A unified convergence theory for $LR$ and $QR$ algorithms applied to symmetric eigenvalue problems

Peluso, R. I., and Piazza, G.. "A unified convergence theory for $LR$ and $QR$ algorithms applied to symmetric eigenvalue problems." Bollettino dell'Unione Matematica Italiana 5-B.3 (2002): 561-584. <http://eudml.org/doc/195191>.

@article{Peluso2002,
abstract = {In this paper we consider the eigenvalue problem for positive definite symmetric matrices. Convergence properties for the zero shift $QR$ method and the shift $LR$ Cholesky method both in restoring and in non restoring version are deduced from the convergence properties of triangular matrices sequences. For general matrices we obtain some results on the convergence speed of the Cholesky method as a function of the chosen shift. These results follow from the absolute convergence of numerical series associated to matrices sequences. Concerning this theory we derive also convergence properties of the $QR$ method for the computation of the eigenvalues of normal matrices and of the $QR$ method for the computation of the singular values of complex matrices. For each method, together with the sequences of associated matrices, we consider a convergent sequence of diagonal matrices. Convergence properties of the methods follow since the matrices series defined by the differences of the terms of the two sequences are absolutely convergent.},
author = {Peluso, R. I., Piazza, G.},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {convergence; eigenvalue; positive definite symmetric matrices; zero shift QR method; shifted LR Cholesky method; normal matrices; singular values},
language = {eng},
month = {10},
number = {3},
pages = {561-584},
publisher = {Unione Matematica Italiana},
title = {A unified convergence theory for $LR$ and $QR$ algorithms applied to symmetric eigenvalue problems},
url = {http://eudml.org/doc/195191},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Peluso, R. I.
AU - Piazza, G.
TI - A unified convergence theory for $LR$ and $QR$ algorithms applied to symmetric eigenvalue problems
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/10//
PB - Unione Matematica Italiana
VL - 5-B
IS - 3
SP - 561
EP - 584
AB - In this paper we consider the eigenvalue problem for positive definite symmetric matrices. Convergence properties for the zero shift $QR$ method and the shift $LR$ Cholesky method both in restoring and in non restoring version are deduced from the convergence properties of triangular matrices sequences. For general matrices we obtain some results on the convergence speed of the Cholesky method as a function of the chosen shift. These results follow from the absolute convergence of numerical series associated to matrices sequences. Concerning this theory we derive also convergence properties of the $QR$ method for the computation of the eigenvalues of normal matrices and of the $QR$ method for the computation of the singular values of complex matrices. For each method, together with the sequences of associated matrices, we consider a convergent sequence of diagonal matrices. Convergence properties of the methods follow since the matrices series defined by the differences of the terms of the two sequences are absolutely convergent.
LA - eng
KW - convergence; eigenvalue; positive definite symmetric matrices; zero shift QR method; shifted LR Cholesky method; normal matrices; singular values
UR - http://eudml.org/doc/195191
ER -