1. 2. 3 Algebraic Solvers

Software for the solution of algebraic equations is required after the discretization of a given differential equation has been assembled. The typical and mostly explored field is the solution of linear equation systems. For such systems various implementations are available. An established standard for the treatment of linear equations is LAPACK [46], which defines a set of interaction functions at different levels.

PETSC [47,48] and Trilinos [4] solvers have a general purpose interface for accessing and solving linear equation systems and eigenvalue equation systems. Many different operations for assembly are possible and various solution methods can be used and exchanged without great effort, for instance, by changing a parameter.

Even though the diversity of different methods used for the solution of eigenvalue problems is considerable and special methods are used for special problems, there are approaches for an intelligent and highly automatized approach towards the solution of algebraic problems. First, the large variety of methods of PETSc and Trilinos allows to test and evaluate different solution methods and parameter settings. In addition, special expert systems like EigAdept [49] have been developed to automatically solve algebraic equations using the best available method. Such environments enable to use an algebraic solver as a black box, while the user does not necessarily need to specify the solution method.