Geodesic
Localization of the eigenvalues of a pencil of positive definite matrices
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 11, pp. 1923-1931 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $A$ and $B$ be real square positive definite matrices close to each other. A domain $S$ on the complex plane that contains all the eigenvalues $\lambda$ of the problem $Az=\lambda Bz$ is constructed analytically. The boundary $\partial S$ of $S$ is a curve known as the limacon of Pascal. Using the standard conformal mapping of the exterior of this curve (or of the exterior of an enveloping circular lune) onto the exterior of the unit disc, new analytical bounds are obtained for the convergence rate of the minimal residual method (GMRES) as applied to solving the linear system $Ax=b$ with the preconditioner $B$. 
@article{ZVMMF_2008_48_11_a0,
     author = {I. E. Kaporin},
     title = {Localization of the eigenvalues of a~pencil of positive definite matrices},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1923--1931},
     year = {2008},
     volume = {48},
     number = {11},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_11_a0/}
}
TY  - JOUR
AU  - I. E. Kaporin
TI  - Localization of the eigenvalues of a pencil of positive definite matrices
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2008
SP  - 1923
EP  - 1931
VL  - 48
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_11_a0/
LA  - ru
ID  - ZVMMF_2008_48_11_a0
ER  - 
%0 Journal Article
%A I. E. Kaporin
%T Localization of the eigenvalues of a pencil of positive definite matrices
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2008
%P 1923-1931
%V 48
%N 11
%U http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_11_a0/
%G ru
%F ZVMMF_2008_48_11_a0
I. E. Kaporin. Localization of the eigenvalues of a pencil of positive definite matrices. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 11, pp. 1923-1931. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_11_a0/

[1] Kaporin I., “High quality preconditioning of a general symmetric positive matrix based on its $U^\mathrm TU+U^\mathrm TR+R^\mathrm TU$-decomposition”, Numer. Linear Algebra Appl., 5 (1998), 484–509 | 3.0.CO;2-7 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[2] Saad Y., Schultz M. H., “GMRES: A generalized minimal residual algorithm for solving nonsymmetric systems of linear equations”, SIAM J. Scient. and Statist. Comput., 7 (1986), 856–869 | DOI | MR | Zbl

[3] Faddeev D. K., Faddeeva V. H., Vychislitelnye metody lineinoi algebry, Lan, SPb., 2002

[4] Suetin P. K., Ryady po mnogochlenam Fabera, Nauka, M., 1984 | MR | Zbl

[5] Beckermann B., Goreinov S. A., Tyrtyshnikov E. E., “Some remarks on the Elman estimate for GMRES”, SIAM J. Matrix Analys. and Appl., 27 (2006), 772–778 | DOI | MR | Zbl

[6] Gurvits A., Kurant P., Teoriya funktsii, Nauka, M., 1968 | MR

[7] Szegő G., “Conformai mapping of the interior of an ellipse onto a circle”, Amer. Math. Monthly, 57 (1950), 474–478 | DOI | MR | Zbl

[8] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. Ellipticheskie i avtomorfnye funktsii. Funktsii Lame i Mate, Nauka, M., 1967 | MR

[9] Kantorovich L. V., Krylov V. I., Priblizhennye metody vysshego analiza, Gostekhteorizdat, M., L., 1950 | Zbl