A NEWTON METHOD FOR SOLVING LOCALLY DEFINITE MULTIPARAMETER EIGENVALUE PROBLEMS BY MULTI-INDEX

We present a new approach to compute eigenvalues and eigenvectors of locally definite multiparameter eigenvalue problems by their signed multi-index. The method has the interpretation of a semismooth Newton method applied to certain functions that have a unique zero. We can therefore show local quadratic convergence, and for certain extreme eigenvalues even global linear convergence of the method. Local definiteness is a weaker condition than right and left definiteness, which is often considered for multiparameter eigenvalue problems. These conditions are naturally satisfied for multiparameter Sturm--Liouville problems that arise when separation of variables can be applied to multidimensional boundary eigenvalue problems.