa)$.
  • The projected function evaluation can be done by diagonalizing $A_U$ and $B_V$, costing $O(k^3)$ plus $k^2$ scalar function evaluations rather than $O(k^6)$.

Reading between the lines

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

  • A natural stress test is the non-normal case the authors leave open: if the field of values lies in the right half-plane, the same Zolotarev intervals should still yield some convergence, but the sharp rates and the diagonalization trick for the projected problem would need separate treatment.
  • The bounds can be read as a stopping-criterion blueprint: since the error after $\ell$ steps is dominated by an explicit power of $\rho$, one can choose $\ell$ in advance from the spectrum and target tolerance, and the nested pole sequences let the same choice be revised online.
  • The singular-value decay theorems imply that solutions of parameter-dependent Sylvester equations inherit low-rank approximability uniformly in the shift parameter, which is exactly what makes low-rank time-stepping for fractional diffusion feasible on tensor grids.
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Editorial analysis

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

Referee Report

0 major / 4 minor

Summary. The manuscript develops pole selection strategies for rational Krylov subspace methods that approximate x = f(M)v when f is a Laplace-Stieltjes or Cauchy-Stieltjes function and M is positive definite. For a Hermitian positive definite A with spectrum contained in [a,b], the authors prove explicit convergence bounds: Laplace-Stieltjes functions converge like O(rho^{ell/2}_{[a,b]}) using Zolotarev poles (Corollary 3.16) and Cauchy-Stieltjes functions converge like O(rho^ell_{[a,4b]}) using a Mobius-transformed Zolotarev set (Corollary 3.18). They also construct nested pole sequences via equidistributed sequences (Section 3.5). For Kronecker-structured matrices M = I otimes A - B^T otimes I with low-rank right-hand sides, they analyze tensorized rational Krylov subspaces and prove analogous rates: O(rho^{ell/2}_{[a,b]}) for Laplace-Stieltjes functions (Corollary 4.1) and O(rho^ell_{[a,2b]}) for Cauchy-Stieltjes functions (Corollary 4.8). Finally, they derive singular-value decay bounds for the solution matrix X, extending known low-rank approximability results for Sylvester equations (Theorems 4.10 and 4.11). Numerical experiments on finite-difference matrices, including fractional diffusion and exponential integrator examples, support the predicted rates.

Significance. If the results are correct, this is a valuable contribution to the rational Krylov literature. The paper provides explicit, non-fitted convergence rates and concrete pole sets for two important classes of matrix functions, and it extends the analysis to Kronecker-structured problems with low-rank right-hand sides, yielding both practical tensorized rational Krylov algorithms and new low-rank approximability bounds. The proofs are built on standard machinery -- Zolotarev theory, skeleton approximation, inverse Laplace transform estimates, and ADI error analysis -- and the constants are explicit rather than fitted to numerical data. The authors are appropriately candid about the limits of the analysis: Remark 3.7 acknowledges that the constant gamma_{ell,kappa} in Lemma 3.5 is likely not sharp, and Section 5 states that the non-normal case is left open. These caveats do not undermine the main rate claims, because gamma_{ell,kappa} only affects constants and not the exponential factor rho^{ell/2}.

minor comments (4)
  1. [Section 3.2, Lemma 3.5 and Remark 3.7] The constant gamma_{ell,kappa} is presented as an upper bound, but Remark 3.7 reports numerical evidence suggesting that the true constant is 1 for all admissible poles. Since Corollaries 3.16, 4.1, and Theorem 4.10 all inherit this constant, the authors should state more explicitly that (i) the exponential rate in Table 1.1 is unaffected by gamma, and (ii) the logarithmic factor in gamma is subexponential and therefore does not change the O(rho^{ell/2}) classification when ell is treated as the growing parameter.
  2. [Section 4.3, proof of Corollary 4.8] The phrase 'By setting I_A = I, I_B = -I in the statement of Theorem 4.6' is confusing: I_A and I_B are intervals, not identity matrices. Please rewrite this sentence using the normalized intervals, e.g. I_A = [\tilde{a},1] and I_B = [-1,-\tilde{a}], so that the Mobius transformation argument is unambiguous.
  3. [Section 3.6 and Section 5] The numerical experiments on tridiag(-1,2,-1) use the exact spectral interval, so they do not probe how the method behaves when the available bounds on [a,b] are over- or underestimated. Since all theorems are stated in terms of the interval used to construct the poles, adding a sentence in Section 5 on practical estimation of [a,b] (e.g., by Lanczos or Ritz values) would help the reader apply the bounds in realistic settings.
  4. [Throughout] There are a few minor typos: the acknowledgment reads 'The author wish' instead of 'The authors wish', and the introduction contains a duplicated 'of of' in the phrase 'the solution of of Sylvester equations' (Section 1.1, item 5). These should be corrected in a final pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence bounds are derived from external rational-approximation results and the Stieltjes integral representation, with no fitted constants or load-bearing self-citations.

full rationale

The paper's central claims are upper bounds on rational Krylov errors for Laplace-Stieltjes and Cauchy-Stieltjes matrix functions. The derivation chain is: (i) the Stieltjes integral representation (1.1) expresses f(A)v as an integral of matrix exponentials or resolvents; (ii) Theorem 3.4 bounds resolvent projection errors via the skeleton approximation from Oseledets [26] and Druskin et al. [14]; (iii) Theorem 3.6 bounds exponential projection errors via the Laplace transform and the inverse-Laplace estimate of Lemma 3.5, whose proof is included in Appendix B; (iv) the resulting integrals are bounded using the Zolotarev rational approximation estimate from Beckermann/Beckermann-Townsend [3,5] and Mobius invariance. The pole sets are chosen to minimize or nearly minimize explicit rational approximation problems, and the error bounds are parameter-free upper bounds: no constant is fitted to the reported numerical errors, and the rates are not used to define the poles. The numerical experiments compare measured errors against the theoretical bounds, but the bounds are not calibrated to the data. The self-citations [23-25] appear only in motivational paragraphs about low-rank solvers and are not used in the convergence theorems. Remark 3.7 openly notes that the constant gamma_{ell,kappa} in Lemma 3.5 may not be sharp and that experiments suggest gamma = 1; this is a potential looseness in the bound, not a circularity, because the bound remains a proven upper bound independent of the numerics. The skeptical concern about the saddle-point constant is therefore a sharpness question rather than a reduction of the prediction to its inputs. Overall, the derivation is self-contained given the cited external mathematical results, and no prediction is equivalent to an input by construction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to the reported experiments; the pole sets come from Zolotarev theory and equidistributed sequences, and the constants gamma_{ell,kappa} are proven in Appendix B. The only user-chosen numbers are the EDS seed and the shift eta, neither of which changes the asymptotic rates.

free parameters (2)
  • Equidistributed sequence seed zeta = 1/sqrt(2)
    Chosen by hand for the nested pole sequence in Section 3.5; any irrational zeta works asymptotically, so this is not fitted to data and does not change the convergence rate.
  • Shift parameter eta for Laplace-Stieltjes functions with f(0+)=infinity = user-chosen in (0,a)
    Introduced in Remark 3.17 to apply Corollary 3.16 to functions such as z^{-alpha}; the rate rho^{ell/2}_{[a,b]} is preserved as eta tends to a, but the constant f(eta) depends on the user's choice.
assumptions (6)
  • standard math Zolotarev's solution of the third Zolotarev problem for [-b,-a] union [a,b] with the estimate rho_{[a,b]} = exp(-pi^2/log(4b/a))
    Used to bound theta_ell in Corollaries 3.14, 3.15, 4.8 and the singular value bounds. The version used here is the logarithmic form, slightly suboptimal to the Groetzsch form as noted in Remark 3.11.
  • standard math Bernstein's theorem identifying Laplace-Stieltjes functions with completely monotonic functions
    Used in Section 2.1 to characterize the function class.
  • standard math The quasi-optimality bound (3.1) for Galerkin projection onto rational Krylov subspaces (Guettel [18])
    Links the projection error to a rational approximation problem; this is the engine of all the error bounds.
  • standard math The skeleton approximation error formula 1-(t+lambda)f_skel(t,lambda)=r(lambda)/r(-t) (Oseledets [26])
    Used in the proof of Theorem 3.4 to reduce the resolvent error to a rational approximation problem.
  • standard math The inverse Laplace representation (3.3) and the ADI error representation for Sylvester equations (Beckermann [3], Beckermann-Townsend [5])
    Used in the proofs of Theorem 3.6 and Theorem 4.11.
  • domain assumption A (and -B) are Hermitian positive definite with spectra contained in [a,b]
    The analysis and pole selection depend on the spectral interval; extension to non-normal matrices is left open in Section 5.

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Pith. "Pith review of Rational Krylov for Stieltjes matrix functions: convergence and pole selection." pith.science (2026). https://pith.science/paper/S6HOYCVZ

@misc{pith2026190802032,
  author       = {Pith},
  title        = {Pith review of: Rational Krylov for Stieltjes matrix functions: convergence and pole selection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6HOYCVZ}},
  note         = {Machine review of arXiv:1908.02032}
}
abstract

Evaluating the action of a matrix function on a vector, that is $x=f(\mathcal M)v$, is an ubiquitous task in applications. When $\mathcal M$ is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating $x$ when $f(z)$ is either Cauchy-Stieltjes or Laplace-Stieltjes (or, which is equivalent, completely monotonic) and $\mathcal M$ is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case $\mathcal M=I \otimes A - B^T \otimes I$, and $v$ obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of $x$. Pole selection strategies with explicit convergence bounds are given also in this case.

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