pith. sign in

arxiv: 2605.01276 · v1 · submitted 2026-05-02 · 🧮 math.NA · cs.NA

A class of low-rank short recurrences for nonsymmetric linear matrix equations

Davide Palitta , Catherine E. Powell , Valeria Simoncini This is my paper

Pith reviewed 2026-05-09 18:37 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords nonsymmetric linear matrix equationsshort recurrenceslow-rank approximationsrank truncationrandomizationsubspace projectioniterative methods
0
0 comments X

The pith

Low-memory short recurrences solve nonsymmetric matrix equations

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

This paper proposes short matrix recurrences for nonsymmetric equations of the form A1 X B1 plus similar terms equaling C D transpose. The methods integrate local subspace projection for faster convergence alongside rank truncation and randomization to reduce memory needs. Computational tests on a benchmark case and a mixed diffusion equation discretization with random inputs show the approach works well. Sympathetic readers would value this if it enables solving bigger problems in applications where full storage is impossible.

Core claim

We propose a new class of short matrix recurrences for the solution of nonsymmetric linear equations of the type A1 X B1 + … + Ap X Bp = C D^T. These iterative methods combine local subspace projection to speed up convergence with rank truncation strategies and randomization procedures to limit memory consumption. Computational experiments on a benchmark problem as well as a challenging discretized mixed formulation of a diffusion equation with random inputs illustrate the potential of the proposed methodology.

What carries the argument

short matrix recurrences that incorporate local subspace projection together with rank truncation and randomization for equations of the form sum Ai X Bi = C D^T

If this is right

  • The iterative methods achieve faster convergence through local subspace projection on the given equations.
  • Rank truncation and randomization keep memory consumption low even for large-scale instances.
  • The approach applies to both standard benchmark problems and discretized diffusion equations with random inputs.
  • These techniques extend the applicability of iterative solvers to nonsymmetric matrix equations that are otherwise storage-intensive.

Where Pith is reading between the lines

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

  • If the accuracy holds under truncation, the methods could scale to higher-dimensional problems arising from uncertainty quantification.
  • The randomization component might enable straightforward parallel or distributed implementations across multiple processors.
  • Similar truncation and projection ideas could be tested on related structured linear systems not covered in the current examples.

Load-bearing premise

Rank truncation and randomization preserve sufficient accuracy and convergence rate without introducing unacceptable bias or instability for the target class of nonsymmetric equations.

What would settle it

If the proposed methods on the benchmark problem either exceed expected memory limits or fail to reach the accuracy of reference full-rank solutions within a comparable number of steps.

read the original abstract

We propose a new class of short matrix recurrences for the solution of nonsymmetric linear equations of the type $\mathbf{A}_1\mathbf{X}\mathbf{B}_1+\ldots+\mathbf{A}_p\mathbf{X}\mathbf{B}_p=CD^T$. These iterative methods combine local subspace projection to speed up convergence with rank truncation strategies and randomization procedures to limit memory consumption. Computational experiments on a benchmark problem as well as a challenging discretized mixed formulation of a diffusion equation with random inputs illustrate the potential of the proposed methodology.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a new class of short matrix recurrences for solving nonsymmetric linear matrix equations of the form A1 X B1 + … + Ap X Bp = C D^T. The iterative methods combine local subspace projection to accelerate convergence with rank truncation strategies and randomization procedures to control memory consumption. Computational experiments on a benchmark problem and a discretized mixed formulation of a diffusion equation with random inputs are used to illustrate the approach.

Significance. If the approximations preserve the short-recurrence properties and convergence behavior, the work could provide a practical tool for large-scale nonsymmetric matrix equations with low-rank right-hand sides, particularly in uncertainty quantification settings. The combination of projection, truncation, and randomization addresses memory and computational bottlenecks, but the absence of supporting analysis reduces the potential impact relative to purely empirical demonstrations.

major comments (2)
  1. [Core algorithm derivation] The derivation of the short recurrences (in the section presenting the core algorithm) assumes exact local subspace projections and exact low-rank factors to maintain the underlying Krylov-like properties. Insertion of rank truncation and randomization produces inexact projected operators; for nonsymmetric A_i and B_i this can destroy the short-recurrence guarantee or introduce instability, yet no perturbation analysis, error bounds, or counter-example tests are supplied to control the deviation.
  2. [Numerical experiments] The numerical experiments section reports results on a benchmark and a random-input diffusion problem but supplies no quantitative error analysis, convergence-rate tables under varying truncation ranks, or ablation studies isolating the effect of randomization. Without these, it is impossible to verify whether the claimed preservation of accuracy and stability holds beyond the two specific illustrations.
minor comments (2)
  1. [Introduction] Notation for the multi-term sum A1 X B1 + … + Ap X Bp is introduced without an explicit equation number; adding one would improve readability when referring back to the problem class.
  2. [Abstract] The abstract states that the experiments 'illustrate the potential' but does not mention any observed convergence metrics or memory savings; a brief quantitative phrase would better align the abstract with the results section.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed report. We address each major comment below, indicating the revisions planned for the manuscript.

read point-by-point responses

  1. Referee: The derivation of the short recurrences (in the section presenting the core algorithm) assumes exact local subspace projections and exact low-rank factors to maintain the underlying Krylov-like properties. Insertion of rank truncation and randomization produces inexact projected operators; for nonsymmetric A_i and B_i this can destroy the short-recurrence guarantee or introduce instability, yet no perturbation analysis, error bounds, or counter-example tests are supplied to control the deviation.

    Authors: We agree that truncation and randomization introduce inexactness that may affect the short-recurrence property in the nonsymmetric setting. In the revised manuscript we will add a new subsection that discusses this issue and presents numerical diagnostics (e.g., monitoring of recurrence residuals and orthogonality loss under controlled truncation levels) on the benchmark problem. A complete perturbation theory, however, would require substantial additional analysis that is outside the scope of the present work, which centers on the algorithmic construction and its empirical behavior. revision: partial

  2. Referee: The numerical experiments section reports results on a benchmark and a random-input diffusion problem but supplies no quantitative error analysis, convergence-rate tables under varying truncation ranks, or ablation studies isolating the effect of randomization. Without these, it is impossible to verify whether the claimed preservation of accuracy and stability holds beyond the two specific illustrations.

    Authors: We accept that the current numerical section would benefit from more quantitative detail. The revised version will expand the experiments with tables of convergence rates and final residuals for several truncation ranks, together with an ablation study that compares the randomized variant against its deterministic truncation counterpart on both test problems. These additions will allow readers to assess accuracy and stability more systematically. revision: yes

standing simulated objections not resolved
  • A full perturbation analysis and associated error bounds for the effect of inexact projections on the short-recurrence property in the nonsymmetric case.

Circularity Check

0 steps flagged

No circularity: algorithmic construction supported by experiments

full rationale

The paper proposes a new class of short matrix recurrences for nonsymmetric linear matrix equations of the form A1 X B1 + … + Ap X Bp = C D^T. It combines local subspace projection, rank truncation, and randomization as methodological choices to control memory and convergence. No equations, fitted parameters, or derivations are shown that reduce any claimed prediction or first-principles result to the inputs by construction. The central contribution is an algorithmic framework illustrated by computational experiments on benchmark and application problems; there are no load-bearing self-citations, self-definitional steps, or ansatzes smuggled via prior work that would create circularity. This is a standard methodological paper whose validity rests on external numerical validation rather than internal reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the method implicitly assumes that low-rank structure is exploitable and that randomization does not destroy convergence, but these are not quantified.

pith-pipeline@v0.9.0 · 5379 in / 1144 out tokens · 45312 ms · 2026-05-09T18:37:52.742522+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

81 extracted references · 81 canonical work pages

  1. [1]

    Petersen, K. B. and Pedersen, M. S. , biburl =. The Matrix Cookbook , url =

    work page

  2. [2]

    2014 , publisher=

    An Introduction to Computational Stochastic PDEs , author=. 2014 , publisher=

    work page 2014

  3. [3]

    and Powell, C.E

    Ernst, O.G. and Powell, C.E. and Silvester, D.J. and Ullmann, E. , title =. SIAM Journal on Scientific Computing , volume =

    work page

  4. [4]

    SIAM Journal on Matrix Analysis and Applications , volume =

    Gould, Nick and Orban, Dominique and Rees, Tyrone , title =. SIAM Journal on Matrix Analysis and Applications , volume =. 2014 , doi =

    work page 2014

  5. [5]

    Keller, Carsten and Gould, Nicholas I. M. and Wathen, Andrew J. , title =. SIAM Journal on Matrix Analysis and Applications , volume =. 2000 , doi =

    work page 2000

  6. [6]

    Adaptive stochastic

    Eigel, Martin and Gittelson, Claude Jeffrey and Schwab, Christoph and Zander, Elmar , journal=. Adaptive stochastic. 2014 , publisher=

    work page 2014

  7. [7]

    and Powell, Catherine E

    Lee, Kookjin and Elman, Howard C. and Powell, Catherine E. and Lee, Dongeun , journal=. Enhanced alternating energy minimization methods for stochastic. 2022 , publisher=

    work page 2022

  8. [8]

    Powell, C. E. and Silvester, D. and Simoncini, V. , title =. SIAM Journal on Scientific Computing , volume =. 2017 , doi =

    work page 2017

  9. [9]

    A priori error analysis of stochastic

    Bespalov, Alexei and Powell, Catherine E.\ and Silvester, David J.\ , journal=. A priori error analysis of stochastic. 2012 , publisher=

    work page 2012

  10. [10]

    SIAM journal on numerical analysis , volume=

    Solution of sparse indefinite systems of linear equations , author=. SIAM journal on numerical analysis , volume=. 1975 , publisher=

    work page 1975

  11. [11]

    Simoncini, Valeria and Hao, Yue , title =. SIAM. J. Matrix Anal. & Appl. , volume =. 2023 , doi =

    work page 2023

  12. [12]

    Ashby and Thomas A

    Steven F. Ashby and Thomas A. Manteuffel and Paul E. Saylor , journal =

    work page

  13. [13]

    Wachspress , Publisher =

    E. Wachspress , Publisher =. 2013 , doi =

    work page 2013

  14. [14]

    and Simoncini, V

    Palitta, D. and Simoncini, V. , title =. Vietnam J. Math. , volume =. 2020 , doi =

    work page 2020

  15. [15]

    2009 , doi =

    Journal of Computational and Applied Mathematics , volume =. 2009 , doi =

    work page 2009

  16. [16]

    and Breiten, T

    Benner, P. and Breiten, T. , title =. Numer. Math. , volume =. 2013 , doi =

    work page 2013

  17. [17]

    and Martinsson, P

    Halko, N. and Martinsson, P. G. and Tropp, J. A. , title =. SIAM Review , volume =. 2011 , doi =

    work page 2011

  18. [19]

    and Piccinini, L

    Simoncini, V. and Piccinini, L. , journal =. TRUNCATED. 2024 , doi =

    work page 2024

  19. [20]

    and Tobler, C

    Kressner, D. and Tobler, C. , title =. SIAM. J. Matrix Anal. & Appl. , volume =. 2011 , doi =

    work page 2011

  20. [21]

    and Mele, G

    Jarlebring, E. and Mele, G. and Palitta, D. and Ringh, E. , title =. Num. Lin. Alg. Appl , volume =. doi:10.1002/nla.2176 , year =

    work page doi:10.1002/nla.2176

  21. [22]

    Hao and V

    Hao, Y. and Simoncini, V. , title =. Num. Lin. Alg. Appl , volume =. doi:10.1002/nla.2384 , year =

    work page doi:10.1002/nla.2384

  22. [23]

    Powell, C. E. and Silvester, D. and Simoncini, V. , title =. SIAM J. Sci. Comput. , volume =. 2017 , doi =

    work page 2017

  23. [24]

    and K\"urschner, P

    Palitta, D. and K\"urschner, P. , title =. Numer Algor , volume =. 2021 , doi =

    work page 2021

  24. [25]

    and Kressner, D

    Bioli, I. and Kressner, D. and Robol, L. , title =. 2024 , institution =

    work page 2024

  25. [26]

    Bioli, Ivan and Kressner, Daniel and Robol, Leonardo , title =. SIAM J. Scientific Computing , volume =. 2025 , doi =

    work page 2025

  26. [27]

    , title =

    Voet, Y. , title =. 2023 , institution =

    work page 2023

  27. [28]

    Powell, C. E. and Elman, H. C. , title =. IMA J. Numer. Anal. , year =

    work page

  28. [29]

    , title =

    Ullmann, E. , title =. SIAM J. Sci. Comput. , year =

    work page

  29. [30]

    GAMM-Mitt

    Benner, P. and Saak, J. , title =. GAMM-Mitteilungen , volume =. doi:https://doi.org/10.1002/gamm.201310003 , year =

    work page doi:10.1002/gamm.201310003

  30. [31]

    , title =

    Simoncini, V. , title =. SIAM Review , volume =. 2016 , doi =

    work page 2016

  31. [32]

    Hestenes, M. R. and Stiefel, E. , title =. J. Research Nat. Bur. Standards , year =

    work page

  32. [33]

    1980 , doi =

    The block conjugate gradient algorithm and related methods , journal =. 1980 , doi =

    work page 1980

  33. [34]

    A. C. Antoulas , Publisher =. Approximation of large-scale. 2005 , Address =

    work page 2005

  34. [35]

    Shank and Valeria Simoncini and Daniel B

    Stephen D. Shank and Valeria Simoncini and Daniel B. Szyld , Journal =. Efficient low-rank solutions of Generalized. 2016 , Number =

    work page 2016

  35. [36]

    BIT Numer

    Matrix-equation-based strategies for convection-diffusion equations , Author =. BIT Numer. Math. , Year =. doi:10.1007/s10543-015-0575-8 , File =

    work page doi:10.1007/s10543-015-0575-8

  36. [37]

    Low-Rank Tensor

    Daniel Kressner and Christine Tobler , Journal =. Low-Rank Tensor. 2011 , Number =

    work page 2011

  37. [38]

    G. J. Lord and C. E. Powell and T. Shardlow , Publisher =. An introduction to computational stochastic. 2014 , Owner =

    work page 2014

  38. [39]

    Stoll and T

    M. Stoll and T. Breiten , journal =. A low-rank in time approach to. 2015 , number =

    work page 2015

  39. [40]

    C. E. Powell and D. Silvester and V. Simoncini , journal =. An Efficient Reduced Basis Solver for Stochastic. 2017 , number =

    work page 2017

  40. [41]

    Silvester and A

    D.J. Silvester and A. Bespalov and C.E. Powell. S-IFISS version 1.0. 2014

    work page 2014

  41. [42]

    On the eigenvalue decay of solutions to operator

    Luka Grubi. On the eigenvalue decay of solutions to operator. Systems & Control Letters , Year =

    work page

  42. [43]

    Beckermann and D

    B. Beckermann and D. Kressner and Ch. Tobler , Journal =. AN ERROR ANALYSIS OF. 2013 , Number =

    work page 2013

  43. [44]

    A preconditioned low-rank

    Daniel Kressner and Martin Plešinger and Christine Tobler , Journal =. A preconditioned low-rank. 2014 , Number =

    work page 2014

  44. [45]

    2013 , Address =

    Matrix Computations , Author =. 2013 , Address =

    work page 2013

  45. [46]

    Simoncini , Journal =

    Vladimir Druskin and Leonid Knizhnerman and V. Simoncini , Journal =. Analysis of the rational. 2011 , Pages =

    work page 2011

  46. [48]

    Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems , Author =. SIAM J. Control Optim. , Year =

    work page

  47. [49]

    Damm , Journal =

    T. Damm , Journal =. Direct methods and. 2008 , Pages =

    work page 2008

  48. [50]

    2015 , Pages =

    Low-Rank Solution of Unsteady Diffusion Equations with Stochastic Coefficients , Author =. 2015 , Pages =

    work page 2015

  49. [51]

    A low-rank matrix equation method for solving

    Alexandra B\"unger and Valeria Simoncini and Martin Stoll , journal =. A low-rank matrix equation method for solving. 2021 , number =

    work page 2021

  50. [52]

    Computers & Mathematics with Applications , volume =

    An efficient solver for space–time isogeometric. Computers & Mathematics with Applications , volume =. 2020 , issn =. doi:https://doi.org/10.1016/j.camwa.2020.09.014 , author =

    work page doi:10.1016/j.camwa.2020.09.014 2020

  51. [53]

    Dolgov and M

    S. Dolgov and M. Stoll , Journal =. LOW-RANK SOLUTION TO AN OPTIMIZATION PROBLEM CONSTRAINED BY THE. 2017 , Number =

    work page 2017

  52. [54]

    2003 , Edition =

    Iterative methods for sparse linear systems , Author =. 2003 , Edition =

    work page 2003

  53. [55]

    S. R. Vatsya , pages =. 1988 , volume=25, number=4, journal=

    work page 1988

  54. [56]

    A. C. Gilbert and J. Y. Park and M. B. Wakin , Booktitle=

    work page

  55. [57]

    Tropp, J. A. , title =. Advances in Adaptive Data Analysis , volume =. 2011 , doi =

    work page 2011

  56. [58]

    and Grigori, L

    Balabanov, O. and Grigori, L. , title =. SIAM Journal on Scientific Computing , volume =. 2022 , doi =

    work page 2022

  57. [59]

    Computers & Mathematics with Applications , volume =

    Peter Benner and Patrick K\". Computers & Mathematics with Applications , volume =. 2014 , doi =

    work page 2014

  58. [60]

    Palitta, Davide and Iannacito, Martina and Simoncini, Valeria , title =. SIAM J. Matrix Anal. Appl , volume =. 2025 , doi =

    work page 2025

  59. [61]

    Bit Numer Math , volume =

    Palitta, Davide and Simoncini, Valeria , title =. Bit Numer Math , volume =. 2026 , doi =

    work page 2026

  60. [62]

    , title =

    Morgan, Ronald B. , title =. SIAM Journal on Matrix Analysis and Applications , volume =. 2000 , doi =

    work page 2000

  61. [63]

    David. S. Watkins , Publisher =. The matrix eigenvalue problem. 2007 , Owner =

    work page 2007

  62. [64]

    1982 , Address =

    Iterative methods for large sparse nonsymmetric systems of linear equations , Author =. 1982 , Address =

    work page 1982

  63. [65]

    2026 , eprint=

    Subspace gradient descent method for linear tensor equations , author=. 2026 , eprint=

    work page 2026

  64. [66]

    Simoncini and D

    V. Simoncini and D. B. Szyld , Journal =. Recent computational developments in. 2007 , Number =

    work page 2007

  65. [67]

    L. Luk. Indefinitely preconditioned inexact. Num. Linear Algebra and Appl. , Year =

    work page

  66. [68]

    2025 , volume=91, journal=

    Piccinini, Lorenzo and Simoncini, Valeria , pages =. 2025 , volume=91, journal=

    work page 2025

  67. [69]

    Krylov subspace methods for saddle point problem with indefinite preconditioning , Author =. SIAM J. Matrix Analysis and Appl , Year =

    work page

  68. [70]

    2017 , Series =

    Model reduction and approximation: theory and Algorithms , Editor =. 2017 , Series =

    work page 2017

  69. [71]

    2013 , Number =

    Peter Benner and Tobias Breiten , Journal =. 2013 , Number =

    work page 2013

  70. [72]

    Numerical Linear Algebra with Applications

    Jianjun Zhang and James G Nagy , title=. Numerical Linear Algebra with Applications

    work page

  71. [73]

    Numerical Linear Algebra with Applications , Year =

    Truncated low-rank methods for solving general linear matrix equations , Author =. Numerical Linear Algebra with Applications , Year =. doi:10.1002/nla.1973 , File =

    work page doi:10.1002/nla.1973 1973

  72. [74]

    Isogeometric preconditioners based on fast solvers for the

    G Sangalli and M Tani , Journal =. Isogeometric preconditioners based on fast solvers for the. 2016 , Number =

    work page 2016

  73. [75]

    Ullmann , title=

    E. Ullmann , title=. SIAM J. Scientific Computing , volume=32, year= 2010, pages=

    work page 2010

  74. [76]

    Numerical Linear Algebra with Applications , volume =

    Voet, Yannis , title =. Numerical Linear Algebra with Applications , volume =. doi:https://doi.org/10.1002/nla.70020 , year =

    work page doi:10.1002/nla.70020

  75. [77]

    ESAIM: Mathematical Modelling and Numerical Analysis , Year =

    An ultraweak space-time variational formulation for the wave equation: analysis and efficient numerical solution , Author =. ESAIM: Mathematical Modelling and Numerical Analysis , Year =. doi:https://doi.org/10.1051/m2an/2022035 , pages =

    work page doi:10.1051/m2an/2022035

  76. [78]

    Simoncini and Y

    V. Simoncini and Y. Hao , title=

    work page

  77. [79]

    2006 , Note =

    Spectral methods , Author =. 2006 , Note =

    work page 2006

  78. [80]

    Ellner and Eugene L

    Nancy S. Ellner and Eugene L. Wachspress , Journal =. 1991 , Number =

    work page 1991

  79. [81]

    2009 , author =

    Journal of Computational and Applied Mathematics , volume =. 2009 , author =

    work page 2009

  80. [82]

    2024 , issn =

    Fast randomized numerical rank estimation for numerically low-rank matrices , journal =. 2024 , issn =. doi:https://doi.org/10.1016/j.laa.2024.01.001 , author =

    work page doi:10.1016/j.laa.2024.01.001 2024

Showing first 80 references.