First publicly announced version

[jarlebring]

This issue is checklist/wishlist of things we should plan to do before we feel we can start announcing the existance of the packade to colleagues (maybe version 0.1). Feel free to add if you think it should be included.

• NLArnoldi
  • [x] Modernization of orthogonalization
  • [x] Stay real when problem is real
  • [x] One version of restarting
  • [x] Proper function comments
  • [x] Unit testing
  • [x] Throw NoConvergenceException if it does not find the specified number of eigenvalues
  • [x] Selection around disk
• IAR etc
  • One version of restart
  • [x] Chebyshev version (as different function maybe)
  • [x] Extraction by projection (instead of Hessenberg)
  • [x] Proper function comments
  • [x] Proper function comments infbilanczos
• TIAR
  • [x] Extraction by projection
  • [x] Modernize orthogonalization
  • [x] Proper function comments
• SG
  • [x] Reimplementation of such that we can compute kth eigenvalue (for minmax problems)
  • [x] Proper function comments
• JD
  • [x] Reimplementation with new ProjectedNEP
  • [x] Reimplementation with modern orthogonalization
  • [x] Reimplement the linear solve to get rid of explicit matrix access (computed via Mlincomb)
  • [x] Implement so that it can be used to compute more than one eigenvalue
  • [x] Proper function comments
  • [x] Deflation a.l.a effenberger(?)
• Newton methods
  • [x] Proper function comments augnewton
  • [x] Proper function comments resinv
  • [x] Proper function comments newton
  • [x] Proper function comments rfi
  • [x] Proper function comments quasinewton
  • [x] Proper function comments newtonqr
  • [x] Proper function comments implicitdet
  • [x] Proper function comments mslp
• Block Newton method
  • [x] Basic implementation
  • [x] Proper function comments
• WEP
  • [x] Direct LinSolver on Schur complement
• Contour integral methods
  • Beyns version
  • [x] trapezoidal rule
  • [x] Proper function comments
  • [x] Eigenvector extraction
  • SLEPc version
  • Basic implementation
  • Proper function comments
• Transformations
  • [x] Effenberger deflation
• Core and types
  • [x] REP as SPMFs
  • [x] Isolate Matlab dependence from main package (LinSolver)
  • Proper function comments for
  • [x] compute_Mder, compute_Mlincomb, compute_MM
  • [x] compute_rf
  • [x] SPMF,PEP, REPs
  • [x] Projectable NEPs
  • [x] nep transformations
  • [x] linear system solvers
  • [x] linear eigenvalue problem solvers
  • [x] companion
• Practical
  • [x] Introduce a copyright/copying licence
  • [x] Proper first page
  • [x] A short/technical manuscript which we can put on arxiv such that it can be cited.

By proper function comments I mean:

• Function definition with optional parameters
• Description of function (reference newton() for the common parameters)
• A self-contained example.
• References

I have already done "proper function comments" for newton() and resinv(). We should do it using Markdown, i.e., the rules here https://docs.julialang.org/en/stable/manual/documentation/

Migrated from https://gitr.sys.kth.se/nep-pack/nep-pack-alpha/issues/26

[meleg]

The method iar_chebyshev is now fully functional. You are welcome to test it, report bugs and give advices for improving it. Documentation is still needed, I will add it soon.

[jarlebring]

Nice work. I like that you show how one can form a sum of two NEPs.

I think selecting γ not equal to one does not work for the SPMF (and also not work for the ComputeY0Cheb_QDEP). I added two tests. I would suggest we try to write the derivation of the SPMF-formula in the manuscript in a clear way, before we try to include scaling.

[meleg]

@jarlebring thanks for the bug reporting! Now the method iar_chebyshev supports shift and scaling for SPMF format as well (and the test passes). Recall that this cannot be incorporated in the method itself but the problem is explicitly transformed.

However, by reading your comment it feels you want a deeper change, i.e., include the shift and scaling in the method. Is this correct? We had a discussion about that for the general case and I remember it was not easy. Do you think is simpler for the SPMF format?

About the case ComputeY0Cheb_QDEP, the point is that the the user-provided method compute_y0_method does not depend on γ and it is implicitly assumed γ=1. I can fix this function or change back γ=1 in order to pass this test.

[jarlebring]

change, i.e., include the shift and scaling in the method. Is this correct? We had a discussion about that for the general case and I remember it was not easy. Do you think is simpler for the SPMF format?

Yes. I don't see why the SPMF-computation formula would not be able to handle shift and scale. As I wrote in my above, I suggest to clean up the derivation and put it into the MS.

[meleg]

I decided to remove the test for ComputeY0Cheb_QDEP with shift and scale (commit e3d76c1728ec524f97951aa560a283edf410bfdf). I did it because the function compute_y0_method for the QDEP is a combination of the functions compute_y0_method for PEP and DEP. Recall that for DEP and PEP we do not have formulas for compute_y0_method if the shift and scale are not standard (shift=0, scale=1).

[eringh]

A naive eigenvector extraction in Beyn 4f32fedd3f759e1f5ff4dacc48a84d1b4a5e6c4f

[eringh]

Some documentation for rfi, newtonqr, and implicitdet fixed in, e.g. 01eee8d8b1211465064a1b545643835a355ea0ad, da0f1b3947038a83b78de299f2c8ea467cdf0e86, and ac1d31b611c73e224807554595838e0a9ff15b67.

[jarlebring]

Last checkbox, short manuscript describing main features: https://arxiv.org/abs/1811.09592