On approximating the eigenvalues and eigenvectors of linear continuous operators - Cătinaş, Emil; Păvăloiu, I (Rev. Anal. Numér. Théor. Approx., 1997)
We consider the computing of an eigenpair (an eigenvector \(v=(v^(i))_i=1,n\) and an eigenvalue \(λ\)) of a matrix \(AR^n× n\), by considering a supplementary condition (we call it norming function for the eigenvector), represented by a polynomial of degree 2. A usual choice is equation G(v):= extstyle12 _i=1^n( v^(i))^2-1=0. equation We propose here a new choice: equation G(v):= extstyle12n _i=1^n( v^(i))^2-1=0, 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: \[ F( x) :=Av-λ v G(v)-1 =0, \] 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 \(A:V→ V\), \(V\) 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: align* x_k+1 =&x_k-Γ_kF( x_k) Γ_k+1 =&Γ_k( 2I-F^′ ( x_k+1) Γ_k) , k=0,1,…, align* where \(x_0X\), \(Γ_0L( X)\), and \(I\) is the identity operator of \(L( X)\). We obtain semilocal convergence conditions which show that the method has r-convergence order 2.