# Large scale nonsymmetric eigenvalue problems in practical applications

I am trying to determine need for solving large scale nonsymmetric eigenvalue problems in both industry and academia.

I am interested in any kind of problem where we cannot assume that matrices are symmetric. This includes the standard eigenvalue problem, the generalized eigenvalue problem as well as the quadratic eigenvalue problem. I am interested in sparse and dense problems.

MINOR EDIT: I am not interested in problems that only require the computation of a few eigenpairs. This excludes cases where, say, a shifted inverse power iteration will suffice.
The ideal answer to this question includes a reference to at least one application where it is absolute necessary to solve a large scale nonsymmetric eigenvalue problem.

The vast majority of existing libraries only support symmetric standard and generalized eigenvalue problems. I think that there are two good reasons for this:

1. A large number of applications produce symmetric eigenvalue problems.
2. The nonsymmetric eigenvalue problem is more difficult than the symmetric eigenvalue problem.

LAPACK contains a complete stack for solving dense nonsymmetric and generalized eigenvalue problems, but the routines for computing eigenvectors are based on BLAS level 1 and level 2 functions. ScaLAPACK never completed the support for nonsymmetric eigenvalue problems and the existing routines for computing eigenvectors are either vulnerable to overflow or do not scale at all. The new StarNEig library can compute all the eigenvalues/eigenvectors for nonsymmetric standard and generalized eigenvalue problems. It applies to matrices/matrix pencils that are real and have real/complex eigenvalue/eigenvectors. It can be substantially faster than LAPACK with parallel BLAS as well as ScaLAPACK.

I have isolated some applications where symmetry is lost.

Example 1 In structural engineering we find the quadratic eigenvalue problem $$(\lambda^2 M + \lambda C + K)x = 0.$$ This problem is equivalent to the generalized eigenvalue problems $$Ax = \lambda B x$$ where $$A = \begin{bmatrix} 0 & K \\ K & C \end{bmatrix}, \quad B = \begin{bmatrix} K & 0 \\ 0 & - M \end{bmatrix}, \quad y = \begin{bmatrix} x \\ \lambda x \end{bmatrix}.$$ We see that $$A$$ and $$B$$ are symmetric when the mass matrix $$M$$, the stiffness matrix $$K$$ and the dampning matrix $$C$$ are all symmetric. However, the eigenvalue problem for a damped gyroscopic system takes the form $$(\lambda^2 M + \lambda (C + G) + K)x = 0$$ where $$G$$ is a skew-symmetric matrix. We see that the symmetric matrix $$C$$ has been replaced with a general nonsymmetric matrix $$C + G$$. We see that the equivalent generalized eigenvalue problem is no longer symmetric.

I imagine that rotating equipment such as jet engines, gas turbines and propellers can be represented using damped gyroscopic systems, but I do not have the knowledge to say this for a fact.

Example 2 There electrical engineering we find many examples of eigenvalue problems that not symmetric. Two generalized eigenvalue problems can be found in this paper which surveys the available software. The examples are power networks from Europe. A circuit for which the nodal conductance matrix is nonsymmetric due to the presence of a voltage controlled current source is presented in this paper. In this paper, the authors need all the left/right eigenvectors of a standard eigenvalue problem.

My fundamental problem is that I lack the engineering knowledge and the key words necessary to track down papers that deal with large scale nonsymmetric eigenvalue problems. I can build small toy examples that are nonsymmetric, but I cannot say with authority that their larger siblings are remotely relevant in practice. This is where I hope that this community can help me.

• A whole bunch of those is here: D. A. Baver, J. R. Myra, M. V. Umansky, "Linear eigenvalue code for edge plasma in full tokamak X-point geometry", Comput. Phys. Commun. 182(8): 1610-1620 (2011) – Maxim Umansky Mar 12 at 15:51
• Might explore search engine indexing, e.g., PageRank. – A rural reader Mar 12 at 15:58
• The modes of any oscillating system with losses are characterized by complex eigenvalues and non-self-adjoint operators. Vibrating strings, modes in a microwave, in an optical fiber, or metallic waveguide, etc. They all have self-adjoint operators only if material/radiation losses are neglected. – Amit Hochman Mar 12 at 16:08
• @MaximUmansky Thank you for the reference. I understand that you used Slepc to solve a generalized eigenvalue problem. I cannot determine that it was nonsymmetric. I do not have the physics background that will allow me to navigate this paper with ease. Can you point to the part where it becomes clear that the eigenvalue problem is nonsymmetric. – Carl Christian Mar 12 at 16:57
• @Aruralreader Quite so, Pagerank is good example that fits the question as it is currently written. I should have emphasized that I care for more than a few eigenpairs. – Carl Christian Mar 12 at 17:00