Geodesic
A gradient descent method for solving of one class of nonlinear multiparameter eigenvalue problems
Trudy Instituta matematiki, Tome 22 (2014) no. 1, pp. 122-130.

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In the real Hilbert space the nonlinear multiparameter spectral problem is put in accordance to the variation problem on zero minimum of some functional. The equivalence of spectral and variation problems is proved. On the base of gradient procedure the numerical algorithm of finding its eigenvalues and eigenvectors is proposed. The local convergence of this algorithm is proved. The practical application of the algorithm is illustrated on example of nonlinear two-parameter eigenvalue problem. 
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V. V. Khlobystov; B. M. Podlevskyi; O. S. Yaroshko. A gradient descent method for solving of one class of nonlinear multiparameter eigenvalue problems. Trudy Instituta matematiki, Tome 22 (2014) no. 1, pp. 122-130. http://geodesic.mathdoc.fr/item/TIMB_2014_22_1_a10/

[1] Abramov A. A., Ul'yanova V. I., Yukhno L. F., “A method for solving the multiparameter eigenvalue problem for certain systems of differential equations”, Comput. Math. Meth. Phys., 40:1 (2000), 18–26

[2] Atkinson F. V., Multiparameter Eigenvalue Problems. Matrices and Compact Operators, v. 1, Academic Press, New York–London, 1972

[3] Blum E. K., Curtis A. R., “A Convergent Gradient Method for Matrix Eigenvector-Eigentuple Problems”, Numer. Math., 31:3 (1978), 247–263 | DOI

[4] Browne P. J., Sleeman B. D., “A numerical technique for multiparameter eigenvalue problems”, IMA J. Numer Anal., 2:3 (1982), 451–457 | DOI

[5] Khlobystov V. V., Podlevskyi B. M., “Numerical method of finding bifurcation points of linear two-parameter eigenvalue problems”, Comput. Meth. Appl. Math., 9:4 (2009), 332–338 | DOI

[6] Khlobystov V. V., Podlevskyi B. M., “Variation-gradient method of the solution of one class of nonlinear multiparameter eigenvalue problems”, J. Numer. Appl. Math., 2009, no. 1(97), 70–78

[7] Müller R. E., “Numerical Solution of Multiparameter Eigenvalue Problems”, ZAMM, 62:12 (1982), 681–686 | DOI

[8] Podlevskyi B. M., “A variational aproach for solving the linear multiparameter eigenvalue problems”, Ukrainian Math. Journal, 61:9 (2009), 1247–1256

[9] Podlevskyi B. M., “On some nonlinear two-parameter spectral problems of mathematical physics”, Mathematical modeling, 22:5 (2010), 131–145

[10] Podlevskyi B. M., Khlobystov V. V., “A gradient method for solving the nonlinear multiparameter spectral problems”, Reports NAS of Ukraine, 2012, no. 8, 22–27 (in Ukraine)

[11] Podlevskyi B. M., Khlobystov V. V., “On one approach to finding eigenvalue curves of linear two-parameter spectral problems”, J. Mathematical Sciences, 167:1 (2010), 96–106 | DOI

[12] Sleeman B. D., Multiparameter spectral theory in Hilbert space, Pitnam Press, London–San Francisco–Melbourne, 1978

[13] Volkmer H., Multiparameter eigenvalue problems and expansion theorem, Lect. Notes Math., 1336, 1988

[14] Berezin I. S., Zydkov N. P., Calculation methods, v. 1, 3-th edition, Nauka, M., 1966