On approximating the eigenvalues and eigenvectors of linear continuous operators

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

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

We propose here a new choice:

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

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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