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. ,., ,  , .,. ., ,, [ ] Ai Bi Yi+1 = Wi Yi, i = 0, 1,...,, N, Wi =, (1) Ci D i 2n 2n :

[ ] 0I WiT JWi = J, J =, i = 0,..., N.

I (1)  :,  2n, n N,  Sturm C. Mmoire sur une classe d'Equations dirences partielles // J. Math. Pures Appl. 1836.

Vol. 1. P. 373444.

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Y (t) = [X(t)T U (t)T ]T (2),. 1967. 508.

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Pp. 127135.

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, 1998. 351.

$ Dol O., Kratz W. Oscillation theorems for symplectic dierence systems // Journal of Dierence Equations and Applications. 2007. no. 13. Pp. 585605.

(1962), F. Atkinson (1964), W. Reid (1980)), & Greenberg L., Marletta M. Numerical methods for higher order SturmLiouville problems // Journal of Computational and Applied Mathematics. 2000. Vol. 125. Pp. 367383.

' Marletta M. Numerical solution of eigenvalue problems for Hamiltonian systems // Advances in Computational Mathematics. 1994. Vol. 2, no. 2. Pp. 155. 2001.. 41.. 2938.

MATLAB.

( 07-07-00213).

"(, 1998;, 2000); 6th International Conference on Dierence Equations (Augsburg, Germany, 2001); V International congress on mathematical modelling (Dubna, 2002); VI - (, 2003); 8th International Conference on Dierence Equations and Applications, ICDEA 2003 (Brno, Czech Republic, 2003); VI International congress on mathematical modeling (N. Novgorod, 2004); 10th International Conference on Dierence Equations and Applications, ICDEA 2005 (Munich, Germany, 2005); European Advanced Studies Conference, EASC7 (Homburg, Germany, 2006);12th International Conference on Dierence Equations and Applications, ICDEA 2007, (Lisbon, 2007); Progress on dierence equations, International conference PODE 2008 (Laufen/Salzach and Salzburg, Germany, 2008); "  "(, 2008); 14th International Conference on Dierence Equations and Applications, ICDEA 2009 (Estoril, Portugal, 2009);

"(, 2011);

, : [1]-[3],[5],[8],[10]-[12], :[4],[6]-[7],[9]  "Web of Science: Citation Index Expanded"(   Kratz W. Quadratic Functionals in Variational Analysis and Control Theory. Berlin: Akademie Verlag, 1995. 293 p.

(3):

3. k (Z[0 I]T, Z[0 I]T ) = (Z 1 [0 I]T, Z 1 Z[0 I]T ), k = 1, 2.

4. 2 (Z[0 I]T, Z[0 I]T ) = (Z[0 I]T, Z[0 I]T ), 7. 0 (Y, Y ) min(rangw, rangX), w (W Z[0 I]T, W Z[0 I]T ) = (Z[0 I]T, Z[0 I]T ) + (W Z[0 I]T, W [0 I]T ) (12), 3. (W, W ) = rang(B) rang(B) + (W, W ) (W Z[0 I]T, W Z[0 I]T ) = (Z[0 I]T, Z[0 I]T ) + (W Z[0 I]T, W [0 I]T ) (1).

10 ().

11 ().

, Yi = Zi [0, I]T, Yi = Zi [0, I]T.

() (17),(19).

 Bohner M., O. Dol, Kratz W. Sturmian and spectral theory for discrete symplectic systems // Transactions of the American Mathematical Society. 2009. Vol. 361. Pp. 31093123.

! Bohner M., O. Dol, Kratz W. Inequalities and asymptotics for Riccati matrix dierence operators // Journal of Mathematical Analysis and Applications. 1998. Vol. 221, no. 1. Pp. 262286.

[i, i + 1).

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Dol O., Kratz W. A Sturmian separation theorem for symplectic dierence systems // Journal of Mathematical Analysis and Applications. 2007. Vol. 325. Pp. 333341.

(2) B(t) 0, C(t) 0, Y (t), JY (t)  ind(AT Ci ) = ind(Ai BiT ) i, Yi, JYi  (a, b] $ Teschl G. Jacobi Operators and Completely Integrable Nonlinear Lattices. Providence: AMS Mathematical Surveys and Monographs, 1999. 351 p.

(23), ZN +1 (b)[0 I] = Z0 (a)[0 I] = [0 I].

#(Z (a), Z p = l (Y R0 R0 = R0 R0 T, RN +1 RN +1 = RN +1 RNT, rang[R0 R0 ] = rang[RN +1 RN +1 ] N0 [0, N + 1].

24 (31), (32).

//. 1999.. 2.. 1525.

 Liu Xue-Shen, Qi Yue-Ying, He Jian-Feng, Ding Pei-Zhu. Recent Progress in Symplectic Algorithms for Use in Quantum Systems // Communications in computational physics. 2007. Vol. 2, no. 1. Pp. 153.

10 1.280672477702526e-004 2.242991865505127e- 103 1.280810673797161e-006 2.242029906268870e- 104 1.280812403183678e-008 5.326204188804695e- (35) Qj(i) = Uj(i) X Ci Qj(i+1) Ai + Di Qj(i) Qj(i+1) Bi Qj(i) = 0, Wi = NT [Vi ], (38).

(40), (42). 2330.

. 1999.. 39, 2.. 187194.

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// J. Elyseeva. On oscillation and nonoscillation domains for dierence Riccati equation // Mathematical Models of Non-Linear Excitations, Transfer, Dy namics and Control in Condensed Systems and Other Media / Ed. by Uvaro va L.A., Arinstein A.E., Latyshev A. V. Kluwer Academic/Plenum publishers, 1999. Pp. 157169.

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20. Elyseeva J. Symplectic Factorizations and the Denition of a Focal Point // Proceedings of the Eighth International Conference on Dierence Equations and Applications, 2003 / Ed. by Elaydi S., Aulbach B., Ladas G. Boca Raton:

Chartman Hall/CRC, 2003. Pp. 127135.

21. Elyseeva J. A transformation for the Riccati dierence operator // Proceed ings of the Sixth International Conference on Dierence Equations, 2001 / Ed. by Elaydi S., Aulbach B., Ladas G. Boca Raton: Chartman Hall/CRC, 2005. Pp. 417424.

22. Elyseeva J. The comparative index for conjoined bases of symplectic dierence systems // Dierence equations, Special functions and Orthogonal polinomi als, Proceedings of the International Conference Munich, Germany, 25  July 2005 / Ed. by Elaydi S., Cushing J., Lasser R., all. Singapore: World Scientic, 2007. Pp. 168177.

23. Schmeidel E., Elyseeva J. Generalized Kiguradze's lemma on time scales with application for oscillation of higher order nonlinear dynamic equations // .....: -, 2008.. 11.. 1821.

24. Elyseeva J. Index results in Sturmian Theory of symplectic dierence 27. Elyseeva J. The comparative index and the number of focal points for con joined bases of symplectic dierence systems // Discrete dynamics and dif ference equations, Proceedings of the Twelfth International Conference on Dierence Equations and Applications / Ed. by Elaydi S., Oliveira H., Fer reira J., Alves J. Singapore: World Scientic, 2010. Pp. 231238.

28. Elyseeva J., Bondarenko A. Calculating eigenvalues of discrete fourth order Sturm-Liouville problems // Mathematical Models of Non-linear Phenomena, Processes and Systems: From Molecular Scale to Planetary Atmosphere / Ed.

by Nadykto A. B., Uvarova L.A., Latyshev A. V. Nova Science Publishers, 2010. Pp. 443452.



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