Pith. sign in

REVIEW 4 major objections 10 minor 34 references

A rational Krylov methods for large scale linear multidimensional dynamical systems

T0 review · 4 major / 10 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read This paper claims that tensor rational block Arnoldi and Lanczos algorithms reduce multidimensional linear time-invariant systems and solve large-scale Lyapunov tensor equations by projection onto rational Krylov subspaces.

desk verdict The tensor rational Lanczos sections rest on an identity that fails for the algorithm's own starting block; the paper is honest about its lineage but the Lyapunov result is currently unsupported. read the letter →

arxiv 2411.18210 v1 pith:GGZWDJHF submitted 2024-11-27 math.NA cs.NA

Houda Barkouki , Khalide Jbilou This is my paper · ORCID
classification math.NAcs.NA MSC 15A6993B11
keywords tensorrationalKrylovsubspaceEinsteinproductmodelorderreductionmultilineardynamicalsystemsLyapunovequationsblockArnoldialgorithmLanczosadaptiveinterpolationpoints
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

Multidimensional linear time-invariant systems, whose state, input, and output are tensors, grow too large for direct simulation and control design. This paper proposes to reduce such systems by projecting them onto tensor rational Krylov subspaces built with the Einstein product, and introduces two algorithms: tensor rational block Arnoldi and tensor rational block Lanczos. The claim is that these projections produce low-order tensor systems whose transfer functions approximate the original, with error bounds, and that the Lanczos variant also gives low-rank approximate solutions to the Lyapunov tensor equations needed for balanced truncation. An adaptive rule for choosing interpolation points from residual-based error estimates is part of the proposed method.

What carries the argument

The central object is the tensor rational block Krylov subspace $\mathcal{K}_m(A,B,\Sigma_m)=\operatorname{Range}\{(A-\sigma_1 I)^{-1}*B,\dots,(\prod_{k=1}^m (A-\sigma_k I)^{-1})*B\}$, built one block at a time. In the Arnoldi variant each new block is $(A-\sigma_j I)^{-1}*\mathcal{V}_j$, orthogonalized against previous blocks by a tensor QR step; in the Lanczos variant the same operation is applied to $A^T$ with a second start tensor $C^T$, and bi-orthonormality is enforced by QR and SVD normalizations. The whole construction uses only the Einstein product and block-tensor notation, so the reduced system has the same tensor format as the original. These bases are what convert a large MLTI system or Lyapunov equation into a small projected problem.

What would settle it

Run Algorithm 4 on a small random stable tensor $A$ and random $B$ with $K_1=K_2=2$, form $X_m=\mathcal{V}_m*Y_m*\mathcal{V}_m^T$ from the reduced equation (6.5), compute the true residual $R_m=A*X_m+X_m*A^T+B*B^T$, and compare $\|R_m\|$ with the bound $2\|\Gamma_{m,A}*X_m*\mathcal{V}_m^T\|$ from Theorem 6.1; if the bound is violated, or if $\mathcal{W}_m^T*(R_m*\mathcal{W}_m)$ is not near zero, the Galerkin reduction is not valid as stated. Also check numerically whether $\mathcal{V}_m*\mathcal{W}_m^T*B$ equals $B$; for generic data it will not.

Watch

Extended reading notes

Core claim

The central claim is that rational Krylov projection can be carried out entirely in tensor form, without unfolding the system into a matrix. For a system $\dot{X}(t)=A*X(t)+B*U(t)$, $Y(t)=C*X(t)$, the paper builds bases $\mathcal{V}_m$ and $\mathcal{W}_m$ for coupled rational Krylov subspaces $\mathcal{K}_m(A,B,\Sigma)$ and $\mathcal{K}_m(A^T,C^T,\Sigma)$, forms reduced tensors $A_m=\mathcal{W}_m^T*(A*\mathcal{V}_m)$, $B_m=\mathcal{W}_m^T*B$, $C_m=C*\mathcal{V}_m$, and states that the reduced transfer function approximates the original to the accuracy controlled by the error bounds in Theorems 3.1–3.3. The same Lanczos bases are used to project the continuous Lyapunov tensor equation $A*X+X*A^T+B*B^T=O$ to a small equation $T_m*Y_m+Y_m*T_m^T+B_m*B_m^T=O$, with the residual norm bounded in Theorem 6.1 and a factored low-rank approximation proposed. The numerical examples with random sparse tensors and a discretized 2D heat equation are offered as evidence that the reduced transfer functions track the original frequency response.

Load-bearing premise

The load-bearing assumption in the Lyapunov part is that the input tensor $B$ lies in the low-dimensional range of the Lanczos basis $\mathcal{V}_m$, so that $\mathcal{V}_m*\mathcal{W}_m^T*B=B$; Algorithm 4 begins with $(A-\sigma_1 I)^{-1}*B$, so this is generally not true, and the reduced Lyapunov equation follows from the Galerkin condition only if that assumption is repaired.

Editorial extensions

If this is right

  • If the algorithms perform as claimed, a package that handles tensor arithmetic can reduce an MLTI system without ever forming the large unfolded matrix, saving memory.
  • The adaptive interpolation-point rule can be reused in both Arnoldi and Lanczos tensor settings, and it removes the need for a user-supplied set of shifts.
  • The Lyapunov solver produces a factored low-rank approximation $X_m \approx Z_1*Z_2^T$, the form balanced truncation needs to compute Hankel singular values.
  • The error estimates in Theorems 3.1 and 3.3 give a stopping criterion: iterate until the residual estimate falls below a tolerance, as used in Algorithm 5.

Reading between the lines

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

  • A reader checking the proof of Theorem 6.1 will find the step $\mathcal{V}_m*\mathcal{W}_m^T*B=B$ used to eliminate $B*B^T$ from the residual; Algorithm 4 starts from $(A-\sigma_1 I)^{-1}*B$ rather than $B$, so this identity is not automatic and the residual bound and the reduced Lyapunov equation may need an extra term.
  • The same tensor rational Krylov construction should extend to order $N>2$ by the block-tensor definitions in Section 2; the paper only runs experiments for 4th-order tensors but the algebraic relations are written for general $N$.
  • A natural next experiment is to isolate the benefit of the adaptive shift rule by comparing it to fixed shifts on the same heat-equation example, reporting transfer error versus total number of solves; the paper's tables compare the rational Lanczos method to the classical block Lanczos method but not to non-adaptive rational shifts.
  • The error estimates in Table 3.1 are heuristic simplifications of the exact residual expression; a practical implementation would need to test which of the six estimates tracks the true transfer-function error, since the paper's experiments do not report that comparison.
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 / 10 minor

Summary. This manuscript extends rational block Krylov subspace methods to tensors via the Einstein product, targeting multidimensional linear time-invariant (MLTI) systems. It presents a tensor rational block Arnoldi algorithm (Algorithm 3) and a tensor rational block Lanczos algorithm (Algorithm 4), together with projection-based reduced models, transfer-function error estimates (Theorems 3.1 and 3.2), an adaptive shift-selection procedure (Section 4), and an application to continuous-time Lyapunov tensor equations with a residual bound (Theorem 6.1). Numerical experiments in Section 7 compare the proposed methods with classical tensor Krylov variants and report residual norms, iteration counts, and runtimes. The paper concludes that the adaptive tensor rational methods are effective for model reduction and for Lyapunov tensor equations.

Significance. The paper addresses a timely problem: extending rational Krylov techniques to tensor-structured dynamical systems, where the Einstein product formulation avoids explicit matricization. The tensor rational block Arnoldi construction in Section 3.1 and the associated transfer-function error bound in Theorem 3.1 are structurally plausible and could be useful for MLTI model reduction. The paper also proposes a concrete Lanczos-type algorithm and reports experiments, and it attempts to provide a cheap residual estimator for Lyapunov tensor equations. However, the analysis of the rational Lanczos variant contains a load-bearing projection error that invalidates several theoretical claims, and several implementation details in Section 7 are not reproducible as written. If the Lanczos error estimates and the Lyapunov residual bound can be repaired, the TRBA portion and the overall framework would be a worthwhile contribution; in its present form the paper cannot be recommended.

major comments (4)
  1. [Section 6.1, Theorem 6.1] The proof of Theorem 6.1 cancels the term B * B^T by invoking the identity Vm * Wm^T * B = B. Under Algorithm 4, the initial block is S0 = (A - sigma1 I)^{-1} * B and V1 is obtained from the QR factorization of S0 (steps 3-4), so B is not generically in Range(V_m). The identity is therefore false for this algorithm, and the residual expansion in the proof is missing the term B * B^T - Vm * Bm * Bm^T * Vm^T. Consequently the bound ||Rm|| <= 2 ||Gamma_m,A * Xm * Vm^T|| in (6.6) is unsupported, and the residual-based stopping criterion in Algorithm 5 is invalid. I note that equation (6.5) does follow directly from the Galerkin condition (6.4) without this identity, so the reduced Lyapunov equation itself is not in question; only the residual estimator and its use in the numerical experiments are.
  2. [Section 3.2, equations (3.11)-(3.12) and Theorem 3.2] The residuals RB(s) and RC(s) in (3.11)-(3.12) are derived by cancelling B - Vm * Bm, which requires B = Vm * Wm^T * B. Because Algorithm 4 starts with (A - sigma1 I)^{-1} * B rather than with B itself, this cancellation is not valid; the residual contains the additional term B - Vm * Wm^T * B (and its transpose analogue), which is nonzero in general. As a result, the simplified error expression (3.13), Theorem 3.3, and the adaptive shift selections (4.1)-(4.2) built on these residuals are not established for the rational Lanczos reduced model. The authors should either change Algorithm 4 to start with B and C^T, as is standard in the matrix rational Lanczos literature (e.g., Ref. [4]), or re-derive the residual formulas without this cancellation.
  3. [Section 7, Examples 2 and 3] The text says 'We use Algorithm 3 to solve the two continuous Lyapunov equations (5.1)' in both Example 2 and Example 3. Algorithm 3 is the tensor rational block Arnoldi algorithm for model reduction and does not solve Lyapunov equations; no variant of Algorithm 3 for Lyapunov equations is described in the paper. If Algorithm 5 or Algorithm 6 was intended, the reference must be corrected. In addition, the convergence criterion in those algorithms relies on Theorem 6.1, whose residual bound is invalid per the comment above, so the residual values in Table 7.1 cannot be interpreted as evidence of convergence.
  4. [Section 4] The adaptive shift selection in (4.1)-(4.2) maximizes over a set S that is never defined. The paper should specify whether S is a finite grid, a subset of the imaginary axis, or a frequency interval, and how it is chosen and updated during the iteration; without this specification the adaptive procedure is not reproducible.
minor comments (10)
  1. [Title and Abstract] The title contains a grammatical error ('A rational Krylov methods'); it should be 'Rational Krylov methods' or 'A rational Krylov method'. The keyword 'Model recuction' on the first page is a typo for 'Model reduction'.
  2. [Section 4] The heading '4. pole selection' should be capitalized: 'Pole selection'.
  3. [Algorithm 5] Step 5 of Algorithm 5 instructs the reader to solve the continuous-time Lyapunov equation (6.5) with the MATLAB function dlyap, which solves the discrete-time Lyapunov equation; the continuous-time solver lyap (or another appropriate routine) should be used.
  4. [Section 7] The example numbering is inconsistent: 'Example 1.1' appears twice, while the second experiment is later referenced as 'Example 1.2' in the captions of Figures 7.3 and 7.4. Also, the captions of Figures 7.2 and 7.4 mix the notations F and H for the transfer function.
  5. [Section 3.1] The definition of the tensor rational Krylov subspace in (3.1) lists only (A - sigma1 I)^{-1} * B and the full product of shifted inverses applied to B, but Algorithm 3 generates a basis that also contains B itself (via the initial QR of B) and intermediate products; the definition should be aligned with the algorithm.
  6. [Theorem 3.2] Theorem 3.2 is stated without proof, and the text then moves to a different residual formulation; a proof or a precise reference to the matrix analogue should be provided.
  7. [Section 5] In (5.3), the dimensions 'r2' and 'N2' are not defined; presumably r and J1*J2 are intended. The phrase 'weakly symmetric positive-definite square tensor' should be defined or cited precisely.
  8. [Section 7] For reproducibility, the random tensors used in the experiments should be generated with a fixed seed, and the tensorization of the matrix A should be described precisely.
  9. [Theorem 6.1 proof] In the proof of Theorem 6.1, 'As Ym is a symmetric matrix' should read 'symmetric tensor'.
  10. [Throughout] There are many typographical errors ('F orj', 'end F or', 'in this cas', 'Lya punov matrix equations' in Section 2.2); a careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main flaw in Theorem 6.1 is an unjustified projection identity, which is a correctness issue rather than a circular reduction.

full rationale

The derivation chain does not reduce any claimed prediction to its inputs by construction. The tensor rational block Arnoldi and Lanczos algorithms are presented as tensor analogues of matrix rational Krylov methods, and the paper explicitly justifies the transfer via the unfolding isomorphism Psi; this is a generalization or translation, not a self-definitional loop. The adaptive shift selection in Section 4 selects sigma_{k+1} by maximizing an error estimate built from the current reduced model; that is a standard greedy heuristic, not a fitted parameter relabeled as a prediction. The only serious defect is in Section 6.1 and Theorem 6.1: after Algorithm 4 initializes S0 = (A - sigma1 I)^{-1} * B, the proof cancels the B * B^T term using 'Vm * W_m^T * B = B'. This identity is not guaranteed and is generally false, so the reduced Lyapunov equation (6.5) and the residual bound (6.6) are not established by the given argument. That is a mathematical correctness flaw, not an instance of circularity: the claimed result is not true by construction from the assumptions; it is simply unsupported or incorrect. The self-citations, such as [4] for the matrix Lanczos results, are explicit external references, and the numerical experiments are checked against exact errors and a classical block Lanczos solver, so no circular self-citation chain carries the central claim.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim depends on the unfolding isomorphism that identifies tensor Einstein products with matrix multiplication; on the existence of shifts outside the spectrum; on exact arithmetic for orthogonality; and on stability and uniqueness conditions for the Lyapunov equations. The rational shifts and truncation tolerance are free parameters, chosen by an underspecified heuristic and never reported in the experiments. No new entities are introduced.

free parameters (2)
  • Rational shifts sigma_k = not reported in experiments
    The adaptive selection (4.1)-(4.2) depends on an unspecified set S; the reported experiments do not give the chosen shifts, so they behave as hand-tuned parameters.
  • Truncation tolerance dtol = not specified numerically
    Used in Remark 6.1 and Algorithm 5 to retain dominant singular values of the reduced Lyapunov solution; the value is never stated in the paper.
assumptions (5)
  • standard math The unfolding isomorphism Psi maps the Einstein product to standard matrix multiplication.
    Stated in Section 2.1 and used throughout; it makes the tensor algorithms equivalent to matrix rational Krylov methods.
  • domain assumption The inverses (A - sigma_k I)^{-1} exist for all chosen shifts.
    Required in Algorithms 3 and 4; no condition on shifts relative to the spectrum of A is stated.
  • domain assumption Exact arithmetic is assumed so that orthogonality and bi-orthogonality of the generated bases hold.
    All relations in (2.16), (3.7) rely on exact orthogonality; finite precision effects are not discussed.
  • standard math The tensor Lyapunov equation has a unique solution, i.e., lambda_i(A) + lambda_j(A) is nonzero.
    Invoked in Section 6.1 citing [22], a standard sufficient condition for the matrix Lyapunov equation.
  • domain assumption The MLTI system is asymptotically stable.
    Balanced truncation in Section 5 requires stable Gramians; the stability condition is given in Proposition 2.7.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A rational Krylov methods for large scale linear multidimensional dynamical systems." pith.science (2026). https://pith.science/paper/GGZWDJHF

@misc{pith2026241118210,
  author       = {Pith},
  title        = {Pith review of: A rational Krylov methods for large scale linear multidimensional dynamical systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GGZWDJHF}},
  note         = {Machine review of arXiv:2411.18210}
}
read the original abstract

In this paper, we investigate the use of multilinear algebra for reducing the order of multidimensional linear time-invariant (MLTI) systems. Our main tools are tensor rational Krylov subspace methods, which enable us to approximate the systems solution within a low-dimensional subspace. We introduce the tensor rational block Arnoldi and tensor rational block Lanczos algorithms. By utilizing these methods, we develop a model reduction approach based on projection techniques. Additionally, we demonstrate how these approaches can be applied to large-scale Lyapunov tensor equations, which are critical for the balanced truncation method, a well-known technique for order reduction. An adaptive method for choosing the interpolation points is also introduced. Finally, some numerical experiments are reported to show the effectiveness of the proposed adaptive approaches.

Figures

Figures reproduced from arXiv: 2411.18210 by the authors.

Figure 7
Figure 7. [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 7
Figure 7. [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 7
Figure 7. [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figures from the paper (1 more)
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [4]

    Barkouki, A

    H. Barkouki, A. H. Bentbib and K. Jbilou, An adaptive rati onal block Lanczos-type algorithm for model reduction of large scale dynamical systems, J. Sci. Comput. , 67, 221–236 (2016)

    work page 2016

  2. [1]

    ABIDI, M

    O. ABIDI, M. HACHED AND K. JBILOU, Adaptive rational bloc k Arnoldi methods for model reductions in large-scale MIMO dynamical systems, New Trend. Math., 4 ( 2016) 227-239

    work page 2016

  3. [2]

    A. C. Antoulas, D. C. Sorensen, and S. Gugercin, A survey o f model reduction methods for large scale systems, Contemp. Math., 280, 193-219 (2001)

    work page 2001

  4. [3]

    Z. Bai, D. Day, and Q. Ye, ABLE: An adaptive block Lanczos m ethod for non-Hermitian eigenvalue problems, SIAM J. Matrix Anal. Appl., 20, 1060-1082 (1999)

    work page 1999

  5. [5]

    Barraud, A numerical algorithm to solve atxa- x = q, IEE E Transac

    A. Barraud, A numerical algorithm to solve atxa- x = q, IEE E Transac. Auto. Contr., 22 (1977), pp. 883–885

    work page 1977

  6. [6]

    M. J. Brazell, N. LI, C. Navasca, and C. Tamon, Solving mul tilinear systems via tensor inversion, SIAM J. Matrix Ana. Appl., 34 (2013), pp. 542–570

    work page 2013

  7. [7]

    C. Chen, A. Surana, A. Bloch, and I. Rajapakse, Multiline ar time invariant system theory, In 2019 Pro- ceedings of the Conference on Control and its Applications, SIAM, (2019), pp. 118–125

    work page 2019

  8. [8]

    Data-Driven Model Reduction for Multilinear Control Systems via Tensor Trains

    C. Chen, Data-driven model reduction for multilinear co ntrol systems via tensor trains, submitted, https://arxiv.org/pdf/1912.03569.pdf, (2020) 23

    work page Pith review arXiv 2020

Show all 34 references
  1. [9]

    C. Chen, A. Surana, A. Bloch, and I. Rajapakse, Multiline ar time invariant system theory, SIAM J. Contr. Optim.,, 59(1) (2021), pp. 749–776

    2021

  2. [10]

    El Guide, A

    M. El Guide, A. El Ichi, K. Jbilou and F. P. A. Beik, Tensor Krylov subspace methods via the Einstein product with applications to image and video processing, Ap p. Numer. Math., 181 (2022), 347-363

    2022

  3. [11]

    Gallivan, E

    K. Gallivan, E. Grimme, and P. Van Dooren, Pad´ e Approxi mation of Large-Scale Dynamic Systems with Lanczos Methods, in Proceedings of the 33rd IEEE Conference on Decision and Control, 443-448 1994

    1994

  4. [12]

    Gallivan, E

    K. Gallivan, E. Grimme, and P. Van Dooren, A rational Lan czos algorithm for model reduction, Num. Alg., 12, 33-63 (1996)

    1996

  5. [13]

    Gugercin and A

    S. Gugercin and A. C. Antoulas, A survey of model reducti on by balanced truncation and some new results, Internat. J. Control, 77(8) (2003), pp. 748–766

    2003

  6. [14]

    Gugercin, and A

    S. Gugercin, and A. C. Antoulas, Model reduction of larg e scale systems by least squares, Lin. Alg. Appl., 415(2-3), 290-321 (2006)

    2006

  7. [15]

    Glover, All optimal Hankel-norm approximation of li near multivariable systems and their L∞ -error bounds, Int

    K. Glover, All optimal Hankel-norm approximation of li near multivariable systems and their L∞ -error bounds, Int. J. Contr., 39 (6), 1115-1193 (1984)

    1984

  8. [16]

    G. H. Golub and C. F. Van Loan, Matrix Computations. John s Hopkins University Press., Baltimore, (1996)

    1996

  9. [17]

    Grimme, Krylov Projection Methods for Model Reducti on, PhD thesis, ECE Dept., University of Illinois, Urbana-Champaign, 1997

    E. Grimme, Krylov Projection Methods for Model Reducti on, PhD thesis, ECE Dept., University of Illinois, Urbana-Champaign, 1997

    1997

  10. [18]

    Grimme, K

    E. Grimme, K. Gallivan, and P. Van Dooren, Rational Lanc zos algorithm for model reduction II: Interpo- lation point selection, Technical report, University of Il linois at Urbana Champaign, 1998

    1998

  11. [19]

    HAMADI, K

    M.A. HAMADI, K. JBILOU , AND A. RATNANI, A model reductio n method for large-scale linear mul- tidimensional dynamical systems, submitted, https://arx iv.org/abs/2305.09361

    arXiv

  12. [20]

    Heyouni, K

    M. Heyouni, K. Jbilou, A. Messaoudi, and K. Tabaa, Model reduction in large scale MIMO dynamical systems via the block Lanczos method, Comp. Appl. Math., 27( 2), 211-236 (2008)

    2008

  13. [21]

    Heyouni and K

    M. Heyouni and K. Jbilou, An extended block arnoldi meth od for large matrix riccati equations, Elect. Trans. Numer. Anal., 33 (2009), pp. 53–62

    2009

  14. [22]

    Horn and C.R

    R.A. Horn and C.R. Johnson, Topics in Matrix Analysis. C ambridge University Press, Cambridge, (1991)

    1991

  15. [23]

    Frangos, and I

    M. Frangos, and I. M. Jaimoukha, Adaptive rational Kryl ov algorithms for model reduction, In Proc. European Control Conference, 4179-4186 (2007)

    2007

  16. [24]

    Frangos, and I

    M. Frangos, and I. M. Jaimoukha, Adaptive rational inte rpolation: Arnoldi and Lanczos-like equations, European Journal of Control, 14(4), 342-354 (2008)

    2008

  17. [25]

    T. G. Kolda and B. W. Bader, Tensor decompositions and ap plications, SIAM review, 51 (2009), pp. 455–500

    2009

  18. [26]

    H. J. Lee, C. C. Chu, and W. S. Feng, An adaptive-order rat ional Arnoldi method for model-order reduc- tions of linear time-invariant systems, Linear Algebra App l., 415, 235-261 (2006)

    2006

  19. [27]

    Liang and B

    M. Liang and B. Zheng, Further results on Moore–Penrose inverses of tensors with application to tensor nearness problems, Computers and Math. with App., 77(5) (20 19), 0898-1221

  20. [28]

    C. B. Lizhu Sun, Baodong Zheng and Y. W ei, Moore–penrose inverse of tensors via einstein product, Linear and Multilinear Algebra, 64 (2016), pp. 686–698

    2016

  21. [29]

    Mehrmann and T

    V. Mehrmann and T. Stykel, Balanced truncation model re duction for large-scale systems in descriptor form, Lecture Notes in Computational Science and Engineeri ng, 45 (2005), pp. 83–115

    2005

  22. [30]

    B. C. Moore, Principal component analysis in linear sys tems: controllability, observability and model reduction, IEEE Trans. Auto. Cont., AC-26 (1981), pp. 17–32

    1981

  23. [31]

    M. Nip, J. P. Hespanha, and M. Khammash, Direct numerica l solution of algebraic lyapunov equations for large-scale systems using quantized, in 2013 52nd IEEE Conf erence on Decision and Control (CDC), (2013), pp. 1950–1957

    2013

  24. [32]

    Rogers, L

    M. Rogers, L. Li and S. J. Russell, Multilinear Dynamica l Systems for Tensor Time Series, Part of Advances in Neural Information Processing Systems 26, (2013)

    2013

  25. [33]

    Simoncini, D

    V. Simoncini, D. B. Szyld, and M. Marlliny, On two numeri cal methods for the solution oflarge-scale algebraic riccati equations, IMA Journal of Numerical Anal ysis, 34 (2014), pp. 904–920

    2014

  26. [34]

    Yuchao and W

    W. Yuchao and W. Yimin, Generalized eigenvalue for even order tensors via Einstein product and its applications in multilinear control systems, Comput. Appl . Math., 41(8) (2022). 24

    2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.