Editing Schur decomposition (section)

== Computation ==
The Schur decomposition of a given matrix is numerically computed by the [[QR algorithm]] or its variants. In other words, the roots of the [[characteristic polynomial]] corresponding to the matrix are not necessarily computed ahead in order to obtain its Schur decomposition. Conversely, the [[QR algorithm]] can be used to compute the roots of any given [[characteristic polynomial]] by finding the Schur decomposition of its [[companion matrix]]. Similarly, the [[QR algorithm]] is used to compute the eigenvalues of any given matrix, which are the diagonal entries of the upper triangular matrix of the Schur decomposition. Although the [[QR algorithm]] is formally an infinite sequence of operations, convergence to machine precision is practically achieved in [[Big O notation|<math>\mathcal{O}(n^3)</math>]] operations.<ref>{{Cite book|last1=Trefethen|first1=Lloyd N.|url=https://www.worldcat.org/oclc/36084666 | title=Numerical linear algebra|last2=Bau|first2=David|date=1997|publisher=Society for Industrial and Applied Mathematics |isbn=0-89871-361-7 |location=Philadelphia|pages=193–194|oclc=36084666}}</ref>
See the Nonsymmetric Eigenproblems section in [[LAPACK]] Users' Guide.<ref>{{cite book| last1=Anderson|first1=E| last2=Bai|first2=Z| last3=Bischof|first3=C| last4=Blackford|first4=S| last5=Demmel|first5=J| last6=Dongarra|first6=J| last7=Du Croz|first7=J| last8=Greenbaum|first8=A| last9=Hammarling|first9=S| last10=McKenny|first10=A| last11=Sorensen|first11=D| title=LAPACK Users guide| date=1995| publisher=Society for Industrial and Applied Mathematics| location=Philadelphia, PA| isbn=0-89871-447-8| url=http://www.netlib.org/lapack/lug/}}</ref>