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Continuous methods for extreme and interior eigenvalue problems. (English) Zbl 1092.65029

The authors suggest to apply some continuous methods (actually their discrete analogs) for a possible numerical solution of the partial eigenvalue problems $Ax=\lambda x$ with a symmetric matrix $A$; the eigenvalues are numerated in increasing order and are assumed to be smaller than a constant $c$; as an example they make use of $c={\parallel A\parallel }_{1}+1$. Instead of the standard minimization of the Rayleigh quotient on the unit sphere for approximations of ${\lambda }_{1}$, they consider the minimization of the merit function $f\left(x\right)\equiv \left(Ax,x\right)-{c\parallel x\parallel }^{2}$ on the unit ball (a closed convex set). They write: “Therefore, it is much easier to solve”. In reality they construct a dynamical system such that $f\left(x\right)$ is monotonically nonincreasing along the solution of the system. More precisely, the system ${x}^{\text{'}}\left(t\right)=-\left[x-P\left(x-\nabla f\left(x\right)\right)\right]$ is applied where $P$ is projection operator on the unit ball (the use of such equations for minimization problems is a very old idea in mathematics but which can not replace the theory of iterative methods). The authors concentrate on the applicability of classical theorems for the differential systems under consideration including the study of the asympotical behavior of the solutions. Similar topics are discussed when the minimization of the merit function $F\left(x\right)\equiv \left(\left(A-aI\right)\left(A-bI\right)x,x\right)$ is applied (it helps to deal with eigenvalues in $\left[a,b\right]$).

Numerical examples are given for model matrices of order $n\in \left[1000,7500\right]$. The ordinary differential equation solver used is ODE45 (a nonstiff medium order method). The computational time “grows at a rate of ${n}^{2+\epsilon }$ ”. The authors write that “our new methods are very effective and attractive” but also that “the investigation of the method is still not enough for a side-by-side comparison with the existing linear algebra methods”.

##### MSC:
 65F15 Eigenvalues, eigenvectors (numerical linear algebra)