BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Numerical Aspects of Quadratic Pad&\;eacute\; A
pproximation - Nick Hale (Stellenbosch University)
DTSTART;TZID=Europe/London:20191209T140000
DTEND;TZID=Europe/London:20191209T143000
UID:TALK135442AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/135442
DESCRIPTION:A classical (linear) Padé\; approximant is a
rational approximation\, F(x) = p(x)/q(x)\, of a
given function\, f(x)\, chosen so the Taylor serie
s of F(x) matches that of f(x) to as many terms as
possible. If f(x) is meromorphic\, then F(x) ofte
n provides a good approximation of f(x) in the com
plex plane beyond the radius of convergence of the
original Taylor series. A generalisation of this
idea is quadratic Padé\; approximation\, whe
re now polynomials p(x)\, q(x)\, and r(x) are chos
en so that p(x) + q(x)f(x) + r(x)f^2(x) = O(x^{max
}). The approximant\, F(x)\, can then be found by
solving p(x) + q(x)F(x) + r(x)F^2(x) = 0 by\, for
example\, the quadratic formula. Since F(x) now co
ntains branch cuts\, it typically provides better
approximations than linear Padé\; approximan
ts when f(x) is multi-sheeted\, and may be used to
estimate branch point locations as well as poles
and roots of f(x). In this talk we focus not on a
pproximation properties of Padé\; approximan
ts\, but rather on numerical aspects of their comp
utation. In the linear case things are well-unders
tood. For example\, it is well-known that the ill
conditioning in the linear system satisfied by p(x
) and q(x) means that these are computed with poor
relative error\, but that in practice\, F(x) itse
lf still has good relative accuracy. Luke (1980) f
ormalises this for linear Padé\; approximant
s\, and we show how this analysis extends to the q
uadratic case. We discuss a few different algorith
ms for computing a quadratic Padé\; approxim
ation\, explore some of the problems which arise i
n the evaluation of the approximant\, and demonstr
ate some example applications.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
END:VEVENT
END:VCALENDAR