Geodesic
Hermitian Spectral Pseudoinversion and Its Applications
Matematičeskie zametki, Tome 96 (2014) no. 1, pp. 101-115.

Voir la notice de l'article provenant de la source Math-Net.Ru

The present paper is devoted to the Hermitian spectral pseudoinversion and its applications to analysis, the solution and reduction of Hermitian differential-algebraic systems. New explicit formulas for the solutions of such systems and the solutions of related generalized Lyapunov equations are proposed. Attainable upper bounds for the norms of the solutions are obtained. A realization of the balanced truncation method not requiring computations involving projections onto reducing subspaces is proposed.
Keywords: Hermitian spectral pseudoinversion, Hermitian differential-algebraic system, Lyapunov equation, balanced truncation method, reducing subspace, matrix pencil, initial value problem, controllability Gramian, observability Gramian. 
@article{MZM_2014_96_1_a8,
     author = {Yu. M. Nechepurenko},
     title = {Hermitian {Spectral} {Pseudoinversion} and {Its} {Applications}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {101--115},
     publisher = {mathdoc},
     volume = {96},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_96_1_a8/}
}
TY  - JOUR
AU  - Yu. M. Nechepurenko
TI  - Hermitian Spectral Pseudoinversion and Its Applications
JO  - Matematičeskie zametki
PY  - 2014
SP  - 101
EP  - 115
VL  - 96
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2014_96_1_a8/
LA  - ru
ID  - MZM_2014_96_1_a8
ER  - 
%0 Journal Article
%A Yu. M. Nechepurenko
%T Hermitian Spectral Pseudoinversion and Its Applications
%J Matematičeskie zametki
%D 2014
%P 101-115
%V 96
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2014_96_1_a8/
%G ru
%F MZM_2014_96_1_a8
Yu. M. Nechepurenko. Hermitian Spectral Pseudoinversion and Its Applications. Matematičeskie zametki, Tome 96 (2014) no. 1, pp. 101-115. http://geodesic.mathdoc.fr/item/MZM_2014_96_1_a8/

[1] G. H. Golub, C. F. Van Loan, Matrix Computations, The John Hopkins Univ. Press, London, 1991

[2] S. K. Godunov, Sovremennye aspekty lineinoi algebry, Nauchnaya kniga, Novosibirsk, 1997

[3] Yu. M. Nechepurenko, G. V. Ovchinnikov, “Verkhnie otsenki norm reshenii ermitovykh sistem obyknovennykh differentsialnykh i algebraicheskikh uravnenii”, Ufimsk. matem. zhurn., 1:4 (2009), 125–132 | Zbl

[4] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, 1986

[5] M. Celik, L. Pileggi, A. Odabasioglu, IC Interconnect Analysis, Kluwer Acad. Publ., Norwell, MA, 2002

[6] B. C. Moore, “Principal component analysis in linear systems: controllability, observability, and model reduction”, IEEE Trans. Automat. Control, 26:1 (1981), 17–32 | DOI | MR | Zbl

[7] K. Glover, “All optimal Hankel-norm approximations of linear multivariable systems and their $L^\infty$-error bounds”, Internat. J. Control, 39:6 (1984), 1115–1193 | DOI | MR | Zbl

[8] T. Stykel, “Balanced truncation model reduction for semidiscretized Stokes equation”, Linear Algebra Appl., 415:2-3 (2006), 262–289 | DOI | MR | Zbl

[9] M. Heinkenschloss, D. C. Sorensen, K. Sun, “Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations”, SIAM J. Sci. Comput., 30:2 (2008), 1038–1063 | DOI | MR | Zbl

[10] I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Comput. Sci. Appl. Math., Academic Press, New York, 1982 | MR | Zbl

[11] P. K. Suetin, Klassicheskie ortogonalnye mnogochleny, Nauka, M., 1979 | MR

[12] Templates for the Solution of Algebraic Eigenvalue Problems, Software Environ. Tools, 11, eds. Zh. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. Van der Vorst, SIAM, Philadelphia, PA, 2000 | MR | Zbl

[13] Yu. M. Nechepurenko, G. V. Ovchinnikov, “An estimation of voltage settling time for RC-circuits”, Russ. J. Numer. Anal. Math. Modelling, 25:3 (2010), 253–259 | DOI | MR | Zbl

[14] J.-R. Li, J. White, “Low-rank solution of Lyapunov equations”, SIAM Rev., 46:4 (2004), 693–713 | MR | Zbl

[15] V. Druskin, L. Knizhnerman, V. Simoncini, “Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation”, SIAM J. Numer. Anal., 49:5 (2011), 1875–1898 | DOI | MR | Zbl