Inexact Newton preconditioning techniques for large symmetric eigenvalue problems (Q1279539)

Variants of Newton's method are applied to the equations \((*)\) \(F_1(x)=0\) and \((**)\) \(F_2(x,\lambda)=0\), where \(F_1(x) = A x-x\langle x| Ax\rangle\) and \[ F_2(x,\lambda)= \binom{Ax-\lambda x} {\frac 12 (1-\| x\|^2)}, \] in order to obtain an eigenvalue \(\lambda\) and an eigenvector \(x\) of a symmetric \(n\times n\) matrix \(A\). These methods are compared to each other and to the Rayleigh quotient iteration, and conditions for \(DF_1(x)\) and \(DF_2(x,\lambda)\) to be regular at the solutions of \((*)\) and \((**)\), respectively, are given. Further, it is discussed how approximately solving the linear equations appearing in these methods may be used to extend bases in Davidson's method. Computational results are presented.