Editing QR algorithm (section)

{{Short description|Algorithm to calculate eigenvalues}}
In [[numerical linear algebra]], the '''QR algorithm''' or '''QR iteration''' is an [[eigenvalue algorithm]]: that is, a procedure to calculate the [[eigenvalues and eigenvectors]] of a [[Matrix (mathematics)|matrix]]. The QR algorithm was developed in the late 1950s by [[John G. F. Francis]] and by [[Vera N. Kublanovskaya]], working independently.<ref>J.G.F. Francis, "The QR Transformation, I", ''[[The Computer Journal]]'', '''4'''(3), pages 265&ndash;271 (1961, received October 1959). [[doi:10.1093/comjnl/4.3.265]]</ref><ref>{{cite journal |first=J. G. F. |last=Francis |title=The QR Transformation, II |journal=The Computer Journal |volume=4 |issue=4 |pages=332–345 |year=1962 |doi=10.1093/comjnl/4.4.332 |doi-access=free }}</ref><ref>
Vera N. Kublanovskaya, "On some algorithms for the solution of the complete eigenvalue problem," ''USSR Computational Mathematics and Mathematical Physics'', vol. 1, no. 3, pages 637–657 (1963, received Feb 1961). Also published in: ''Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki'', vol.1, no. 4, pages 555–570 (1961). [[doi:10.1016/0041-5553(63)90168-X]]</ref>  The basic idea is to perform a [[QR decomposition]], writing the matrix as a product of an [[orthogonal matrix]] and an upper [[triangular matrix]], multiply the factors in the reverse order, and iterate.