Generalization of Hermitian Case

Similar to the Lanczos methods and the Arnoldi method, the Jacobi-Davidson method constructs a subspace onto which the given eigenproblem is projected. The subspace is constructed with approximate shift-and-invert steps, instead of forming a Krylov subspace. In §4.7 we have explained the method in detail, and the generalization to the non-Hermitian case for the basic algorithm, described in Algorithm 4.13 (p. ), is quite straightforward. In fact, the changes are:

1. The construction of the matrix ${M}$ has to take into account that ${M}$ is non-Hermitian, hence the corresponding action in (7)-(9) has to be replaced by

   
    				       for $i=1,\ldots, {m}-1$ 
   
   ${M}_{i,{m}}={v}_{i}^\ast v^A_{m}$,     ${M}_{{m},i}={v}_{m}^\ast v^A_i$
   
   ${M}_{{m},{m}}={v}_{m}^\ast v^A_{m}$
   

2. In (10) a routine has to be selected for the non-Hermitian dense matrix ${M}$

If the correction equation (in step (15) of the algorithm) is solved exactly, then the approximate eigenvalues have quadratic convergence towards the eigenvalues of $A$.