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
Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis)
Ion 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.
Scanned paper: on the journal website.
Latex-pdf version of the paper. (soon)
About this paper
Publisher Name
Paper on journal website
Print ISSN
1222-9024
Online ISSN
?
MR
?
ZBL
2457-8126
Google Scholar citations
[1] M.P. Anselone, L.B. Rall, The solution of characteristic value-vector problems by Newton method, Numer. Math. 11 (1968), 38–45.
CrossRef
[2] E. Catinas, I. Pavaloiu, On the Chebyshev method for approximating the eigenvalues of linear operators, Rev. Anal. Numer. Theor. Approx. 25 (1996) nos. 1–2, pp. 43–56.
article post, article on the journal website
[3] P.G. Ciarlet, Introduction a l’analyse numerique matricielle et a l’optimisation, Mason, Paris, 1990.
[4] F. Chatelin, Valeurs propres de matrices, Mason, Paris, 1988.
[5] L. Collatz, Functionalanalysis und Numerische Mathematik, Springer-Verlag, Berlin, 1964.
[6] A. Diaconu, On the convergence of an iterative proceeding of Chebyshev type, Rev. Anal. Numer. Theor. Approx. 24 (1995) nos. 1–2, pp. 91–102.
article on the journal website
[7] A. Diaconu, I. Pavaloiu, Sur quelque methodes iteratives pour la resolution des equations operationelles, Rev. Anal. Numer. Theor. Approx. 1 (1972) no. 1, pp. 45–61.
article on the journal website
[8] V.S. Kartısov, F.L. Iuhno, O nekotorıh Modifikatiah Metoda Niutona dlea Resenia Nelineinoi Spektralnoi Zadaci, J. Vıcisl. matem. i matem. fiz. 33 (1973) 9, pp. 1403–1409.
[9] I. Lazar, On a Newton type method, Rev. Anal. Numer. Theor. Approx. 23 (1994) no. 2, pp. 167–174.
article on journal website
[10] I. Pavaloiu, Observations concerning some approximation methods for the solutions of operator equations, Rev. Anal. Numer. Theor. Approx. 23 (1994) no. 2, pp. 185–196.
article on the journal website
[11] I. Pavaloiu, Sur les procedes iteratifs a un ordre eleve de convergence, Mathematica (Cluj) 12 (35) (1970) no. 2, pp. 309–324.
post
[12] R.A. Tapia, L.D. Whitley, The projected Newton method has order 1+√2 for the symmetric eigenvalue problem, SIAM J. Numer. Anal. 25 (1988) no. 6, pp. 1376–1382.
CrossRef
[13] F.J. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall Inc., Englewood Cliffs, N. J., 1964.
[14] S. Ul’m, On the iterative method with simultaneous approximation of the inverse of the operator, Izv. Acad. Nauk. Estonskoi S.S.R. 16 (1967) no. 4, pp. 403–411.
[15] T. Yamamoto, Error bounds for computed eigenvalues and eigenvectors, Numer. Math. 34 (1980), pp. 189–199.
CrossRef