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Multiprecision computations with Schwarz methods

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Pith's one-line read Replacing exact subdomain solves in Schwarz methods with sign-informed lower-precision solves preserves convergence for M-matrices when two local rounding-error conditions hold; in tested model problems four to five digits of local…

desk verdict First convergence analysis of Schwarz local solves in lower precision; the theory is clean but covers an idealized exact-inverse model, and the practical claim rests on an unstated robustness assumption. read the letter →

arxiv 2509.20937 v3 pith:CP73AJZY submitted 2025-09-25 math.NA cs.NA

Michal Outrata , Daniel B. Szyld This is my paper · ORCID
classification math.NAcs.NA MSC 65F1065F0865N55
keywords multiprecisionarithmeticmixedprecisionSchwarzmethodsdomaindecompositionM-matrixadditivemultiplicativelow-precisionlocalsolves
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

Schwarz domain-decomposition methods split a large linear system into overlapping subdomain problems; this paper asks how few digits the subdomain solves can use before the outer iteration stops converging. The authors prove that for M-matrices—nonsingular matrices with nonpositive off-diagonal entries and nonnegative inverse—the answer is controlled by two local conditions. If each subdomain matrix is rounded with a sign-aware rule (positive entries up, negative entries toward zero) and if $\|A_i^{-1}F_i\|<1$ together with $A_i^{-1}\ge A_i^{-1}F_iA_i^{-1}$ hold, then the multiprecision damped additive, restricted additive, and multiplicative Schwarz iterations converge. In the numerical experiments on discretized reaction-advection-diffusion equations, the conditions are met with about four to five accurate decimal digits, making single precision sufficient for the local solves and half precision viable for smaller problems. The practical payoff is that the expensive local solves can be run at reduced precision, with the required precision chosen from computable algebraic conditions rather than by trial and error.

What carries the argument

The load-bearing object is the sign-informed rounding rule $\operatorname{roundMmtrx}(X,u_\ell)$: positive entries are rounded upward and negative entries toward zero, so that $\tilde A_i$ differs from $A_i$ by an entrywise nonnegative error matrix $F_i$. This makes the splitting $A_i=\tilde A_i-(\tilde A_i-A_i)$ weak regular once the inverse perturbation is controlled. The analysis expands $\tilde A_i^{-1}=A_i^{-1}(I+F_iA_i^{-1})^{-1}$ as a Neumann series; condition (16) guarantees the series converges and condition (19) guarantees every term keeps the inverse nonnegative. The row/column rescaling of the subdomain systems into the target precision's exponent range, taken from the cited half-precision rescaling algorithm, is what allows the conditions to be met in practice and prevents overflow.

What would settle it

Take one of the paper's model problems with $N=108900$ and two overlapping subdomains, implement the low-precision local solves in actual IEEE fp16 hardware or a simulator that models underflow and overflow, and check condition (19) explicitly; if there is a precision at which $\|A_i^{-1}F_i\|<1$ holds but $A_i^{-1}\ge A_i^{-1}F_iA_i^{-1}$ fails and the multiprecision multiplicative Schwarz iteration has spectral radius at least 1 or fails to converge in the error norm, then the sufficiency claim and the Remark 3 assumption that (16) is generally binding would be refuted for that case.

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Extended reading notes

Core claim

The central claim is Theorem 3: for an M-matrix $A$ partitioned into overlapping subdomains, replacing each exact local solve $A_i^{-1}$ by $\tilde A_i^{-1}$, where $\tilde A_i$ is the sign-informed low-precision rounding of the rescaled subdomain matrix, yields convergent multiprecision Schwarz methods provided (16) and (19) hold for every $i$: $\|A_i^{-1}F_i\|<1$ in some norm and $A_i^{-1}\ge A_i^{-1}F_iA_i^{-1}$ elementwise, with $F_i=\tilde A_i-A_i$. The convergence is $\rho(\tilde T_{\mathrm{MS}})<1$, $\rho(\tilde T_{\mathrm{RAS}})<1$, and $\rho(\tilde T_{\mathrm{AS},\theta})<1$ for damping $\theta<1/q$, where $q$ is the coloring number of the subdomain overlap graph. The rounding rule is engineered so that $F_i\ge0$, which turns the splitting into a weak regular splitting; the two conditions make the Neumann expansion of $\tilde A_i^{-1}$ valid and nonnegative. In the finite-difference experiments on three nonsymmetric and three symmetric M-matrix problems, the first precision at which the conditions hold is $d_\ell=4$ or 5 digits, the observed convergence factor is then essentially identical to double precision, and the same threshold stabilizes the iteration count of preconditioned GMRES. Theorem 4 gives the analogous SPD/Stieltjes statement with an additional eigenvalue condition $\lambda_{\min}(A_i)\ge 2|\lambda_{-\infty}(F_i)|$.

Load-bearing premise

The load-bearing premise is that the simulated low precisions used in the experiments faithfully reproduce real half/single precision arithmetic (rounding directions, exponent range, underflow/overflow), and that for large problems the unverified condition (19) is indeed implied by the verified condition (16), an assertion the paper leaves as a remark.

Editorial extensions

If this is right

  • For the tested M-matrix model problems, four to five accurate decimal digits in the local solves satisfy the theoretical conditions, and further precision buys almost no visible improvement in the first 60 iterations.
  • Single precision is therefore sufficient for the subdomain solves in these problems, and half precision is viable for small $N$; on hardware with native low-precision arithmetic this can reduce the cost of the dominant part of the algorithm without meaningful convergence loss.
  • The multiprecision Schwarz methods remain effective as left preconditioners for GMRES: iteration counts to reach a relative residual of $10^{-12}$ are stable once the same precision thresholds are met.
  • Increasing the local precision cannot worsen the spectral radius of the multiprecision additive, restricted additive, or multiplicative Schwarz iteration matrices, by the monotonicity result (22).
  • The convergence guarantees hold for the methods as stand-alone stationary iterations, with separate sufficient conditions for nonsymmetric M-matrices (Theorem 3) and for symmetric Stieltjes matrices (Theorem 4).

Reading between the lines

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

  • Editorial inference: if condition (16) is the binding one, as the paper's Remark 3 suggests but does not prove, then for large problems the precision threshold could be checked cheaply from $\|A_i^{-1}\|\,\|F_i\|$ or a norm estimate, avoiding the dense-matrix verification of (19) that becomes untenable as $N$ grows.
  • Editorial inference: the sign-informed rounding recipe is not tied to Schwarz specifically; it should transfer to other splittings or multisplittings whose convergence rests on weak regular splittings, such as block Jacobi or overlapping block smoothers, giving them a similarly principled mixed-precision mode.
  • Editorial inference: the experiments suggest a testable rule of thumb: choose the local precision so that its unit roundoff is near $10^{-5}$, essentially independent of the subdomain condition numbers ($10^3$ to $10^{11}$ in the tests); checking this on three-dimensional problems with many subdomains and actual fp16/fp32 hardware would show whether the rule survives outside two-subdomain model pr
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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

4 major / 4 minor

Summary. This paper develops an algebraic convergence analysis for Schwarz methods in which the local subdomain solves are performed at a lower precision than the working precision. The authors propose sign-informed rounding of the rescaled subdomain matrices so that the rounding error matrix F_i is nonnegative, and they derive sufficient conditions, namely ||A_i^{-1}F_i||<1 in (16) and A_i^{-1} >= A_i^{-1}F_i A_i^{-1} in (19), under which the multiprecision damped additive, restricted additive, and multiplicative Schwarz iterations converge (Theorem 3); a Weyl-type eigenvalue condition (29) is added for the Stieltjes case (Theorem 4). The numerical experiments with the chop and advanpix toolboxes suggest that about four or five decimal digits suffice for the tested model problems, and the paper also contains perturbation bounds and pseudospectral statements for the use of these methods as preconditioners. The central theoretical scaffolding is coherent, but there is a gap between the idealized solver analyzed in the theorems and the actual low-precision LU solves used in the experiments, and for larger N only condition (16) is verified.

Significance. If the gap between theory and implementation is closed, this would be a useful contribution: it provides the first algebraic convergence conditions explicitly designed for multiprecision Schwarz methods, the conditions are derived rather than fitted, and the paper makes its code available. The sign-informed rounding idea is simple and effective, and the experiments are careful in distinguishing precisions for which the sufficient conditions hold from those for which they do not. The numerical evidence that single precision suffices for the tested M-matrix problems is interesting and reproducible. At present, however, the main theorem covers only exact inverses of the rounded matrices, which weakens the stated headline that the analysis identifies the minimum number of digits for the computed local solves.

major comments (4)
  1. [Section 4.1, Theorem 3; Figures 1-2] Theorem 3 analyzes the iteration obtained by replacing A_i^{-1} with the exact inverse of the sign-rounded matrix \tilde A_i = A_i + F_i, but the numerical experiments are run with chop and advanpix, which round every operation in the LU factorization and in the triangular solves. The computed local solution therefore satisfies a perturbed system with an additional unstructured backward error \Delta_i of size O(u_l ||\tilde A_i||), and conditions (16) and (19) give no control on \Delta_i because they constrain only the sign-structured entry-rounding error F_i. As stated, the theorem does not cover the implemented algorithm; the paper should either prove that the solve-time rounding is dominated by or absorbable into F_i, or explicitly restrict the convergence claims and the interpretation of Figures 1-2 to the idealized exact-inverse-of-rounded-matrix method plus an unproved robustness assumption.
  2. [Section 4.1, Remark 3; Figure 4] For N > 2500 the paper verifies only condition (16) and not condition (19); the text itself says this tests whether the observation in Remark 3 holds for larger N. Consequently the large-N claim that 'four or five digits suffice' rests on the unproven assertion that (16) is generally the binding condition. The paper should provide support for this assertion, either by verifying (19) with a cheaper componentwise estimate, by proving a relevant implication from (16) to (19) under additional assumptions, or by explicitly labeling the large-N statements as heuristic and outside the scope of Theorem 3.
  3. [Section 5, Proposition 1, Eq. (41)] The statement of Proposition 1 says that for any \delta \in (0,\epsilon) the bound (41) holds, but the right-hand side of (41) contains the factor 1/(\delta - \epsilon), which is negative or undefined on that interval. The correct condition should be \delta > \epsilon; please fix the interval and re-check the subsequent use of the bound in the discussion of pseudospectral stability.
  4. [Section 4.1, Eq. (13)] Equation (13) and the expression for E_i use (D_r^i)^{-1} on both sides of \tilde A_i, but the rescaling in (8) is A_i := \mu D_r^i A_i D_c^i. In the general non-symmetric case the right factor must be (D_c^i)^{-1}; as printed, the formula for E_i and the equivalence in (14) are valid only when the scaling is symmetric, and the general derivation should be corrected.
minor comments (4)
  1. [Abstract] In the abstract, 'proecision' should be 'precision'.
  2. [Section 5 (near Eq. (38)-(39))] The sentence 'taking \theta=1/2 still guarantees convergence convergence' contains a duplicated word and should read 'still guarantees convergence'.
  3. [Captions of Figures 3 and 8] Both captions contain 'are are analogous'; the duplicated word should be removed.
  4. [Section 4.1, notation around Eq. (13)] The notation \tilde A_i is used both for the rounded rescaled subdomain matrix and for the rounding of the original matrix; please introduce distinct symbols or state the convention explicitly to avoid ambiguity in the derivation of (13)-(19).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: conditions (16) and (19) are derived from standard weak-splitting theory and verified independently in experiments.

full rationale

The derivation chain is self-contained. The paper's sufficient conditions (16) and (19) are obtained from the weak-regular splitting criterion (12), with \tilde A_i^{-1} expanded by a Neumann series under (16), and F_i >= 0 enforced by sign-informed rounding (21). None of these ingredients is fitted to observed convergence; the conditions are a priori inequalities on A_i^{-1}F_i. The numerical experiments verify the inequalities and the convergence separately, so the convergence claim is not built into the hypotheses. The authors' earlier works [3,17,18] are cited for the classical algebraic Schwarz convergence theorems and the weak-splitting comparison theorems; these are parameter-free published results whose assumptions do not include the multiprecision target, so under the stated rules they count as independent support rather than load-bearing circularity. The skeptical concern that the experiments use full lower-precision LU solves whereas Theorem 3 analyzes exact inverses of rounded matrices is a model-mismatch and correctness risk, not a circularity: the theory is conditional on exact inverse replacement, and the numerical claim about roughly five digits is an empirical observation, not a consequence of the theorem by construction. The large-N reliance on condition (16) alone is an acknowledged limitation, not circular reasoning.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central theorems are conditional on A being an M-matrix (or Stieltjes) and on two or three quantitative conditions (16), (19), and (29) that involve the rounding error F_i. The paper does not prove these conditions hold for a given precision a priori; it verifies them numerically for small problems and only partially for larger ones. No new physical entities are introduced. The rescaling parameter nu = 0.1 is a tuning choice inherited from [32] and affects the empirical thresholds rather than the conditional theorem.

free parameters (1)
  • Rescaling parameter nu = 0.1
    Chosen following [32, Table 4.5] to control how close rescaled matrices are to the maximum range of the lower precision. It is an experimental tuning choice that affects the empirical precision thresholds in Figures 1 to 4, but not the conditional theorem.
assumptions (5)
  • domain assumption A is a nonsingular M-matrix (off-diagonal entries non-positive, A^{-1} >= 0); in Section 4.2 also a Stieltjes matrix (symmetric M-matrix).
    Theorems 3 and 4 and the sign-informed rounding are only valid for M-matrices; the model problems are constructed so that A is an M-matrix or Stieltjes matrix.
  • standard math Weak regular splitting theorem for M-matrices: condition (12) is sufficient for convergence of damped AS, RAS, and MS.
    Quoted from [17,18,3]; underpins Theorem 3.
  • standard math P-regular splitting theorem and Weyl's eigenvalue perturbation bound.
    Used to derive condition (29) in Theorem 4.
  • ad hoc to paper Directed rounding functions rdu and ruu are available in the lower precision, and roundMmtrx yields F_i >= 0.
    The proof of F_i >= 0 relies on this implementable rounding routine; hardware support for directed rounding in every low precision is assumed.
  • ad hoc to paper The Neumann series convergence condition ||A_i^{-1} F_i|| < 1 (16) and the componentwise condition (19) hold for the chosen precision.
    These are the hypotheses of Theorem 3; the paper verifies them numerically but does not prove they hold for a given precision a priori.

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Pith. "Pith review of Multiprecision computations with Schwarz methods." pith.science (2026). https://pith.science/paper/CP73AJZY

@misc{pith2026250920937,
  author       = {Pith},
  title        = {Pith review of: Multiprecision computations with Schwarz methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CP73AJZY}},
  note         = {Machine review of arXiv:2509.20937}
}
read the original abstract

Schwarz methods and preconditioners, where the approximate solution of the local problems is performed at a lower precision, i.e., with fewer digits of accuracy than in the underlying (double precision) computation. Conditions for the appropriate round-off criteria for the lower precision are presented. It is found experimentally that for the model problems about 5 digits of accuracy are sufficient to achieve the theoretical restrictions, and thus, single precision suffices for the local solves. Several numerical experiments illustrate the obtained results.

Figures

Figures reproduced from arXiv: 2509.20937 by the authors.

Figure 1
Figure 1. Top: the 2-norm of the error of the multiplicative Schwarz method for different choices of [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Top: the 2-norm of the error of the multiplicative Schwarz method for different choices of [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. The errors of the multiplicative Schwarz method used for Problem 1 with [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: We show ρconv of the Schwarz methods for dℓ = 1, 2, . . . , 16 and different problem sizes N. We highlight the first dℓ for which the condition (16) is satisfied (it is also satisfied for all the following ones). see that after the first dℓ such that (16) is satisfied …
Figure 5
Figure 5. Figure 5: The number of the preconditioned GMRES iterations to reduce the relative residual below [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Top: the 2-norm of the error of the multiplicative Schwarz method for different choices [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 3
Figure 3. Figure 3: Our experience with dAS and RAS is fairly similar. [PITH_FULL_IMAGE:figures/full_fig_p020_3.png]
Figure 7
Figure 7. Figure 7: Left: the 2-norm of the error of the multiplicative Schwarz method for different choices [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: The errors of the multiplicative Schwarz method used for Problem 5 with [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: We show ρconv of the Schwarz method for dℓ = 1, 2, . . . , 16 and different problem sizes N. We highlight the first dℓ for which the conditions (16) and (29) are satisfied by (or by ). and denoting Ei := D (c) i [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: The number of the preconditioned GMRES iterations to reduce the relative residual [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]

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Works this paper leans on

50 extracted references · 46 canonical work pages

  1. [1]

    doi: 10.1109/IEEESTD.2019.8766229

    IEEE Standard for floating-point arithmetic.IEEE Std 754-2019 (Revision of IEEE 754-2008), pages 1–84, 2019. doi: 10.1109/IEEESTD.2019.8766229

    arXiv 2019

  2. [2]

    H. Anzt, J. Dongarra, G. Flegar, N. J. Higham, and E. S. Quintana-Ortí. Adaptive precision in block-Jacobi preconditioning for iterative sparse linear system solvers.Concurrency and Computation: Practice and Experience, 31:e4460, 2019. 29

    work page 2019

  3. [3]

    Benzi, A

    M. Benzi, A. Frommer, R. Nabben, and D. B. Szyld. Algebraic theory of multiplicative Schwarz methods.Numerische Mathematik, 89:605–639, 2001

    work page 2001

  4. [4]

    Berman and R

    A. Berman and R. J. Plemmons.Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York, 1979. Reprinted by SIAM, Philadelphia, 1994

    work page 1979

  5. [5]

    J. Blechta. Stability of linear GMRES convergence with respect to compact perturbations. SIAM Journal on Matrix Analysis and Applications, 42:436–447, 2021

    work page 2021

  6. [6]

    Buttari, N

    A. Buttari, N. J. Higham, T. Mary, and B. Vieublé. A modular framework for the backward error analysis of GMRES.IMA Journal of Numerical Analysis; to appear, 2025

    work page 2025

  7. [7]

    Cai and M

    X.-C. Cai and M. Sarkis. A restricted additive Schwarz preconditioner for general sparse linear systems.SIAM Journal on Scientific Computing, 21:792–797, 1999

    work page 1999

  8. [8]

    X.-C. Cai, M. Dryja, and M. Sarkis. Restricted additive Schwarz preconditioners with harmonic overlap for symmetric positive definite linear systems.SIAM Journal on Numerical Analysis, 41:1209–1231, 2003

    work page 2003

Show all 50 references
  1. [9]

    Carson and X

    E. Carson and X. Chen. Pychop: Emulating low-precision arithmetic in numerical methods and neural networks, 2025. URLhttps://arxiv.org/abs/2504.07835

    2025 arXiv

  2. [10]

    Carson and N

    E. Carson and N. J. Higham. Accelerating the solution of linear systems by iterative refinement in three precisions.SIAM Journal on Scientific Computing, 40:A817–A847, 2018

    2018

  3. [11]

    Cuvelier, M

    F. Cuvelier, M. J. Gander, and L. Halpern. Fundamental coarse space components for Schwarz methods with crosspoints. In S. C. Brenner, E. T. S. Chung, A. Klawonn, F. Kwok, J. Xu, and J. Zou, editors,Domain Decomposition Methods in Science and Engineering XXVI, volume 145 ofLec...

    2023

  4. [12]

    Dolean, M

    V. Dolean, M. J. Gander, and L. Gerardo-Giorda. Optimized Schwarz methods for Maxwell’s equations.SIAM Journal on Scientific Computing, 31:2193–2213, 2009

    2009

  5. [13]

    Dolean, M

    V. Dolean, M. J. Gander, S. Lanteri, J.-F. Lee, and Z. Peng. Effective transmission conditions fordomaindecompositionmethodsappliedtothetime-harmoniccurl–curlMaxwell’sequations. Journal of Ccomputational Physics, 280:232–247, 2015

    2015

  6. [14]

    SIAM, Philadelphia, 2015

    Victorita Dolean, Pierre Jolivet, and Frédéric Nataf.An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation. SIAM, Philadelphia, 2015

    2015

  7. [15]

    Efstathiou and M

    E. Efstathiou and M. J. Gander. Why restricted additive Schwarz converges faster than additive Schwarz.BIT Numerical Mathematics, 43:945–959, 2003

    2003

  8. [16]

    M. Embree. How descriptive are GMRES convergence bounds?, 2023. arXiv preprint: 2209.01231

    2023 arXiv

  9. [17]

    Frommer and D

    A. Frommer and D. B. Szyld. Weighted max norms, splittings, and overlapping additive Schwarz iterations.Numerische Mathematik, 83:259–278, 1999

    1999

  10. [18]

    Frommer and D

    A. Frommer and D. B. Szyld. An algebraic convergence theory for restricted additive Schwarz methods using weighted max norms.SIAM Journal on Numerical Analysis, 39:463–479, 2001. 30

    2001

  11. [19]

    Frommer and D

    A. Frommer and D. B. Szyld. On the convergence of randomized and greedy relaxation schemes for solving nonsingular linear systems of equations.Numerical Algorithms, 92:639–664, 2023

    2023

  12. [20]

    M. J. Gander. Optimized Schwarz methods.SIAM Journal on Numerical Analysis, 44:699–731, 2006

    2006

  13. [21]

    Schwarzmethodsoverthecourseoftime.Electronical Transactions on Numerical Analysis, 31:228–255, 2008

    M.J.Gander. Schwarzmethodsoverthecourseoftime.Electronical Transactions on Numerical Analysis, 31:228–255, 2008

    2008

  14. [22]

    M. J. Gander and L. Halpern. Piece-wise constant, linear and oscillatory: a historical intro- duction to spectral coarse spaces with focus on Schwarz methods. In Z. Dostál, T. Kozubek, A. Klawonn, L. Ulrich, L. F. Pavarino, J. Šístek, and O. B. Widlund, editors,Domain Decom- p...

    2024

  15. [23]

    M. J. Gander and M. Outrata. On algebraic bounds for POSM and MRAS. In Z. Dostál, T. Kozubek, A. Klawonn, L. Ulrich, L. F. Pavarino, J. Šístek, and O. B. Widlund, editors, Domain Decomposition Methods in Science and Engineering XXVII, volume 149 ofLecture notes in Computer Sci...

    2024

  16. [24]

    M. J. Gander and H. Zhang. A class of iterative solvers for the Helmholtz equation: factoriza- tions, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods.SIAM Review, 61:3–76, 2019

    2019

  17. [25]

    M. J. Gander and H. Zhang. Schwarz methods by domain truncation.Acta Numerica, 31: 1–134, 2022

    2022

  18. [26]

    M. J. Gander, S. Loisel, and D. B. Szyld. An optimal block iterative method and preconditioner for banded matrices with applications to PDEs on irregular domains.SIAM Journal on Matrix Analysis and Applications, 33:653–680, 2012

    2012

  19. [27]

    Giraud, A

    L. Giraud, A. Haidar, and L. T. Watson. Mixed-precision preconditioners in parallel domain decomposition solvers. In U. Langer, M. Discacciati, D. Keyes, O. Widlund, and W. Zulehner, editors,Domain Decomposition Methods in Science and Engineering XVII, volume 60 ofLecture note...

    2008

  20. [28]

    Glusa, E

    C. Glusa, E. G. Boman, E. Chow, S. Rajamanickam, and D. B. Szyld. Scalable asynchronous domain decomposition solvers.SIAM Journal on Scientific Computing, pages C384–C409, 2020

    2020

  21. [29]

    N. J. Higham.Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia, PA, USA, 2002

    2002

  22. [30]

    N. J. Higham and T. Mary. Mixed precision algorithms in numerical linear algebra.Acta Numerica, 31:347–414, 2022

    2022

  23. [31]

    N. J. Higham and S. Pranesh. Simulating low precision floating-point arithmetic.SIAM Journal on Scientific Computing, 41:C585–C602, 2019. 31

    2019

  24. [32]

    N. J. Higham, S. Pranesh, and M. Zounon. Squeezing a matrix into half precision, with an application to solving linear systems.SIAM Journal on Scientific Computing, 41:A2536–A2551, 2019

    2019

  25. [33]

    R. A. Horn and C. R. Johnson.Topics in Matrix Analysis. Cambridge University Press, Cambridge, Cambridge, 1994

    1994

  26. [34]

    Liesen and Z

    J. Liesen and Z. Strakoš.Krylov Subspace Methods: Principles and Analysis. Oxford University Press, Oxford, 2013

    2013

  27. [35]

    P. L. Lions. On the Schwarz alternating method. In R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, editors,Domain Decomposition Methods for Partial Differential Equations, pages 1–42. SIAM, 1988

    1988

  28. [36]

    Magoulès, D

    F. Magoulès, D. B. Szyld, and C.Venet. Asynchronous optimized Schwarz methods with and without overlap.Numerische Mathematik, 137:199–227, 2017

    2017

  29. [37]

    Advanpix LLC., Yokohama, Japan

    Multiprecision Computing Toolbox for MATLAB 5.2.5.15470. Advanpix LLC., Yokohama, Japan

  30. [38]

    J. M. Ortega.Numerical Analysis: A Second Course. Classics in Applied Mathematics. SIAM, Philadelpia, 1990

    1990

  31. [39]

    Ozaki, T

    K. Ozaki, T. Ogita, S. Oishi, and S. Rump. Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications.Numerical Algorithms, 59: 95–118, 2012

    2012

  32. [40]

    Ozaki, T

    K. Ozaki, T. Ogita, S. Oishi, and S. M. Rump. Generalization of error-free transformation for matrix multiplication and its application.Nonlinear Theory and Its Applications, 4:2–11, 2013

    2013

  33. [41]

    Ozaki, Y

    K. Ozaki, Y. Uchino, and T. Imamura. Ozaki scheme II: A GEMM-oriented emulation of floating-point matrix multiplication using an integer modular technique. arXiv preprint arXiv:2504.08009, 2025

    2025 arXiv

  34. [42]

    Saad.Iterative Methods for Sparse Linear Systems

    Y. Saad.Iterative Methods for Sparse Linear Systems. Other Titles in Applied Mathematics. SIAM, Philadelphia, Second edition, 2003. ISBN 978-0-89871-534-7

    2003

  35. [43]

    Sarkis and M

    M. Sarkis and M. Dryja. Convergence bounds for one-dimensional ASH and RAS. In Z. Dostál, T. Kozubek, A. Klawonn, L. Ulrich, L. F. Pavarino, J. Šístek, and O. B. Widlund, editors, Domain Decomposition Methods in Science and Engineering XXVII, Lecture notes in Computer Science ...

    2023

  36. [44]

    Schneck, M

    J. Schneck, M. Weiser, and F. Wende. Impact of mixed precision and storage layout on additive Schwarz smoothers.Numerical Linear Algebra with Applications, 28:e2366, 2021

    2021

  37. [45]

    J. A. Sifuentes, M. Embree, and R. B. Morgan. GMRES convergence for perturbed coefficient matrices, with application to approximate deflation preconditioning.SIAM Journal on Matrix Analysis and Applications, 34:1066–1088, 2013. 32

    2013

  38. [46]

    Smith, Petter E

    Barry F. Smith, Petter E. Bjørstad, and William D. Gropp.Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge, New York, Melbourne, 1996

    1996

  39. [47]

    N. Tian, S. Huang, and X. Xu. Mixed precision block-Jacobi preconditioner: algorithms, per- formance evaluation and feature analysis.CCF Transactions on High Performance Computing, 7:114–128, 2025

    2025

  40. [48]

    Springer, Berlin, Heidelberg, New York, 2005

    Andrea Toselli and Olof Widlund.Domain Decomposition Methods - Algorithms and Theory, volume 34 ofSeries in Computational Mathematics. Springer, Berlin, Heidelberg, New York, 2005

    2005

  41. [49]

    L. N. Trefethen and M. Embree.Spectra and Pseudospectra: The Behaviour of Non-Normal Matrices and Operators. Princeton University Press, Princeton, New Jersey, 2005

    2005

  42. [50]

    Widlund and M

    O. Widlund and M. Dryja. An additive variant of the Schwarz alternating method for the case of many subregions. Technical Report 339, Ultracomputer Note 131, Department of Computer Science, Courant Institute, New York University, 1987. 33

    1987

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