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
\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.