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I apologize if this is not the right forum for my question, but I cannot think of better place to reach a large number of engineers within a short time span.

Question: I am a mathematician working in the field of high performance scientific computing. My goal is to determine the need for solving large scale nonsymmetric dense standard and generalized eigenvalue problems in industry. I would like to

  1. see examples of large scale nonsymmetric eigenvalue problems that you have had to solve in the past and
  2. learn about the performance of the software that you would like to have in the future.

Background: I am one of the authors of a new software library called StarNEig for solving dense nonsymmetric standard and generalized eigenvalue problem. StarNEig can compute all eigenvalues/eigenvectors of a dense nonsymmetric matrix $A$ and all the generalized eigenvalue/eigenvectors of a dense nonsymmetric matrix pencils $A - \lambda B$. StarNEig is currently limited to real matrices and real matrices pencils with real and complex eigenvalues and eigenvectors. StarNEig can be substantially faster than either LAPACK with parallel BLAS and ScaLAPACK, the de facto standard libraries in field of dense numerical linear algebra. StarNEig is freely available from GitHub and some performance figures can be found on our front page, see link above.

Motivation: At this point we request input from potential users of the StarNEig library. This will help us guide the future development of the library. If I can argue that the existing software is outdated and insufficient to fulfill the future needs of industry, then this will dramatically improve our chances to obtain additional funding and provided the community with better software in the long run.

Some possibilities: I expect that the quadratic eigenvalue problem $$(\lambda^2 M + \lambda C + K)x=0$$ will be a rich source of nonsymmetric problems. The quadratic eigenvalue problem is equivalent with the generalized eigenvalue problem $$Ay=\lambda By$$ 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 $B$ symmetric when the mass matrix $M$, the stiffness matrix $K$ and the dampning matrix $C$ are all symmetric. However, I understand that there are many systems where this is not case. In particularly, I gather that there are gyroscopic systems where damping matrix is either skew-symmetric $(C = -C^T)$ or a general nonsymmetric matrix. I imagine that such problems can be found in the design and analysis of rotating mechanical systems such as turbines and engines. I am told that the presence of non-conservative forces produces stiffness matrices that are not symmetric.

I expect the analysis of transients in electrical circuit to be a rich source of nonsymmetric eigenvalue problems. Certainly, I can build small toy examples that lead to nonsymmetric matrices, but I do not know enough to determine what relevant systems look like. I have access to nonsymmetric eigenvalue problems describing power networks.

What would the ideal answer look like? A short list of real applications with a clear need to solve large nonsymmetric eigenvalue problems with references to specific books or papers. A wish list describing the software that you would like to have in the future.

What is a large problem? There is no precise definition. However, a problem is certainly large if you need a multi-node multi-core parallel computer to solve the problem in the time you have available.

Carl Christian
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  • Large matrix problems are very rarely dense, and you are rarely interested in all the eigenpairs of a "large" matrix. A second-order system with gyroscopic (skew symmetric) terms is easy to convert to a Hermitian system, since the gyro term do not dissipate energy and the system eigenvalues are pure real (or pure imaginary, depending exactly how you reformulate the problem). FWIW we were routinely finding the first 3,000 eigenpairs of (sparse) matrices of order 300,000 with commercially available software, 30 years ago. – alephzero Mar 11 '21 at 15:42
  • @alephzero My thanks. Would you elaborate on how to convert a second order system to a Hermitian system? In particular, are you assuming that the dampning matrix is skew-symmetric or can this transformation be done for general dampning? Iterative methods such as the subspace iteration, implicit restarted Arnoldi, Jacobi-Davidsons, etc, typically solve dense eigenvalues as part of every iteration. They are not trivial to use and it is not trivial to determine if you have obtained all desired eigenpairs. Moreover, the dense kernels have to scale well or they become a bottleneck. – Carl Christian Mar 11 '21 at 16:07
  • @alephzero I completely agree that one is often interested in all eigenpairs corresponding to all eigenvalues within a certain region of the complex plane. The safest option is to treat the sparse problem as dense and reduce it Schur form which you then reorder in order to compute the orthogonal projector onto the corresponding eigenspace. Again an iterative method such as an shifted inverse subspace iteration can miss eigenpairs. – Carl Christian Mar 11 '21 at 16:14
  • @alephzero I may have misread your comment. If the mass matrix and the stiffness matrix are symmetric and the dampning matrix is skew-symmetric, then the quadratic eigenvalue problem is equivalent to a Hermitian eigenvalue problem. This is clear to me. Surely, we cannot hope to do this in general, right? – Carl Christian Mar 11 '21 at 17:52
  • Engineers rarely get into the weeds of these problems and will likely not be able to tell you which problems your approach will be good for. I'd suggest you migrate the question to https://scicomp.stackexchange.com/ where you'll find more people who know both sides of the problem. – Biswajit Banerjee Mar 11 '21 at 19:01
  • @BiswajitBanerjee This is good advice. I would like to retain a copy here, but I will make a copy/reformulate the question a bit and set up a cross reference. – Carl Christian Mar 11 '21 at 19:10
  • Damping matrices are (usually) symmetric. Gyroscopic forces (or Coriolis terms if you are working in rotating coordinates) are skew. It depends what the model is used for. For eaxmple if you want to calculate whirling speeds for rotating machinery, including damping terms wouldn't add to the understanding of the output. Also, for large mechanical models, often you don't have an explicit model of damping anyway so damping is approximated in some way - either Rayleigh damping based on M and K, or as modal damping added to the model after you have solved the eigenproblem. – alephzero Mar 12 '21 at 01:28
  • You might also try the new Matter Modelling stack. Not my field at all, but they seem to have a lot of questions about computer modelling. – alephzero Mar 12 '21 at 01:30
  • @alephzero It seems to me that one would necessarily have both rotation and friction present when simulating say a gas turbine or propeller. Is it not reasonable to expect the equivalent second order system to be a damped gyroscopic system? When I search for modal dampning I get the distinct impression that we are not changing the modes/eigenvectors of the model, is this impression correct? Surely there is a subfield of engineering where one uses actual measurements obtained from the finished product to adjust say a FEM model? – Carl Christian Mar 12 '21 at 13:00
  • @alephzero (continued) To me it seem almost inevitable that such a adjustment would involve solving many eigenvalue problems as part of an iterative adjustment procedure. – Carl Christian Mar 12 '21 at 13:01

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