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 Num. Meth. Prog., 2016, Volume 17, Issue 1, Pages 44–54 (Mi vmp814)

An orthogonal power method of solving the partial eigenproblem for a symmetric nonnegative definite matrix

I. V. Kireev

Institute of Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk

Abstract: An efficient version of the conjugate direction method to find a nontrivial solution of a homogeneous system of linear algebraic equations with a singular symmetric nonnegative definite square matrix is proposed and substantiated. A one-parameter family of one-step nonlinear iterative processes to determine the eigenvector corresponding to the largest eigenvalue of a symmetric nonnegative definite square matrix is also proposed. This family includes the power method as a special case. The convergence of corresponding vector sequences to the eigenvector associated with the largest eigenvalue of the matrix is proved. A two-step procedure is formulated to accelerate the convergence of iterations for these processes. This procedure is based on the orthogonalization in Krylov subspaces. A number of numerial results are discussed.

Keywords: eigenvector, eigenvalue, conjugate direction method, Krylov subspaces.

Full text: PDF file (801 kB)
UDC: 519.614

Citation: I. V. Kireev, “An orthogonal power method of solving the partial eigenproblem for a symmetric nonnegative definite matrix”, Num. Meth. Prog., 17:1 (2016), 44–54

Citation in format AMSBIB
\Bibitem{Kir16} \by I.~V.~Kireev \paper An orthogonal power method of solving the partial eigenproblem for a symmetric nonnegative definite matrix \jour Num. Meth. Prog. \yr 2016 \vol 17 \issue 1 \pages 44--54 \mathnet{http://mi.mathnet.ru/vmp814} 

• http://mi.mathnet.ru/eng/vmp814
• http://mi.mathnet.ru/eng/vmp/v17/i1/p44

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This publication is cited in the following articles:
1. I. V. Kireev, “Ortogonalnye proektory i sistemy lineinykh algebraicheskikh uravnenii”, Sib. zhurn. vychisl. matem., 23:3 (2020), 315–324