# LOBPCG bad preconditioned performance for largest eigenpairs

The LOBPCG algorithm finds eigenpairs of the generalized eigenproblem $$Ax = \lambda B x$$ where $$B$$ is symmetric and positive-definite, $$A$$ is symmetric. One of the features that makes LOBPCG so interesting is the fact that one can use preconditioners of $$A$$ to speed up convergence.

When applying LOBPCG to the FEM-discretized Poisson problem (with $$B=I$$), and converging the smallest 6 eigenvalues, the convergence behavior I'm getting is

When using an ML preconditioner, convergence speed increases dramatically:

Now, let's try the same thing for the 6 largest eigenpairs.

Without preconditioner:

With ML preconditioner:

It seems that the preconditioner slows down the convergence! Is this something one would expect? I couldn't find anything in literature about it.

• Framing question -- why do you want the largest eigenvalues? The smaller / mid-range eigenvalues of a FE / FD discretization of an elliptic operator are probably close to those of the true operator, but I'd expect that the largest ones aren't. Jan 23 at 19:09
• @DanielShapero True. I just read in LOBPCG articles that the algorithm should be able to approximate both small and large eigenpairs, and I was wondering if perhaps there's something I'm doing wrong. Small-magnitude eigenvalue are indeed of greater interest though. Jan 23 at 20:18