5. Methods for large matrices

Most of the practical methods used to solve large-scale problems are often a combination of projection techniques, deflation and restart methods and preconditioning techniques. Among the most widely used methods are :

• the simultaneous iteration method, which is a basic, robust and relatively slow method, but which can play a role, for example, in validating the results of other approaches;

• the Arnoldi (or Lanczos) method with polynomial acceleration, such as the method implemented in the ARPACK code [18] ;

• shift-and-invert methods, which generally use direct factorization methods to speed up convergence. This type of method is implemented, for example, in the NASTRAN code