From: Zlatko Drmac
Date: Wednesday, April 15, 2026 at 9:47 AM

First a summary:

X and Y are data that define implicitly a matrix A such that A*X \approx Y. So, in principle the caller can input any X and Y.   

However, if scaling options are active, the subroutine uses the opportunity to check for some inconsistencies - like all X is zero, or some are NaNs. Having all X zero indeed suggests something is wrong with the data. This is again checked after the SVD of X - if largest singular value is zero, error code -8 is returned.

If scaling of Y is selected, detected NaNs will trigger an error message. If all Y are zero, this also suggests something is wrong with the data, although it may implicitly define a zero matrix - this again is something that the caller probably did not want.

So, this is something I call "a political decision" - the roles of X and Y are not symmetric in this case but one can simply define/declare that zero input (X or Y) will be reported as error. This is what is in the proposed correction. Perhaps it should be done. Otherwise, DGEEV will get a zero matrix on input. In DGEEV you have a quick return if N=0. On the other hand, to check for scaling DGEEV computes ANRM and scales if needed but it does not use the opportunity for quick return if ANRM=0.

So, I would say it is OK to return an error code if Y=0. The only "con" is that then the subroutine behaves differently if scaling of Y is not requested and Y=0. 

(For X=0, even without scaling, after the SVD X=0 will be detected and reported error.)

So, in the end, it is a "political decision" 😀

From: Zlatko Drmac
Date: Thursday, April 16, 2026 at 8:51 AM

It is indeed important to discuss all aspects of such "singularities". Well designed behaviour for singular cases may save the day in mission critical applications.  

If Y = 0 then the matrix whose eigenpairs were to be identified is zero matrix (based on the supplied data) so in principle the question is what an eigenvalue subroutine should return if zero matrix is given as input? If this is DGEEV then it should return eigenvalues and eigenvectors (which ones) of the zero matrix. In the DMD case we can say zero matrix is something unheard of and  return error code.  Making the change as suggested is OK, but also some another flag could be raised. E.g., note that DGEDMD uses positive values of INFO to return some diagnostics/warnings. So it would be also wise to update INFO to make sure the caller is well informed.