RFC: Ordering by Generalized Eigenvalues for Generalized Schur methods

[cc7768]

This is related to #9655. In #9655, we were hoping to implement the option to pass a function directly to gges/gees so that LAPACK would do everything in the background. After reading a little more and seeing the work that had been done for ordschur in #8467 with the LAPACK function trsen, we decided that a good step would be to implement the corresponding method for the generalized schur method tgsen.

Provides the same functionality as Matlab's ordqz.

CC: @spencerlyon2

[cc7768]

Is there a reference to selctg in this branch, or are you talking about the other PR (#9655)?

[andreasnoack]

No you are right. I had both PR open in the browser. I'll delete the message here and post it in the other PR.

[andreasnoack]

Why? Is that mentioned somewhere in the LAPACK docs?

[cc7768]

No, just in all of the tests that relate to the Generalized Schur method (both mine and the original generalized Schur tests), they get two matrices by splitting a into 2 matrices -- a[1:5, 1:5], a[6:10, 6:10]. Just thought it was worth mentioning in case some one ever changed it (the alternative could be to index based on n).

[andreasnoack]

Ah. It would be great if you could change the that to something like 1:div(n,2) and div(n,2)+1:n.

[cc7768]

Excellent, I think that will be better anyways. I didn't want to make too many changes to the files, I will fix that.

[cc7768]

I went through lapack.jl and tried to find as many of the stride issues as I could. I think I have fixed at least the majority of them and at least the linalg tests still pass on my computer.

[andreasnoack]

Great. I can see that it needs a rebase. Could you also squash some of the commits? It would be great with two final commits: one for the ordering and one for that fix of the leading dimensions. If that gets too cumbersome it is okay with a single commit.

[cc7768]

I think that I have finally put them all together. I'm not quite a "git-foo" expert, so it took me a few minutes to figure out what I wanted to do (and had a seminar to attend). Before I finished this it looked like it was successfully building on Travis on OSX but not on Linux.

Update: Tests passed with make clean testall on my computer (Ubuntu 14.04)

I created a back up of the branch that I can revert to just in case I shot the whole branch.

[andreasnoack]

Great. Thanks for the rebase.

Last thing. Could you add some documentation such that people would know how to use the added functionality?

[cc7768]

No problem. Just added documentation to doc/stdlib/linalg.rst. Let me know if that is sufficient, I tried to be descriptive enough that it is understandable while being brief enough that it isn't too much.

[andreasnoack]

Great. I'll merge when the lights are green.