  We present a new approach to compute eigenvalues and eigenvectors of locally
definite multiparameter eigenvalue problems by its signed multiindex. 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.
