Editing QR algorithm (section)

== History ==
The QR algorithm was preceded by the ''LR algorithm'', which uses the [[LU decomposition]] instead of the QR decomposition. The QR algorithm is more stable, so the LR algorithm is rarely used nowadays. However, it represents an important step in the development of the QR algorithm.

The LR algorithm was developed in the early 1950s by [[Heinz Rutishauser]], who worked at that time as a research assistant of [[Eduard Stiefel]] at [[ETH Zurich]]. Stiefel suggested that Rutishauser use the sequence of moments ''y''<sub>0</sub><sup>T</sup> ''A''<sup>''k''</sup> ''x''<sub>0</sub>, ''k'' = 0, 1, ... (where ''x''<sub>0</sub> and ''y''<sub>0</sub> are arbitrary vectors) to find the eigenvalues of ''A''. Rutishauser took an algorithm of [[Alexander Aitken]] for this task and developed it into the ''quotient&ndash;difference algorithm'' or ''qd algorithm''. After arranging the computation in a suitable shape, he discovered that the qd algorithm is in fact the iteration ''A''<sub>''k''</sub> = ''L''<sub>''k''</sub>''U''<sub>''k''</sub> (LU decomposition), ''A''<sub>''k''+1</sub> = ''U''<sub>''k''</sub>''L''<sub>''k''</sub>, applied on a tridiagonal matrix, from which the LR algorithm follows.<ref>{{Citation | last1=Parlett | first1=Beresford N. | last2=Gutknecht | first2=Martin H. | title=From qd to LR, or, how were the qd and LR algorithms discovered? | doi=10.1093/imanum/drq003 | year=2011 | journal=IMA Journal of Numerical Analysis | issn=0272-4979 | volume=31 | issue=3 | pages=741–754| hdl=20.500.11850/159536 | url=http://doc.rero.ch/record/299139/files/drq003.pdf }}</ref>