## Abstract

We consider the computing of an eigenpair (an eigenvector \(v=(v^{(i)})_{i=1,n}\) and an eigenvalue \(\lambda\)) of a matrix \(A\in\mathbb{R}^{n\times n}\), by considering a supplementary condition (we call it norming function for the eigenvector), represented by a polynomial of degree 2. A usual choice is

\begin{equation}

G(v):={\textstyle\frac12} \sum_{i=1}^n( v^{(i)})^2-1=0.

\end{equation}

We propose here a new choice:

\begin{equation}

G(v):={\textstyle\frac1{2n}} \sum_{i=1}^n( v^{(i)})^2-1=0,

\end{equation}

which has the advantage that leads to a smaller nonlinearity.

Indeed, for the \(n+1\)-dimensional nonlinear system (a polynomial equation of degree 2) we obtain:

\[

F\left( x\right) :={{Av-\lambda v} \choose {G(v)-1}}

=0,

\]

and our proposed choice leads to a second derivative of \(F\) that has norm 2 (compared to \(n\), for the usual choice).

We consider this problem in a general setting, of a linear continuous operator \(A:V\rightarrow V\), \(V\) Banach space, which leads to a polynomial equation of degree two.

We study the semilocal convergence of an iterative method of Schultz type for this problem; this method has the advantage that does not require the solving of a linear system at each iteration step:

\begin{align*}

x_{k+1} =&x_k-\Gamma_kF\left( x_k\right) \\

\Gamma_{k+1} =&\Gamma_k\left( 2I-F^{\prime }\left( x_{k+1}\right) \Gamma_k\right) , \qquad k=0,1,\ldots,

\end{align*}

where \(x_0\in X\), \(\Gamma_0\in \mathcal{L}\left( X\right)\), and \(I\) is the identity operator of \(\mathcal{L}\left( X\right)\).

We obtain semilocal convergence conditions which show that the method has r-convergence order 2.

## Authors

E. **Cătinaş**

(Tiberiu Popoviciu Institute of Numerical Analysis)

I. **Păvăloiu**

(Tiberiu Popoviciu Institute of Numerical Analysis)

## Keywords

linear continuous operator; Banach space; Newton type method; eigenvector; eigenvalue; eigenpair; Schultz method; r-convergence order.

## Cite this paper as:

E. Cătinaş, I. Păvăloiu, *On approximating the eigenvalues and eigenvectors of linear continuous operators*, Rev. Anal. Numér. Théor. Approx., 26 (1997) nos. 1-2, pp. 19-27.

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## About this paper

##### Publisher Name

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##### Print ISSN

1222-9024

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##### ZBL

2457-8126

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