Naslov
Asymptotic Behavior of Scaled Iterates by Diagonalization Methods

Autori
Matejaš, Josip ; Hari, Vjeran

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Book of abstracts / - , 2002

Skup
IV International Workshop on Accurate Solution of Eigenvalue Problems

Mjesto i datum
Split, Hrvatska, 24.06.2002. - 27.06.2002

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Diagonalization methods; asymptotic convergence; scaled iterates

Sažetak
Diagonalization methods for solving eigenvalue and singular value problems have been lately reconsidered for their accuracy properties. The usual measures of advancing of the processes and the corresponding stopping criteria have been replaced by ones which warrant that all output data are computed with possibly highest relative accuracy. For several diagonalization methods such as: symmetric and Hermitian Jacobi methods, J-symmetric eigenvalue method and Kogbetliantz method for computing SVD of triangular matrices, new improved measures of convergence have been found. When divided by suitably defined relative gaps in the set of eigenvalues or singular values, they actually measure relative distance between eigenvalues or singular values and the corresponding diagonal elements of (scaled) almost diagonal matrices. They also appear in some way in perturbation estimates for eigenvectors or singular vectors. For all mentioned methods, these measures have form $\|DHD\|_F$ where $D$ is diagonal matrix which makes diagonals of $|DHD|$ ones. When one-sided versions of diagonalization methods are used, computing of $\|DHD\|_F$ is costly. It requires around $n^2/2$ dot products. Hence it is of interest to find a quadratic reduction rule for the new measure. When eigenvalues/singular values cluster around zero, classical results are useless, since they use absolute gaps which are tiny. Therefore, beside using the new measure of convergence, one would like to obtain quadratic convergence estimates which use relative gaps instead of absolute gaps. Then such results could be used in predicting the number of sweeps till convergence. In all obtained results, quadratic reduction of the ratio $\|DHD\|_F/\mbox{rel. gap}$ is sharply estimated for scaled diagonally dominant matrices. Hence, additionally, the logarithm of this ratio measures number of correct digits in all diagonal elements as approximations of eigenvalues or singular values. The results hold in the general case of multiple eigenvalues or singular values.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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