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SUMMARY:Infinite dimensional spectral computations and linear algebra: Ext
 ending the QR algorithm to infinite dimensions - Matt Colbrook (University
  of Cambridge)
DTSTART:20180530T150000Z
DTEND:20180530T160000Z
UID:TALK106555@talks.cam.ac.uk
CONTACT:Andrew Celsus
DESCRIPTION:Spectral computations of infinite-dimensional operators are no
 toriously difficult\, yet ubiquitous in the sciences. Indeed\, despite mor
 e than half a century of research it is still unknown which classes of ope
 rators allow for computation of spectra and eigenvectors with convergence 
 rates and error control. Recent progress in classifying the difficulty of 
 spectral problems into complexity hierarchies has revealed that the most d
 ifficult spectral problems are so hard that one needs three limits in the 
 computation and no convergence rates nor error control is possible. This b
 egs the question: which classes of operators allow for computations with c
 onvergence rates and error control?\n\nIn finite dimensions\, the QR algor
 ithm seeks to calculate the eigenvalues and eigenvectors of a matrix. The 
 basic idea is to perform a QR decomposition\, writing the matrix as a prod
 uct of an orthogonal matrix and an upper triangular matrix\, multiply the 
 factors in reverse order\, and iterate. This algorithm has been hailed as 
 one of the top ten influential algorithms of the 20th century and works ex
 ceedingly well in practice even though rigorous results for non-normal ope
 rators are scarce.\n\nIn this talk I shall extend the QR algorithm to infi
 nite dimensions and discuss theorems on convergence to spectra and eigenve
 ctors. The infinite dimensional case is more difficult for two reasons. Fi
 rst\, the existence of essential spectra stops the algorithm computing all
  of the spectrum – it only computes the extremal parts. This is naturall
 y understood through a link with power iterations. Second\, it is impossib
 le to use shifts to speed up convergence (this is crucial in the implement
 ation of the finite dimensional case\, increasing linear convergence to cu
 bic convergence). Nevertheless\, I shall link the QR algorithm to computat
 ional hierarchies\, showing how it can solve/classify spectral problems an
 d be executed on a finite machine. Numerical examples will also be demonst
 rated where it outperforms standard approaches that are notorious for prov
 iding false solutions. Finally\, I shall discuss some exciting new develop
 ments for a cousin of the QR algorithm - the QL algorithm\, which may help
  solve the shift problem. This talk is based on joint work with Anders Han
 sen.
LOCATION:MR14\, Centre for Mathematical Sciences
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