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Lanczos with compression for symmetric matrix Lyapunov equations

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Pith's one-line read This paper shows that Lanczos solvers for symmetric Lyapunov equations can compress the Krylov basis on the fly, matching full-Lanczos accuracy while storing only a small window of vectors.

desk verdict Solid, well-analyzed extension of low-memory Lanczos to Lyapunov equations, with a real but addressable reliability gap in the eigenvalue heuristic that the authors themselves document. read the letter →

arxiv 2505.22498 v1 pith:Y6FP4NL6 submitted 2025-05-28 math.NA cs.NA

Angelo A. Casulli , Francesco Hrobat , Daniel Kressner This is my paper
classification math.NAcs.NA MSC 65F1065F3015A24
keywords LyapunovequationLanczosmethodKrylovsubspacecompressionrationalZolotarevpoleslow-rankapproximationfinite-precisionanalysismatrixequations
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

Large-scale Lyapunov equations $AX + XA = cc^T$ with symmetric positive definite $A$ are routinely solved by the Lanczos method, which needs only matrix-vector products with $A$. The bottleneck is memory: when $A$ is ill-conditioned, Lanczos converges slowly and the stored Krylov basis becomes enormous. This paper claims that the basis can be compressed on the fly by approximating the projected tridiagonal Lyapunov equation inside a rational Krylov subspace with specially chosen Zolotarev poles. The resulting algorithm stores only about $m+2k+1$ vectors of length $N$ — a window that does not grow with the number of Lanczos iterations — while the error stays within the full-Lanczos error plus a term that decays exponentially in the number of poles. If the claim holds, low-memory Lanczos becomes practical for exactly the ill-conditioned problems where it previously failed.

What carries the argument

The machinery is a rational Krylov subspace $Q(T_M, e_1, \xi_k)$ built for the projected tridiagonal matrix $T_M$, with poles $\xi_k$ chosen as Zolotarev poles that minimize the rational error function in (2.3). A recursive construction involving two basis sequences, $W_i$ and $U_i$, builds this subspace cycle by cycle: the block tridiagonal extension $T_{i+1}$ is coupled to $T_i$ by a rank-2 update, and Theorem 3.1 — a low-rank update formula for rational matrix functions — lets the next rational Krylov basis be computed from a small auxiliary matrix $S_{i+1}$. The product $Q_s U_s$ is updated incrementally, so only the current compressed basis, the $m$ new Lanczos vectors, and the next Lanczos vector reside in memory. The stopping criterion (3.8) estimates the residual from these small projected quantities without access to the full basis.

What would settle it

Run the method with a prescribed tolerance on a matrix whose true smallest eigenvalue is far smaller than $0.1$ times the smallest eigenvalue of the first projected tridiagonal matrix; if the stopping criterion fires while the true residual is above tolerance — as happened in the paper's own $N = 79{,}841$ experiment — the practical claim of delivering a prescribed accuracy fails.

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Extended reading notes

Core claim

The central claim is that solving the projected $M \times M$ Lyapunov equation exactly is unnecessary; replacing the exact solve by projection onto the rational Krylov subspace $Q(T_M, e_1, \xi_k)$ with Zolotarev poles adds only the term $(\text{raterr}/\lambda_{\min})\|c\|^2$ to the Lanczos error (Corollary 2.3), and this term decays like the Zolotarev number. The algorithmic contribution is to perform this replacement without ever storing the full basis $Q_M$: the Lanczos iterations are divided into cycles, and low-rank update formulas for rational functions of tridiagonal matrices let the compressed product $Q_M U_{M,k}$ be formed on the fly from at most $m+2k+1$ Lanczos vectors. Algorithm 3 is therefore mathematically equivalent to the reference method that explicitly forms $Q_M U_{M,k}$, but uses a bounded window of memory. Theorem 4.2 extends the statement to finite-precision arithmetic, showing that loss of orthogonality in the Lanczos process only delays convergence to the roundoff level and affects the compression term only through a slightly enlarged eigenvalue interval.

Load-bearing premise

The practical procedure stands or falls on the ad hoc estimate of the smallest and largest eigenvalues of $A$ from the first projected tridiagonal matrix; the error bounds and the stopping test both assume that estimate is reliable.

Editorial extensions

If this is right

  • At most $m+2k+1$ vectors of length $N$ are stored, independent of the total number $M$ of Lanczos iterations, so memory no longer grows with the convergence of the method.
  • The number of poles $k$ grows only logarithmically in the tolerance and the condition number $\lambda_{\max}/\lambda_{\min}$, so a modest $k$ makes the compression error negligible relative to the Lanczos error.
  • The method needs about half the matrix-vector products of the two-pass Lanczos method, since it does not recompute the Lanczos basis.
  • Finite-precision roundoff does not change the qualitative picture: the residual bound matches full Lanczos up to a Zolotarev term evaluated on a slightly enlarged eigenvalue interval.
  • The same cycle-based compression extends to symmetric Sylvester equations $A_1X + XA_2 = c_1c_2^T$, with two Lanczos processes and Zolotarev poles mapped onto symmetric intervals.

Reading between the lines

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

  • Editorial extension: the low-rank update structure is not specific to Lyapunov equations, so the same on-the-fly compression should apply to other matrix equations with low-rank right-hand sides, such as Stein or algebraic Riccati equations.
  • Editorial extension: replacing the ad hoc eigenvalue estimate from the first projected tridiagonal matrix with a certified estimate, for example by using more vectors in the first cycle, would remove the observed premature stopping without changing the error analysis.
  • Editorial extension: because the compression error decays exponentially in $k$ and the memory budget is fixed in advance, the algorithm behaves as an anytime low-rank solver: one fixes the memory, uses it all for poles and cycle length, and lets a single Lanczos run continue until the residual tolerance is met.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper considers the symmetric Lyapunov equation AX + XA = cc^T with A symmetric positive definite, and proposes a low-memory variant of the Lanczos method, Algorithm 3 (called compress), that compresses the Lanczos basis on the fly using rational Krylov subspaces with Zolotarev poles. The authors derive error bounds (Corollary 2.3, Lemma 2.4, Theorems 4.1 and 4.2) that quantify the effect of compression and of finite-precision arithmetic on the residual norm, and they report numerical experiments on a 4D Laplacian problem and two model-order-reduction examples. The experiments show that compress uses roughly half the matrix-vector products of two-pass Lanczos and far fewer than a compress-and-restart method, at comparable or lower runtime. A central practical caveat is that the number of Zolotarev poles and the stopping criterion depend on an ad hoc estimate of the extreme eigenvalues of A; Table 2 contains a case (N = 79,841) where the returned approximation has a scaled residual 2.4e-3, which exceeds the prescribed tolerance of 1e-3.

Significance. If the advertised guarantees are made robust, the paper makes a valuable contribution: it provides a memory-efficient Lanczos-type solver for large-scale Lyapunov equations with a rigorous convergence analysis that takes round-off into account. The Zolotarev-based compression idea is elegant, and the error bounds are explicit and well matched to the algorithmic construction. The availability of the MATLAB code is a strength, and the finite-precision analysis in Section 4 is a useful addition. However, the practical reliability of the algorithm is currently limited by the unvalidated eigenvalue-estimation heuristic in Section 3.3; the paper itself documents a failure to meet the requested tolerance in one of the test cases. This does not invalidate the compression argument, but it is a load-bearing point that needs to be fixed before the headline claim of delivering a prescribed accuracy with bounded memory can be accepted.

major comments (2)
  1. [§3.3 and Table 2 (N = 79,841 row)] The stopping criterion (3.8) and the choice of k and of the Zolotarev poles both depend on the ad hoc estimates λmin ≈ 0.1·λmin(T1) and λmax ≈ 1.1·λmax(T1). The residual bound (2.9) consists of two terms; equation (3.8) only monitors the first term. The second term is controlled only through the choice of k and poles, and if the estimated interval is too narrow (in particular if 0.1·λmin(T1) overestimates λmin(A)), the unmonitored term can exceed tol/2. This is precisely the situation observed in the N = 79,841 row of Table 2, where the scaled residual is 2.4e-3 > tol = 1e-3, an outcome attributed in Section 5.2 to a poor estimate of the smallest eigenvalue of A during the first cycle. Since the headline claim is that compress delivers a prescribed accuracy with bounded memory, this gap is load-bearing. The authors should either use certified eigenvalue bounds, update the interval adaptively during the iteration, or check the full residual bound (2.9) before terminating the algorithm.
  2. [§5.2 (model order reduction, Example 1)] The N = 79,841 experiment is not merely a single outlier; it demonstrates a systematic risk. For that run, the algorithm stops based on the first term of (2.9) being below tol/2, but the actual residual is 2.4·tol. The paper does not provide a remedy or a warning to users beyond the brief note at the end of Section 5.2. Because the practical method is intended for users who do not know the extreme eigenvalues of A, the manuscript should either provide a modified stopping rule that accounts for the possible error in the eigenvalue estimates or state clearly that the method requires reliable estimates of λmin and λmax and give guidance on how to obtain them.
minor comments (5)
  1. [Definition 2.1] In the definition of a rational Krylov subspace, the block vector B should be in R^{M×ℓ} rather than R^{N×ℓ}, since S is an M×M matrix; as written, the notation is inconsistent with the rest of the paper.
  2. [Introduction, Section 1] The sentence 'Second, Theorem 4.1 derives an error bound for Lanczos with compression Theorem 4.2' has a typo: it should read '... with compression; see Theorem 4.2'.
  3. [§5.2] The text says the scaled residual in the N = 79,841 case is 'slightly larger than tol'; 2.4e-3 versus 1e-3 is a factor of 2.4, which is more than 'slight'. Please adjust the wording to reflect the magnitude of the deviation.
  4. [Table 3] For N = 4813, the time ratio compress/two-pass is 1.3, meaning compress is slower than two-pass in that instance. The text mentions that two-pass is competitive but does not explain this specific case; a short explanation would help the reader interpret the comparison.
  5. [Theorem 4.1] The constant C2 is defined as sqrt(1+2ϵ0) M λmax / (λmin − M^{5/2}ϵ2||A||_2); this expression relies on the assumption λmin > M^{5/2}ϵ2||A||_2, which is stated earlier. It would be helpful to also define the exact range of M for which the bound is valid.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity. Lyapunov-specific error/residual bounds rest on external [16], [15], [2], [4]; recursive-basis correctness imports general rational-Krylov lemmas from same-author preprint [12] (load-bearing but independent support). The Section 3.3 eigenvalue heuristic's documented failure in Table 2 (N = 79,841) is a robustness gap, not circularity.

  1. self citation load bearing [Section 3.2, Proposition 3.3 (proof) and Theorem 3.1 (proof)]
    "the claim for U1 = W1 eU1 follows from [12, Proposition 2.3], noting that Q(T1, e1, ξk) ⊆ span(W1) ... Then, by the second inclusion in (3.9), [12, Proposition 2.3] ensures that the matrix Wi+1 defined in (3.2) forms an orthonormal basis for Q(Ti+1, [e1, e(i+1)m+2k], ξk). Similarly, applying [12, Proposition 2.3] to the first inclusion in (3.9) guarantees that an orthonormal basis for Q(Ti+1, e1, ξk) is given by Wi+1 eUi+1, which equals Ui+1."

    Proposition 3.3, which makes Algorithm 3 equivalent to Algorithm 2, is not proven here: its base case and induction step reduce verbatim to [12, Proposition 2.3], and Theorem 3.1 reduces to [12, Corollary 2.6], a preprint sharing first author Casulli. The algorithmic-correctness claim is thus inherited from self-citation. Yet the imported lemmas are general rational-Krylov restriction and low-rank-update statements whose stated assumptions (nested tridiagonal Ti, poles outside spectra) exclude the Lyapunov target; the paper points to published [3] for the same general theory, and the Lyapunov-specific error and residual bounds rest on external [16], [15], [2] and self-contained Lemma 2.4.

full rationale

Walking the derivation chain: (1) Reference-method accuracy. Corollary 2.3 bounds ||X−X_ref||_F by ||X−X_lan||_F plus raterr(ξ_k, λ_min, λ_max)/λ_min · ||c||². Lemma 2.2 supplies the additive term, explicitly cited to external Theorem 4.2 of [16] (Druskin–Knizhnerman–Simoncini), with the needed assumption (spectrum of T_M ⊆ [λ_min, λ_max]) stated; the rest of Corollary 2.3 is a triangle inequality with the isometry Q_M. (2) Compression recursion. The proof that the recursively constructed U_i is an orthonormal basis of Q(T_i, e_1, ξ_k) (Proposition 3.3) and the low-rank update formula for r(T_{i+1}) (Theorem 3.1) import [12, Prop. 2.3] and [12, Cor. 2.6] from a same-first-author arXiv preprint; this is the only flagged item. It is load-bearing for Algorithm 3 ≡ Algorithm 2, but the imported statements are parameter-free rational-Krylov restriction and low-rank-update lemmas whose stated assumptions (nested tridiagonal T_i, poles outside the spectra) exclude the Lyapunov target, and the paper cites published [3] for the same general theory, so per the rubric they are real evidence and the circularity score is not raised. (3) Residual estimation and pole selection. Lemma 2.4 derives the two-term bound (2.9) in a self-contained way from the Lanczos decomposition (2.2) and (2.5); the Zolotarev decay (2.8) is cited to external [4]; the stopping rule balances the two terms rather than renaming an estimate as a prediction. (4) Finite precision. Theorems 4.1 and 4.2 follow external [15], [2], [30], [31], adapting them with the paper's own enlarged-interval argument. Flagged limitation, Section 5.2 and Table 2: for N = 79,841 the computed scaled residual 2.4e-3 exceeds tol = 1e-3, which the paper attributes to 'a poor estimate of the smallest eigenvalue of A during the first cycle'; the Section 3.3 heuristic (0.1·λ_min(T1), 1.1·λ_max(T1)) is thus an unvalidated robustness gap in the practical accuracy guarantee, but the reported residuals are actually computed rather than fitted, so this is a correctness risk, not a circular step. The paper benchmarks against external two-pass [27] and restart [28] implementations and reports its own failure candidly. No fitted parameter is renamed as a prediction and no definition equates an input with the target. Verdict: no significant circularity; score 2 reflects the load-bearing but evidence-backed self-citation to [12].

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard results from numerical linear algebra and on two lemmas from a previous paper by one of the same authors (Casulli and Simunec, arXiv:2403.04390). These are treated as independent support since they are parameter-free derivations, but they are not reproven here. The practical algorithm also introduces a heuristic eigenvalue estimate (factors 0.1 and 1.1) that is not part of the theory and can fail.

free parameters (2)
  • eigenvalue inflation factors = 0.1 (lower), 1.1 (upper)
    Section 3.3: eigenvalues of T1 are multiplied by 0.1 and 1.1 to estimate lambda_min and lambda_max for pole selection and residual stopping. This ad hoc heuristic directly affects the Zolotarev poles and the stopping criterion; the N=79,841 experiment shows a poor estimate led to residual above tol.
  • cycle length m = m = maxmem - 2k - 1 (with maxmem=120 in experiments)
    Defines how many Lanczos iterations occur between compressions. Chosen by user for memory target; does not affect theoretical error bounds but affects runtime and storage.
assumptions (7)
  • standard math Low-rank update formulas for rational matrix functions ([12, Cor 2.6])
    Used to prove Theorem 3.1 for the recursion of rational Krylov bases; from a prior paper with overlapping first author.
  • standard math Rational Krylov subspace basis inclusion ([12, Prop 2.3])
    Used in Proposition 3.3 to show Wi and Ui are orthonormal bases.
  • standard math Error bound for rational Krylov method for Lyapunov equations ([16, Thm 4.2])
    Basis for Lemma 2.2.
  • standard math Convergence bound for rational Galerkin projection for Sylvester equations ([2, Cor 2.5 and Eq (2.11)])
    Used in proof of Theorem 4.1 (inequality (4.10)).
  • standard math Finite-precision Lanczos error bounds ([15, Eq (21),(22)], [30,31])
    Used to bound FM and QM in Section 4.
  • standard math Zolotarev poles are optimal and decay bound (2.8) ([4, Thm 3.3])
    Used for pole selection and exponential decay of raterr.
  • domain assumption A is symmetric positive definite and no breakdown occurs in the Lanczos process (M<N)
    Stated in Section 2.1; guarantees existence/uniqueness and tridiagonal structure.

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Pith. "Pith review of Lanczos with compression for symmetric matrix Lyapunov equations." pith.science (2026). https://pith.science/paper/Y6FP4NL6

@misc{pith2026250522498,
  author       = {Pith},
  title        = {Pith review of: Lanczos with compression for symmetric matrix Lyapunov equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y6FP4NL6}},
  note         = {Machine review of arXiv:2505.22498}
}
abstract

This work considers large-scale Lyapunov matrix equations of the form $AX + XA = \boldsymbol{c}\boldsymbol{c}^T$, where $A$ is a symmetric positive definite matrix and $\boldsymbol{c}$ is a vector. Motivated by the need to solve such equations in a wide range of applications, various numerical methods have been developed to compute low-rank approximations of the solution matrix $X$. In this work, we focus on the Lanczos method, which has the distinct advantage of requiring only matrix-vector products with $A$, making it broadly applicable. However, the Lanczos method may suffer from slow convergence when $A$ is ill-conditioned, leading to excessive memory requirements for storing the Krylov subspace basis generated by the algorithm. To address this issue, we propose a novel compression strategy for the Krylov subspace basis that significantly reduces memory usage without hindering convergence. This is supported by both numerical experiments and a convergence analysis. Our analysis also accounts for the loss of orthogonality due to round-off errors in the Lanczos process.

Figures

Figures reproduced from arXiv: 2505.22498 by the authors.

Figure 1
Figure 1. Graphical representation of the orthonormal basis Qi+1 computed until cycle i + 1. TM = . . . Ti Tb m i+1 m im + 2k im + 2k . . . = Ti+1 (i + 1)m + 2k [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Graphical representation of the tridiagonal matrices Ti+1 generated until cycle i + 1. and both Ti and Tbi+1 are principal submatrices of the full tridiagonal matrix TM; see Section 3.1.1. 3.1.2. Recursive computation of rational Krylov subspace bases. Let us recall that algorithm 2 performs an a posteriori compression of the Lanczos basis QM by mul￾tiplying it with a basis of the rational Krylov subspace Q(TM, e1, … view at source ↗

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