Naslov
Accuracy and stability in numerical linear algebra

Autori
Drmač, Zlatko

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, cjeloviti rad (in extenso), pregledni

Izvornik
Proceedings of the Conference Applied Mathematics and Computation, Dubrovnik, Croatia, September 13--18, 1999. / Rogina, Mladen ; Hari, Vjeran ; Limić, Nedžad ; Tutek, Zvonimir - Zagreb : Matematički odsjek Prirodoslovno-matematičkog fakulteta Sveučilišta u Zagrebu, 2001, 1-8

Skup
Applied Mathematics and Computation

Mjesto i datum
Dubrovnik, Hrvatska, 13.09.1999. - 18.09.1999

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
accuracy; eigenvalues; singular values

Sažetak
We give an overview of our recent work on ccurate computation of generalized eigenvalue and ingular value decompositions of matrix pairs and triplets. Our goal is to develop efficient and highly accurate algorithms and to produce high quality mathematical software. Using error analysis and perturbation theory, we develop templates for accurate floating point computation of the product and quotient induced singular value decompositions, canonical correlations and diagonalization of symmetric pencils $H-\lambda M$, $HM-\lambda I$ with positive definite $H$, $M$. The new algorithms are numerically robust. For instance, the eigenvalues of $H-\lambda M$ and $HM-\lambda I$ are computed with optimal relative error bound: each eigenvalue $\lambda$ is computed with relative error $|\delta\lambda|/\lambda$ of order (up to a factor of the dimension) ${\bf u}\{ \min_{\Delta\in{\cal D}}\kappa_2(\Delta H\Delta) + \min_{\Delta\in{\cal D}}\kappa_2(\Delta M\Delta)\}$, where ${\bf u}$ is the roundoff unit and ${\cal D}$ is the set of nonsingular diagonal matrices. Moreover, the backward error is element-wise small in the sense that finite precision computation corresponds to an exact computation with $H+\delta H$, $M+\delta M$, where for all $i, j$ and some moderate functions $f()$ and $g()$, $|\delta H_{ij}|\leq f(n){\bf u} \sqrt{H_{ii}H_{jj}}$, $|\delta M_{ij}|\leq g(n) {\bf u}\sqrt{M_{ii}M_{jj}}$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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