# 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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1561-4042

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