Skew-circulant preconditioners for systems of LMF-based ODE codes

Daniele Bertaccini; Michael K. Ng

doi:10.1007/3-540-45262-1_12

Skew-circulant preconditioners for systems of LMF-based ODE codes

Daniele Bertaccini, Michael K. Ng

Department of Mathematics

Research output: Chapter in book/report/conference proceeding › Conference proceeding › peer-review

28 Citations (Scopus)

Abstract

We consider the solution of ordinary differential equations (ODEs) using implicit linear multistep formulae (LMF). More precisely, here we consider Boundary Value Methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. I n [6], Chan et al. proposed using Strang block-circulant preconditioners for solving these linear systems. However, as observed in [1], Strang preconditioners can be often ill-conditioned or singular even when the given system is well-conditioned. In this paper, we propose a nonsingular skew-circulant preconditioner for systems of LMF-based ODE codes. Numerical results are given to illustrate the effectiveness of our method.

Original language	English
Title of host publication	Numerical Analysis and Its Applications
Subtitle of host publication	Second International Conference, NAA 2000 Rousse, Bulgaria, June 11-15, 2000. Revised Papers
Editors	Lubin Vulkov, Plamen Yalamov, Jerzy Waniewski
Publisher	Springer Berlin Heidelberg
Pages	93-101
Number of pages	9
Edition	1st
ISBN (Electronic)	9783540452621
ISBN (Print)	9783540418146
DOIs	https://doi.org/10.1007/3-540-45262-1_12
Publication status	Published - 7 Mar 2001
Event	2nd International Conference on Numerical Analysis and Its Applications, NAA 2000 - Rousse, Bulgaria Duration: 11 Jun 2000 → 15 Jun 2000 https://link.springer.com/book/10.1007/3-540-45262-1

Publication series

Name	Lecture Notes in Computer Science
Volume	1988
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	2nd International Conference on Numerical Analysis and Its Applications, NAA 2000
Country/Territory	Bulgaria
City	Rousse
Period	11/06/00 → 15/06/00
Internet address	https://link.springer.com/book/10.1007/3-540-45262-1

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10.1007/3-540-45262-1_12

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Bertaccini, D., & Ng, M. K. (2001). Skew-circulant preconditioners for systems of LMF-based ODE codes. In L. Vulkov, P. Yalamov, & J. Waniewski (Eds.), Numerical Analysis and Its Applications: Second International Conference, NAA 2000 Rousse, Bulgaria, June 11-15, 2000. Revised Papers (1st ed., pp. 93-101). (Lecture Notes in Computer Science; Vol. 1988). Springer Berlin Heidelberg. https://doi.org/10.1007/3-540-45262-1_12

Bertaccini, Daniele ; Ng, Michael K. / Skew-circulant preconditioners for systems of LMF-based ODE codes. Numerical Analysis and Its Applications: Second International Conference, NAA 2000 Rousse, Bulgaria, June 11-15, 2000. Revised Papers. editor / Lubin Vulkov ; Plamen Yalamov ; Jerzy Waniewski. 1st. ed. Springer Berlin Heidelberg, 2001. pp. 93-101 (Lecture Notes in Computer Science).

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abstract = "We consider the solution of ordinary differential equations (ODEs) using implicit linear multistep formulae (LMF). More precisely, here we consider Boundary Value Methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. I n [6], Chan et al. proposed using Strang block-circulant preconditioners for solving these linear systems. However, as observed in [1], Strang preconditioners can be often ill-conditioned or singular even when the given system is well-conditioned. In this paper, we propose a nonsingular skew-circulant preconditioner for systems of LMF-based ODE codes. Numerical results are given to illustrate the effectiveness of our method.",

author = "Daniele Bertaccini and Ng, {Michael K.}",

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Bertaccini, D & Ng, MK 2001, Skew-circulant preconditioners for systems of LMF-based ODE codes. in L Vulkov, P Yalamov & J Waniewski (eds), Numerical Analysis and Its Applications: Second International Conference, NAA 2000 Rousse, Bulgaria, June 11-15, 2000. Revised Papers. 1st edn, Lecture Notes in Computer Science, vol. 1988, Springer Berlin Heidelberg, pp. 93-101, 2nd International Conference on Numerical Analysis and Its Applications, NAA 2000, Rousse, Bulgaria, 11/06/00. https://doi.org/10.1007/3-540-45262-1_12

Skew-circulant preconditioners for systems of LMF-based ODE codes. / Bertaccini, Daniele; Ng, Michael K.
Numerical Analysis and Its Applications: Second International Conference, NAA 2000 Rousse, Bulgaria, June 11-15, 2000. Revised Papers. ed. / Lubin Vulkov; Plamen Yalamov; Jerzy Waniewski. 1st. ed. Springer Berlin Heidelberg, 2001. p. 93-101 (Lecture Notes in Computer Science; Vol. 1988).

Research output: Chapter in book/report/conference proceeding › Conference proceeding › peer-review

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T1 - Skew-circulant preconditioners for systems of LMF-based ODE codes

AU - Bertaccini, Daniele

AU - Ng, Michael K.

N1 - Research supported in part by Italian Ministry of Scientific Research. Research supported in part by Hong Kong Research Grants Council Grant No. HKU 7147/99P and UK/HK Joint Research Scheme Grant No. 20009819. Publisher Copyright: © Springer-Verlag Berlin Heidelberg 2001.

PY - 2001/3/7

Y1 - 2001/3/7

N2 - We consider the solution of ordinary differential equations (ODEs) using implicit linear multistep formulae (LMF). More precisely, here we consider Boundary Value Methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. I n [6], Chan et al. proposed using Strang block-circulant preconditioners for solving these linear systems. However, as observed in [1], Strang preconditioners can be often ill-conditioned or singular even when the given system is well-conditioned. In this paper, we propose a nonsingular skew-circulant preconditioner for systems of LMF-based ODE codes. Numerical results are given to illustrate the effectiveness of our method.

AB - We consider the solution of ordinary differential equations (ODEs) using implicit linear multistep formulae (LMF). More precisely, here we consider Boundary Value Methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. I n [6], Chan et al. proposed using Strang block-circulant preconditioners for solving these linear systems. However, as observed in [1], Strang preconditioners can be often ill-conditioned or singular even when the given system is well-conditioned. In this paper, we propose a nonsingular skew-circulant preconditioner for systems of LMF-based ODE codes. Numerical results are given to illustrate the effectiveness of our method.

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T3 - Lecture Notes in Computer Science

SP - 93

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Bertaccini D, Ng MK. Skew-circulant preconditioners for systems of LMF-based ODE codes. In Vulkov L, Yalamov P, Waniewski J, editors, Numerical Analysis and Its Applications: Second International Conference, NAA 2000 Rousse, Bulgaria, June 11-15, 2000. Revised Papers. 1st ed. Springer Berlin Heidelberg. 2001. p. 93-101. (Lecture Notes in Computer Science). doi: 10.1007/3-540-45262-1_12

Skew-circulant preconditioners for systems of LMF-based ODE codes

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