On approximating the eigenvalues and eigenvectors of linear continuous operators

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: 248

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.

Downloads

Download data is not yet available.

References

M. P. Anselone and B. L. Rall, The solution of characteristic value-vector problems by Newton method, Numer. Math. 11 (1968), pp. 38-45, https://doi.org/10.1007/bf02165469

E. Cătinaş and I. Păvăloiu, On the Chebyshev method for approximating the eigenvalues of linear operators, Rev. Anal. Numer Theorie Approximation 25, 1-2 (1996), pp. 43-56.

P. G. Ciarlet, Introduction à l'analyse numérique matricielle et à l'optimisation, Mason, Paris-Milan-Barcelone-Mexico, 1990.

F. Chatelin, Valeurs propres de matrices, Mason, Paris-Milan-Barcelone-Mexico, 1988.

L. Collatz, Functionalanalysis und numerisch Mathematik, Berlin-Göttingen-Heidleberg, Springer Verla, 1964.

A. Diaconu, On the convergence of an iterative proceeding of Chebyshev type, Rev. Anal. Numér. Théorie Approximation 24, 1-2 (1995), pp. 91-102.

A. Diaconu and I. Păvăloiu, Sur quelques méthodes itératives pour la résolution des équations opérationnelles, Rev. Anal. Numér. Theorie Approximation 1, 1 (1972), pp. 45-61.

V. S. Kartîşov and F. L. Iuhno, O nekatorîh Modifikaţiah Metoda Niutona dlea Resenia Nelineinoi Spektralnoi Zadaci, J. Vîcisl. matmrn. i matem. fiz.33, 9 (1973), pp.1403-1409.

L Lazăr, On a Newton-type method, Rev. Anal. Numér. Theorie Approximation 23, 2 (1994), pp.167-174.

I. Păvăloiu, Obsevations concerning some approximation methods for the solutions of operator equations, Rev. Anal. Numér. Thér¡ie Approximation 23, 2 (1994), pp. 185-196.

I. Păvăloiu, Sur les procédes iteratifs à un ordre élevé de convergence, Mathematica (Cluj)12 (35), 2 (1970), pp. 309-324.

A. R. Tapia and L. D. Whitley, The projected Newton method has order 1+√2 for the symmetric eigenvalue problem, SIAM J. Numer. Anal. 25, 6 (1988), pp. 1376-1382, https://doi.org/10.1137/0725079

F. J. Traub, Iterative methods for the solution of equations, Frentice-Hall Inc., Englowood Cliffs, N.J., 1964.

S. Ul'm, On the iterative method with simultaneous approximation of the inverse of the operator, Izv. Acad,. Nauk. Estonskoi S.S.R. 16, 4 (1967), pp. 403-417.

T. Yamamoto, Error bounds for computed eigenvalues and eigenvectors, Numer. Math. 34 (1980), pp. 189-199, https://doi.org/10.1007/bf01396059

Downloads

Published

1997-08-01

How to Cite

Cătinaş, E., & Păvăloiu, I. (1997). On approximating the eigenvalues and eigenvectors of linear continuous operators. Rev. Anal. Numér. Théor. Approx., 26(1), 19–27. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1997-vol26-nos1-2-art3

Issue

Section

Articles