An Arnoldi Method for Nonlinear Symmetric Eigenvalue Problems

where T (λ) ∈ R is a family of symmetric matrices depending on a parameter λ ∈ J , and J ⊂ R is an open interval which may be unbounded. As in the linear case T (λ) = λI −A a parameter λ is called an eigenvalue of T (·) if problem (1) has a nontrivial solution x 6= 0 which is called a corresponding eigenvector. We assume that the matrices T (λ) are large and sparse. For sparse linear eigenvalue problems most efficient methods are iterative projection methods, where approximations to the wanted eigenvalues and corresponding eigenvectors are obtained from projections of the eigenproblem to subspaces which are expanded in the course of the algorithm. Methods of this type are the Lanczos algorithm for symmetric problems, and Arnoldi’s method and the JacobiDavidson method, e.g., for more general problems. Taking advantage of shift–and– invert techniques in Arnoldi’s method one gets approximate eigenvalues closest to the shift. Ruhe [5] generalized this approach. He suggested the rational Krylov method using several shifts in one run, thus getting good approximations to all eigenvalues in a union of regions around the shifts chosen. In some sense, Ruhe [6] and Hager and Wiberg [3], [2] generalized this approach to sparse nonlinear eigenvalue problems by nesting the linearization of problem (1) by Regula falsi and the solution of the resulting linear eigenproblem by Arnoldi’s method, where the Regula falsi iteration and the Arnoldi recursion are knit together. Similarly as in the rational Krylov process they construct a sequence