On the absolute stability regions corresponding to partial sums of the exponential function

Handle URI:
http://hdl.handle.net/10754/325675
Title:
On the absolute stability regions corresponding to partial sums of the exponential function
Authors:
Ketcheson, David I. ( 0000-0002-1212-126X ) ; Kocsis, Tihamer; Loczi, Lajos ( 0000-0002-7999-5658 )
Abstract:
Certain numerical methods for initial value problems have as stability function the nth partial sum of the exponential function. We study the stability region, i.e., the set in the complex plane over which the nth partial sum has at most unit modulus. It is known that the asymptotic shape of the part of the stability region in the left half-plane is a semi-disk. We quantify this by providing disks that enclose or are enclosed by the stability region or its left half-plane part. The radius of the smallest disk centered at the origin that contains the stability region (or its portion in the left half-plane) is determined for 1 n 20. Bounds on such radii are proved for n 2; these bounds are shown to be optimal in the limit n ! +1. We prove that the stability region and its complement, restricted to the imaginary axis, consist of alternating intervals of length tending to , as n ! 1. Finally, we prove that a semi-disk in the left half-plane with vertical boundary being the imaginary axis and centered at the origin is included in the stability region if and only if n 0 mod 4 or n 3 mod 4. The maximal radii of such semi-disks are exactly determined for 1 n 20.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Numerical Mathematics Group
Publisher:
Oxford University Press (OUP)
Journal:
IMA Journal of Numerical Analysis
Issue Date:
3-Dec-2013
DOI:
10.1093/imanum/dru039
Type:
Article
ISSN:
0272-4979
Additional Links:
http://arxiv.org/pdf/1312.0216.pdf
Appears in Collections:
Articles; Numerical Mathematics Group; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorKetcheson, David I.en
dc.contributor.authorKocsis, Tihameren
dc.contributor.authorLoczi, Lajosen
dc.date.accessioned2014-09-03T11:50:41Z-
dc.date.available2014-09-03T11:50:41Z-
dc.date.issued2013-12-03en
dc.identifier.issn0272-4979en
dc.identifier.doi10.1093/imanum/dru039en
dc.identifier.urihttp://hdl.handle.net/10754/325675en
dc.description.abstractCertain numerical methods for initial value problems have as stability function the nth partial sum of the exponential function. We study the stability region, i.e., the set in the complex plane over which the nth partial sum has at most unit modulus. It is known that the asymptotic shape of the part of the stability region in the left half-plane is a semi-disk. We quantify this by providing disks that enclose or are enclosed by the stability region or its left half-plane part. The radius of the smallest disk centered at the origin that contains the stability region (or its portion in the left half-plane) is determined for 1 n 20. Bounds on such radii are proved for n 2; these bounds are shown to be optimal in the limit n ! +1. We prove that the stability region and its complement, restricted to the imaginary axis, consist of alternating intervals of length tending to , as n ! 1. Finally, we prove that a semi-disk in the left half-plane with vertical boundary being the imaginary axis and centered at the origin is included in the stability region if and only if n 0 mod 4 or n 3 mod 4. The maximal radii of such semi-disks are exactly determined for 1 n 20.en
dc.language.isoenen
dc.publisherOxford University Press (OUP)en
dc.relation.urlhttp://arxiv.org/pdf/1312.0216.pdfen
dc.titleOn the absolute stability regions corresponding to partial sums of the exponential functionen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentNumerical Mathematics Groupen
dc.identifier.journalIMA Journal of Numerical Analysisen
dc.eprint.versionPre-printen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
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