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For a backward-stable evaluation of Y=AV, the standard perturbation bound of numerical linear algebra gives a relative forward error bounded by κ(A) times the input perturbation, so a condition number κ corresponds to a worst-case loss of about log10 κ significant decimal digits.
“For a backward-stable evaluation, the standard perturbation bound of numerical linear algebra [32,33] gives a relative forward error”
From Curvature-Corrected Rotary Position Embeddings: An Entropy-Invariant Temperature for Mitigating Numerical-Rank Collapse in Long-Context Attention (2026) · cites this paper as [33]
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“0 Fµ(ξ)dξ. The proof of Lemma 9.2 is based on the following technical lemma. Lemma 9.5. Let µ1 and µ2 be a positive numbers at most 1/2 such that the following two properties hold for for any ξ,ξ′ ∈ [0,1]. Fµ 1(ξ) ≤ Fµ 2(ξ) 4 (36) and Fµ 1(ξ)Fµ 1(ξ ′) ≤ F µ 2(ξ +ξ ′)2. (37) Furthermore, Z 1 0 Fµ 1(ξ) dξ ≤ o(1). (38) T…”
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“For information on Krylov subspace iterations see (Barrett, et al., 1994), (Dongarra, Du(cid:11), Sorensen and van der Vorst, 1998), (Greenbaum, 1997), (Lehoucq, Sorensen, and Yang, 1998), (Saad, 1992), and (Trefethen and Bau, 1997).”
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“matrix by Tm, and we only need traverse the i-loop in the Arnoldi iteration twice: once for i = j − 1 and once for i = j. The resulting algorithm is known as the Lanczos iteration (Algorithm 2); see Trefethen and Bau (1997, Alg 36.1). It takes 3(2m−1)n flops and m products of A with a vector to compute Tm and Vm when A…”
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“Classical methods for diagonalization typically scale polynomially in the matrix dimensions [24].”
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“Although there are some approximation methods for speeding up this computation [30,31], these could produce undesired errors in the final results.”
MethodDistributed Control of Multirobot Systems With Global Connectivity Maintenance
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“are GMRES, Conjugate Gradient (CG), and Gauss Jacobi (GJ) [ 10 ].”
BackgroundChoosing Basis Functions and Shape Parameters for Radial Basis Function Methods
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“Several techniques, such as Cholesky factorization [ 31 ], can be used for efficiently solving a symmetric positive definite square system.”
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“Also by using offline OLS direct methods [ 25 ], i.e., the”
BackgroundMethodInfluentialFast Computation of Many-Particle Hydrodynamic and Electrostatic Interactions in a Confined Geometry
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Motivations and realizations of Krylov subspace methods for large sparse linear systems
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“[ 6 ] T. Chen and L. Rong, “Delay-independent stability analysis of”
BackgroundMethod
- Authors
- Trefethen, L. N., & Bau, III
- Affiliations
- University of Oxford (GB)
- Microsoft Corporation
- ORCID
- Trefethen, Lloyd N.
- Venue
- Society for Industrial and Applied Mathematics eBooks
- Published
- 1997-01
- OpenAlex
- W4320800818
- Cited by
- 3,301
- References
- 0
- Citations by year
- 2026: 97 · 2025: 179 · 2024: 148 · 2023: 192 · 2022: 194 · 2021: 186 · 2020: 209 · 2019: 174 · 2018: 179 · 2017: 169 · 2016: 145 · 2015: 150 · 2014: 176 · 2013: 164 · 2012: 150
- FWCI
- 5.17
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- Society for Industrial and Applied Mathematics eBooks (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Numerical Methods and Algorithms
- Physical Sciences › Mathematics › Modeling and Simulation › Mathematical and Computational Methods
- Keywords
- numerical linear algebra, linear systems, eigenvalue problem, singular value decomposition, least squares, QR decomposition, LU decomposition, Cholesky decomposition, iterative methods
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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