Eigenvalues and eigenvectors

[fredrik-johansson]

For some reason this didn't have any issue yet. Of course the goal is to support nonsymmetric matrices with possibly repeated/clustered eigenvalues.

Siegfried Rump has work on this: http://www.ti3.tuhh.de/paper/rump/Ru99c.pdf (Also theorem 13.9 in "Verication methods: Rigorous results using oating-point arithmetic")

A more recent and possibly more efficient algorithm by Joris van der Hoeven and Bernard Mourrain: http://www.texmacs.org/joris/certeigen/certeigen.pdf

[fredrik-johansson]

Added initial code for computing general eigenvalues without rigorous error bounds. This is just a straight port of Timo Hartmann's code in mpmath (which becomes up to 150 times faster in Arb).

Worklist for this code:

• [x] Easy: finish the code for computing eigenvectors
• [x] Easy: add more test code
• [ ] Easy: add even more test code
• [ ] Easy: speed up the Hessenberg reduction
• [ ] Easy: add and make public the variant functions (schur, hessenberg)
• [ ] Medium: speed up the qr_step code
• [ ] Hard: consider implementing the original listed todo improvements (balancing, aggressive early deflation)
• [ ] Hard: parallelize it

Of course, the really interesting problem now is to implement one of the certification algorithms.

[fredrik-johansson]

Finally implemented Rump's certification algorithm and a top-level function for diagonalization. Still need some top-level function for clusters of overlapping eigenvalues.

[fredrik-johansson]

Closing this and creating separate issues for the important remaining tasks.