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Iterative refinement implies numerical stability for Gaussian elimination

Robert D. Skeel
1980 Mathematics of Computation  
Because of scaling problems, Gaussian elimination with pivoting is not always as accurate as one might reasonably expect.  ...  It is shown that even a single iteration of iterative refinement in single precision is enough to make Gaussian elimination stable in a very strong sense.  ...  Error bounds are given and numerical stability is discussed for Gaussian elimination with column pivoting.  ... 
doi:10.1090/s0025-5718-1980-0572859-4 fatcat:dw6sh3jgknhvzpqnrf4zpqarm4

Iterative Refinement Implies Numerical Stability for Gaussian Elimination

Robert D. Skeel
1980 Mathematics of Computation  
Because of scaling problems, Gaussian elimination with pivoting is not always as accurate as one might reasonably expect.  ...  It is shown that even a single iteration of iterative refinement in single precision is enough to make Gaussian elimination stable in a very strong sense.  ...  Error bounds are given and numerical stability is discussed for Gaussian elimination with column pivoting.  ... 
doi:10.2307/2006197 fatcat:i2sqcwuagjgpregpy6dt7gbpwe

A model for understanding numerical stability

Folkmar Bornemann
2007 IMA Journal of Numerical Analysis  
By means of nontrivial examples, such as the componentwise backward stability analysis of Gaussian elimination with a single iterative refinement step, we demonstrate that the model even yields quantitative  ...  The model can serve as a convenient tool for teaching or as a heuristic device to discover stability results before entering a further, detailed analysis.  ...  Acknowledgements We are grateful to Nick Higham and Nick Trefethen for commenting on drafts of this paper.  ... 
doi:10.1093/imanum/drl037 fatcat:f54a2swlfra5npczbxcpv4sgpm

Three mysteries of Gaussian elimination

Lloyd N. Trefethen
1985 ACM SIGNUM Newsletter  
For some reason the question of averagecase stability of Gaussian elimination has received far less attention over the years.  ...  The standard error analysis of Gaussian elimination, due to Wilkinson, quickly reduces stability to the question of whether much growth occurs in the size of the elements as the elimination proceeds [  ... 
doi:10.1145/1057954.1057955 fatcat:hmnxjvc33nhz3gzmabpaz3l2ce

Page 1948 of Mathematical Reviews Vol. , Issue 80E [page]

1980 Mathematical Reviews  
This paper should be read by all of those involved in scaling for numerical stability in Gaussian elimination. David R.  ...  The effect of scaling on the stability of Gaussian elimination is analyzed.  ... 

On computation and use of Fourier coefficients for associated Legendre functions

Christian Gruber, Oleh Abrykosov
2016 Journal of Geodesy  
The computation of spherical harmonic series in very high resolution is known to be delicate in terms of performance and numerical stability.  ...  In this article we compare three recursive computations of the associated Legendre functions as trigonometric series, thereby ensuring a defined numerical range for each constituent wave-number, separately  ...  With Gaussian elimination and the backward substitution no scaling factors have to be considered in practice, if threshold values are introduced during the elimination and the process is broken as it asymptotically  ... 
doi:10.1007/s00190-016-0891-z fatcat:xiuqgq7nobd4pekvkccwwvpzlm

Parallel algorithms for solving large linear systems

T.J. Dekker, W. Hoffmann, K. Potma
1994 Journal of Computational and Applied Mathematics  
In particular are considered: Gaussian elimination, Gauss-Jordan elimination and a variant due to Huard (19791, and an algorithm due to Enright (19781, designed in relation to solving (stiff) ODES, such  ...  The solution of linear systems continues to play an important role in scientific computing.  ...  For large systems, however, numerical stability can only be guaranteed with complete pivoting, as shown in [26] .  ... 
doi:10.1016/0377-0427(94)90302-6 fatcat:dt747bh5ivdwvcdrjse7uqcd2m

Page 395 of Mathematical Reviews Vol. , Issue 87a [page]

1987 Mathematical Reviews  
Pairwise pivoting in Gaussian elimination involves adding a multi- ple of one row to another to introduce a zero in a specified column.  ...  Sorensen, Danny C. (1-ANL) 87a:65059 Analysis of pairwise pivoting in Gaussian elimination. IEEE Trans. Comput. 34 (1985), no. 3, 274-278.  ... 

Page 7061 of Mathematical Reviews Vol. , Issue 99j [page]

1999 Mathematical Reviews  
Necessary and sufficient conditions for the stability (orthonormality) of scaling vectors are provided in terms of their two-scale symbols. The paper is based on the results of Z. Shen [SIAM J. Math.  ...  Andreas Rieder (D-KLRH-MD,; Karlsruhe) 99j:65249 65T99 Plonka, Gerlind (D-ROST; Rostock) On stability of scaling vectors.  ... 

Page 6622 of Mathematical Reviews Vol. , Issue 88m [page]

1988 Mathematical Reviews  
This additional capability, known as ‘hardware gather/scatter’, can be used to great effect in general sparse Gaussian elimination.  ...  In this note we present some examples that show the impact of this change in hardware on the choice of algorithms for sparse Gaussian elmi- nation.  ... 

One out-of-core block solving method of large-scale linear equations

Xiaofei Xu, Zongqing Wu, J. Joo
2020 MATEC Web of Conferences  
The huge amount of memory is needed for solving the large-scale linear equations. However, many problems can not be calculated due to the limitation of the computer's physical memory.  ...  The experiments indicate that the method can solve the larger-scale linear equations efficiently, the solving time is only 5.7% more than In-Core method.  ...  It shows that the proposed method has good stability for solving large-scale linear equations.  ... 
doi:10.1051/matecconf/202030903013 fatcat:mghek3gvkfcx3pfknayz2lzl6m

Implicit Hamiltonian Monte Carlo for Sampling Multiscale Distributions [article]

Arya A. Pourzanjani, Linda R. Petzold
2019 arXiv   pre-print
Unlike previous modifications to HMC, our method is generally applicable to highly non-Gaussian distributions exhibiting multiple scales.  ...  Despite this, HMC often performs sub-optimally on distributions with high correlations or marginal variances on multiple scales because the resulting stiffness forces the leapfrog integrator in HMC to  ...  Acknowledgements Funding for this project was provided by the U.S. Army Research Office under Coagulopathy grant W911NF-10-2-0114.  ... 
arXiv:1911.05754v2 fatcat:qcg6ceb37zaenfqga7dwx4ih5i

Page 2697 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews  
In the second phase, this latter system is solved by using symmet- ric Gaussian elimination.  ...  2697 65F 99d:65092 65F05 Huckle, Thomas (D-WRZB-A; Wiirzburg) Symmetric Gaussian elimination for Cauchy-type matrices with application to positive definite Toeplitz matrices. (English summary) Numer.  ... 

Page 3725 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
See also «30012, 30052. 65F Numerical linear algebra 2003e:65034 65F05 Chang, Xiao-Wen (3-MGL-C; Montreal, QC) Some features of Gaussian elimination with rook pivoting.  ...  Summary: “In this paper we develop new techniques for stabilizing factored approximate inverse preconditioners (AINV) using pivot- ing. This method yields stable preconditioners in many cases and  ... 

Numerically stable LDLT-factorization of F-type saddle point matrices

A. C. de Niet, F. W. Wubs
2008 IMA Journal of Numerical Analysis  
We show that a lot of structure of the matrix is preserved during Gaussian elimination. The preserved structure allows us to prove that any feasible ordering for an F -matrix is numerically stable.  ...  Numerical results for F -matrices show that the algorithm is able to produce a factorization with low fill.  ...  We thank Miroslav Tůma for the permission to use his set of F -matrices from (Tůma, 2002) .  ... 
doi:10.1093/imanum/drn005 fatcat:uept72l25zbizm5xw5hx5gsoiy
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