Abstract
A modification of the Danilewski method is presented, permitting the solution of the eigenvalue problem for a constant sparse matrix of large order to be reduced to the solution of the same problem for a polynomial matrix of lower order. Certain solution algorithms are proposed for a partial eigenvalue problem for the polynomial matrix. Questions of the realization of the algorithms on a model PRORAB computer are examined.
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Literature cited
D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra, W. H. Freeman and Co., San Francisco-London (1963).
V. B. Khazanov, “A remark on the Danilewski method,” J. Sov. Math.,7, No. 1, 108–110 (1977).
J. E. Dennis, J. F. Traub, and R. P. Weber, “On the matrix polynomial, lambda-matrix and block eigenvalue problem,” CMU-CS-71 (1971), 110, 117 pp.
V. N. Kublanovskaya, “On the application of Newton's method to the determination of eigenvalues of λ-matrices,” Dokl. Akad. Nauk SSSR,188, No. 5, 1004–1005 (1969).
V. N. Kublanovskaya, “Newton's method for the determination of the eigenvalues and eigenvectors of a matrix,” Zh. Vychisl. Mat. Mat. Fiz.,12, No. 6, 1371–1380 (1972).
T. N. Smirnova and N. B. Malyshev, “On a certain approach to the problem orientation of electronic computers,” J. Sov. Math.,7, No. 1, 95–104 (1977).
L. V. Kantorovich, “Perspectives of the large-block approach in applied mathematics, programming, and computing techniques,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., Akad. Nauk SSSR,48, 5–11 (1974).
V. P. Eremina, N. B. Malyshev, T. N. Smirnova, and N. B. Stsiborskaya, “Realization of matrix operations for the PRORAB computer,” J. Sov. Math.,13, No. 2 (1980).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 92–110, 1976.
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Kublanovskaya, V.N., Smirnova, T.N. & Khazanov, V.B. Solving the eigenvalue problem for sparse matrices. J Math Sci 13, 261–275 (1980). https://doi.org/10.1007/BF01296242
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DOI: https://doi.org/10.1007/BF01296242