TY - JOUR

T1 - On the absolute stability regions corresponding to partial sums of the exponential function

AU - Ketcheson, David I.

AU - Kocsis, Tihamer

AU - Loczi, Lajos

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2014/9/15

Y1 - 2014/9/15

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/325675

UR - http://arxiv.org/pdf/1312.0216.pdf

UR - http://www.scopus.com/inward/record.url?scp=84943238901&partnerID=8YFLogxK

U2 - 10.1093/imanum/dru039

DO - 10.1093/imanum/dru039

M3 - Article

SN - 0272-4979

VL - 35

SP - 1426

EP - 1455

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

IS - 3

ER -