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## 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 has to take into account that is non-Hermitian, hence the corresponding action in (7)-(9) has to be replaced by

```
for
,

```

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

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

Susan Blackford 2000-11-20