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CATEGORIES:Cambridge Analysts' Knowledge Exchange
SUMMARY:Infinite dimensional spectral computations and lin
ear algebra: Extending the QR algorithm to infinit
e dimensions - Matt Colbrook (University of Cambri
dge)
DTSTART;TZID=Europe/London:20180530T160000
DTEND;TZID=Europe/London:20180530T170000
UID:TALK106555AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/106555
DESCRIPTION:Spectral computations of infinite-dimensional oper
ators are notoriously difficult\, yet ubiquitous i
n the sciences. Indeed\, despite more than half a
century of research it is still unknown which clas
ses of operators allow for computation of spectra
and eigenvectors with convergence rates and error
control. Recent progress in classifying the diffic
ulty of spectral problems into complexity hierarch
ies has revealed that the most difficult spectral
problems are so hard that one needs three limits i
n the computation and no convergence rates nor err
or control is possible. This begs the question: wh
ich classes of operators allow for computations wi
th convergence rates and error control?\n\nIn fini
te dimensions\, the QR algorithm seeks to calculat
e the eigenvalues and eigenvectors of a matrix. Th
e basic idea is to perform a QR decomposition\, wr
iting the matrix as a product of an orthogonal mat
rix and an upper triangular matrix\, multiply the
factors in reverse order\, and iterate. This algor
ithm has been hailed as one of the top ten influen
tial algorithms of the 20th century and works exce
edingly well in practice even though rigorous resu
lts for non-normal operators are scarce.\n\nIn thi
s talk I shall extend the QR algorithm to infinite
dimensions and discuss theorems on convergence to
spectra and eigenvectors. The infinite dimensiona
l case is more difficult for two reasons. First\,
the existence of essential spectra stops the algor
ithm computing all of the spectrum – it only compu
tes the extremal parts. This is naturally understo
od through a link with power iterations. Second\,
it is impossible to use shifts to speed up converg
ence (this is crucial in the implementation of the
finite dimensional case\, increasing linear conve
rgence to cubic convergence). Nevertheless\, I sha
ll link the QR algorithm to computational hierarch
ies\, showing how it can solve/classify spectral p
roblems and be executed on a finite machine. Numer
ical examples will also be demonstrated where it o
utperforms standard approaches that are notorious
for providing false solutions. Finally\, I shall d
iscuss some exciting new developments for a cousin
of the QR algorithm - the QL algorithm\, which ma
y help solve the shift problem. This talk is based
on joint work with Anders Hansen.
LOCATION:MR14\, Centre for Mathematical Sciences
CONTACT:Andrew Celsus
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