Abstract
We obtain exact (unimprovable) estimates for the rate of convergence of the s-step method of steepest descent for finding the least (greatest) eigenvalue of a linear bounded self-adjoint operator in a Hilbert space.
References
L. V. Kantorovich, “Functional analysis and applied mathematics,” Usp. Mat. Nauk 3, No. 6, 89–184 (1948).
M. Sh. Birman, “Calculation of eigenvalues by the method of steepest descent,” Zap. Leningrad. Gorn. Inst., 27, No. 1, 209–215 (1952).
A. B. Kovrigin, “Estimation of the rate of convergence of the k-step gradient method,” Vestn. Leningrad. Univ., Issue 13, 34–36 (1970).
V. G. Prikazchikov, “Strict estimates of the rate of convergence of the iteration method for the calculation of eigenvalues,” Zh. Vychisl. Mat. Mat. Fiz. 15, No. 5, 1330–1333 (1975)
A. V. Knyazev, Calculation of Eigenvalues and Eigenvectors in Network Problems: Algorithms and Estimates of Error [in Russian], Department of Computational Mathematics, Academy of Sciences of the USSR, Moscow (1986).
A. F. Zabolotskaya, “On a method of steepest descent,” in: Communications on Applied Mathematics [in Russian], Computer Center, Academy of Sciences of the USSR, Moscow (1988), p. 26.
P. F. Zhuk, “Asymptotic behavior of the s-step method of steepest descent in eigenvalue problems in a Hilbert space,” Mat. Sb. 184, No. 12, 87–122 (1993).
V. S. Charin, Linear Transformations and Convex Sets [in Russian], Vyshcha Shkola, Kiev (1978).
Additional information
Krupskaya Kherson Pedagogic Institute, Kherson. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1694–1699, December, 1997.
Rights and permissions
About this article
Cite this article
Zhuk, P.F., Bondarenko, L.N. Exact estima tes for the rate of convergence of the s-step method of steepest descent in eigenvalue problems. Ukr Math J 49, 1912–1918 (1997). https://doi.org/10.1007/BF02513070
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02513070