Compute specific eigenvalues in the complex plane with Feast?

Question

In physical problems, it's quite common that we need to solve for specific eigenvalues in the complex plane, e.g. with a positive real part and negative imaginary part. In this case, we are looking for eigenvalues and associated eigenvectors in the second quadrant.

I am curious if the FEAST algorithm could deliver faster and more accurate results for this kind of problem (dense as well), with respect to solving the full problem using GEEV routine from LAPACK and do the sorting explicitly.

Does anyone have experience or suggestion?

Answer 1

The key component of using FEAST algorithm for computing eigenvalues is specifying an appropriate search contour. In FEAST, the user can specify circular or elliptical contours (in the newest versions, also custom-shaped ones, which can be important for targeting specific eigenvalues of non-Hermitian problems).

Now, if the size of the contour is too large (not even talking about semi-infiniteness), one would require a very large number of contour points (thus, significantly sacrificing the computation speed). For the whole quadrant, additional issues arise as one has to make a contour finite somehow (probably using some crude estimates for eigenvalue locations).

Limiting the search space to only a quarter of the complex plane (2nd quadrant) does not seem enough to justify using Feast algorithm. I don't see the benefits of using FEAST for computing all eigenvalues that lie in the whole quadrant – a more restrictive search contour is required.

References: FEAST Manual. Section 2.1 The FEAST algorithm; Section 4.2 & 4.3 Defining a search contour