Geodesic
On some applications of bilateral orthogonalization in computational algebra
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 187-205.

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

In this article it is proved that the equations of sequential solution of a number of computational algebra problems are the consequences of equations of counter orthogonalization and biorthogonalization in Hilbert and Euclidean spaces. The basis of these equations is the known sequential method of direct Gram–Sonin–Schmidt orthogonalization. It is considered the problems related to matrix inversions, their triangular factorizations, and solving systems of linear algebraic equations.
Keywords: Gram–Sonin–Schmidt orthogonalization, bilateral orthogonalization, triangular factorization, general matrix inverse, least square method, innovation process, Kalman filter.
Mots-clés : Frobenius formula 
@article{SEMR_2019_16_a82,
     author = {A. O. Egorshin},
     title = {On some applications of bilateral orthogonalization in computational algebra},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {187--205},
     publisher = {mathdoc},
     volume = {16},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2019_16_a82/}
}
TY  - JOUR
AU  - A. O. Egorshin
TI  - On some applications of bilateral orthogonalization in computational algebra
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2019
SP  - 187
EP  - 205
VL  - 16
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2019_16_a82/
LA  - ru
ID  - SEMR_2019_16_a82
ER  - 
%0 Journal Article
%A A. O. Egorshin
%T On some applications of bilateral orthogonalization in computational algebra
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2019
%P 187-205
%V 16
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2019_16_a82/
%G ru
%F SEMR_2019_16_a82
A. O. Egorshin. On some applications of bilateral orthogonalization in computational algebra. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 187-205. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a82/

[1] I.M. Glazman, Ju.I. Ljubic, Finite-Dimensional Linear Analysis, Nauka, M., 1969 (in Russian) | MR

[2] V.G. Bardakov, M.V. Neshchadim, Yu.V. Sosnovsky, “Groups of triangular automorphisms of a free associative algebra and a polynomial algebra”, Journal of Algebra, 362:1 (2012), 201–220 | DOI | MR | Zbl

[3] A.O. Egorshin, “On counter orthogonalization processes”, Numerical Analysis and Applications, 5:4 (2012), 307–319 | DOI | MR | Zbl

[4] I.C. Gohberg, M.G. Krein, Theory and applications of Volterra operators in Hilbert space, Amer. Math. Soc., Providence, 1970 | MR | Zbl

[5] A.O. Egorshin, “Parameters optimization of the stationary models in unitary space”, Automation and Remote Control, 65:12 (2004), 1734–1756 | DOI | MR

[6] A.O. Egorshin, “On a method for estimating of coefficients of simulating equation for sequences”, Sibirskii Zhurnal Industrial'noy Matematiki, 3:2 (2000), 78–96 (in Russian) | MR | Zbl

[7] A.O. Egorshin, “On tracking extremum parameters in the identification variational problem”, Journal of Mathematical Sciences, 195:6 (2013), 791–804 | DOI | MR

[8] A.O. Egorshin, “On one variational smoothing problem”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 4 (2011), 9–22 (in Russian) | DOI | Zbl

[9] A.O. Egorshin, “On one variational problem of the piecewise-linear dynamical approximation”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 4 (2012), 30–45 (in Russian) | DOI | Zbl

[10] R.C.K. Lee, Optimal Estimation, Identification and Control, MIT Press, Cambridge, 1964 | Zbl

[11] F.R. Gantmacher, The Theory of Matrices, Chelsea Publishing Company, New York, 1959 | MR

[12] P. Ye. Elyasberg, Definition of the Motion via Measurement Result, URSS, M., 2011 (in Russian)

[13] T. Kailath, “An innovation approach to least-square estimation, part I: Linear filtering in additive white noise”, IEEE Trans. Automat. Contr., AC-13 (1968), 646–655 | DOI | MR

[14] A.S. Householder, The Theory of Matrices in Numerical Analysis, Dover Publication Inc., New York–Toronto–London, 1964 | MR