On Tue, May 3, 2022 at 8:40 PM Leo Fang wrote:
> Hi, I am catching up with an assigned task that I slipped. I am wondering
> a few things about the numpy.linalg.eig() API (eigvals() is also included
> in this discussion, but for simplicity let's focus on eig()). AFAIU the
> purpose of eig() is for handling general eigen-systems which have left and
> right eigenvectors (that could differ beyond transpose/conjugate).
For this, I have two questions:
>
> 1. What's the history of NumPy adding this eig() routine? My guess would
> be NumPy added it because Matlab had it, but I couldn't tell from a quick
> commit search.
>
It essentially dates back to Numeric, though the name was a bit different.
But the core feature of only computing right-eigenvectors was there.
scipy.linalg.eig() came along and expose both left- and right-eigenvalues
(with close to the present signature, I think). Then numpy took over from
Numeric and adopted the scipy.linalg names for the routines, but cut out a
number of the features that made some of the interfaces complicated. Since
Numeric only did right-eigenvalues (what most people want, even in the
general-matrix case), so did numpy.
> 2. Would it be possible to deprecate/remove this API? From past
> discussions with users and developers, I found:
> - numpy.linalg.eig() only returns the right eigenvectors, so it's not very
> useful:
> * When the left eigenvectors are not needed --- which is the majority of
> the use cases --- eigh() is the right routine, but most users (especially
> those transitioning from Matlab) don't know it's there
>
eigh() is only for symmetric/Hermitian matrices.
> * When the left eigenvectors are needed, users have to use
> scipy.linalg.eig() anyway
>
It seems to me that you are assuming that in the non-Hermitian case, the
right-eigenvectors are useless without the left-eigenvectors. It isn't
obvious to me why that would be the case. It is true that the left- and
right-eigenvectors differ from each other, but it's not clear to me that in
every application with a general matrix I need to compute both.
- A general support for eig() could be challenging (numerical issues), and
> doing SVDs instead is usually the better way out (in terms of both
> numerical stability and mathematical sense).
>
Computing the left-eigenvectors is no real problem. The underlying _GEEV
LAPACK routine can do it if asked. It's the same routine under
scipy.linalg.eig().
--
Robert Kern
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