Rational functions with maximal radius of absolute monotonicity

Handle URI:
http://hdl.handle.net/10754/551012
Title:
Rational functions with maximal radius of absolute monotonicity
Authors:
Loczi, Lajos ( 0000-0002-7999-5658 ) ; Ketcheson, David I. ( 0000-0002-1212-126X )
Abstract:
We study the radius of absolute monotonicity R of rational functions with numerator and denominator of degree s that approximate the exponential function to order p. Such functions arise in the application of implicit s-stage, order p Runge-Kutta methods for initial value problems and the radius of absolute monotonicity governs the numerical preservation of properties like positivity and maximum-norm contractivity. We construct a function with p=2 and R>2s, disproving a conjecture of van de Griend and Kraaijevanger. We determine the maximum attainable radius for functions in several one-parameter families of rational functions. Moreover, we prove earlier conjectured optimal radii in some families with 2 or 3 parameters via uniqueness arguments for systems of polynomial inequalities. Our results also prove the optimality of some strong stability preserving implicit and singly diagonally implicit Runge-Kutta methods. Whereas previous results in this area were primarily numerical, we give all constants as exact algebraic numbers.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Numerical Mathematics Group
Citation:
Lajos Lóczi and David I. Ketcheson (2014). Rational functions with maximal radius of absolute monotonicity. LMS Journal of Computation and Mathematics, 17, pp 159-205. doi:10.1112/S1461157013000326.
Publisher:
Oxford University Press (OUP)
Journal:
LMS Journal of Computation and Mathematics
Issue Date:
19-May-2014
DOI:
10.1112/S1461157013000326
ARXIV:
arXiv:1303.6651
Type:
Article
ISSN:
1461-1570
Sponsors:
This publication is based on work supported by Award No. FIC/2010/05 – 2000000231, made by King Abdullah University of Science and Technology (KAUST).
Additional Links:
http://www.journals.cambridge.org/abstract_S1461157013000326; http://arxiv.org/abs/1303.6651
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorLoczi, Lajosen
dc.contributor.authorKetcheson, David I.en
dc.date.accessioned2015-04-30T12:54:48Zen
dc.date.available2015-04-30T12:54:48Zen
dc.date.issued2014-05-19en
dc.identifier.citationLajos Lóczi and David I. Ketcheson (2014). Rational functions with maximal radius of absolute monotonicity. LMS Journal of Computation and Mathematics, 17, pp 159-205. doi:10.1112/S1461157013000326.en
dc.identifier.issn1461-1570en
dc.identifier.doi10.1112/S1461157013000326en
dc.identifier.urihttp://hdl.handle.net/10754/551012en
dc.description.abstractWe study the radius of absolute monotonicity R of rational functions with numerator and denominator of degree s that approximate the exponential function to order p. Such functions arise in the application of implicit s-stage, order p Runge-Kutta methods for initial value problems and the radius of absolute monotonicity governs the numerical preservation of properties like positivity and maximum-norm contractivity. We construct a function with p=2 and R>2s, disproving a conjecture of van de Griend and Kraaijevanger. We determine the maximum attainable radius for functions in several one-parameter families of rational functions. Moreover, we prove earlier conjectured optimal radii in some families with 2 or 3 parameters via uniqueness arguments for systems of polynomial inequalities. Our results also prove the optimality of some strong stability preserving implicit and singly diagonally implicit Runge-Kutta methods. Whereas previous results in this area were primarily numerical, we give all constants as exact algebraic numbers.en
dc.description.sponsorshipThis publication is based on work supported by Award No. FIC/2010/05 – 2000000231, made by King Abdullah University of Science and Technology (KAUST).en
dc.language.isoenen
dc.publisherOxford University Press (OUP)en
dc.relation.urlhttp://www.journals.cambridge.org/abstract_S1461157013000326en
dc.relation.urlhttp://arxiv.org/abs/1303.6651en
dc.rightsArchived with thanks to LMS Journal of Computation and Mathematicsen
dc.titleRational functions with maximal radius of absolute monotonicityen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentNumerical Mathematics Groupen
dc.identifier.journalLMS Journal of Computation and Mathematicsen
dc.eprint.versionPre-printen
dc.identifier.arxividarXiv:1303.6651en
kaust.authorLoczi, Lajosen
kaust.authorKetcheson, David I.en
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