On approximating the eigenvalues and eigenvectors of linear continuous operators

Emil Cătinaş; I Păvăloiu

Authors

Emil Cătinaş Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania
I Păvăloiu Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania

Abstract views: 311

Abstract

We consider the computing of an eigenpair (an eigenvector $v = (v^{(i)})_{i = 1, n}$ and an eigenvalue $λ$ ) of a matrix $A \in R^{n \times n}$ , by considering a supplementary condition (we call it norming function for the eigenvector), represented by a polynomial of degree 2. A usual choice is
$G (v) := \frac{1}{2} \sum_{i = 1}^{n} (v^{(i)})^{2} - 1 = 0.$
We propose here a new choice:
$G (v) := \frac{1}{2 n} \sum_{i = 1}^{n} (v^{(i)})^{2} - 1 = 0,$
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:
$F (x) := (\binom{A v - λ v}{G (v) - 1}) = 0,$
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 $A : V \to V$ , $V$ 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:
$\begin{aligned} x_{k + 1} = & x_{k} - Γ_{k} F (x_{k}) \\ Γ_{k + 1} = & Γ_{k} (2 I - F^{'} (x_{k + 1}) Γ_{k}), k = 0, 1, \dots, \end{aligned}$
where $x_{0} \in X$ , $Γ_{0} \in L (X)$ , and $I$ is the identity operator of $L (X)$ .

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

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