Remarks on some Newton and Chebyshev-type methods for approximation eigenvalues and eigenvectors of matrices

Abstract

We consider a square matrix \(A\) with real or complex elements. We denote \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C}\) and we are interested in computing \(\lambda \in \mathbb{K}\) such that there exists \(v\in \mathbb{K}^{n}\) such that \(Av-\lambda v=0\), i.e. we are interested in computing the eigenpairs (eigenvalue +eigenvector) of the matrix \(A\). In this sense, we consider the nonlinear system of equations \(F(x) =0\), where \(F(x) =\)\({Av-\lambda v}\choose{Gv-1}\), where \(G\) is a convenient mapping.

In order to solve this system we consider the Newton and the Chebyshev methods, and at each iteration step, the order 1 derivative is approximated by the Schultz method; such an approach does not require the solving of a linear system at each step.

We conditions for local convergence and errors evaluations for the r-convergence order.

Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis)

Keywords

eigenvalue and eigenvector of square matrix; eigenpair; Newton method; Chebyshev method; Schultz method; local convergence theorem; error estimation; linear systems solving-free iterative methods; r-convergence order.

Cite this paper as:

I. Păvăloiu, E. Cătinaş, Remarks on some Newton and Chebyshev-type methods for approximation eigenvalues and eigenvectors of matrices, Comput. Sci. J. Mold., 7 (1999) no. 1, pp. 3-15.

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References

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