eigenmatrix_right totally broken

[williamstein]

sage: a = matrix(CDF,2,[1,2,4,7])
sage: a.eigenmatrix_right()
---------------------------------------------------------------------------
IndexError                                Traceback (most recent call last)

/Users/wstein/sage/build/sage-3.2.2.alpha0/<ipython console> in <module>()

/Users/wstein/sage/build/sage-3.2.2.alpha0/local/lib/python2.5/site-packages/sage/matrix/matrix2.so in sage.matrix.matrix2.Matrix.eigenmatrix_right (sage/matrix/matrix2.c:18170)()

/Users/wstein/sage/build/sage-3.2.2.alpha0/local/lib/python2.5/site-packages/sage/matrix/matrix2.so in sage.matrix.matrix2.Matrix.eigenmatrix_left (sage/matrix/matrix2.c:17965)()

IndexError: list index out of range

CC: @jasongrout

Component: linear algebra

Author: Rob Beezer

Reviewer: Alex Ghitza

Merged: sage-4.4.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/4756

[mwhansen]

comment:1

This is because the eigenvectors_left and eigenvectors_right for CDF and RDF matrices returns the data in a way that is totally incompatible with what the parent is expecting.

I can make a patch for this once I have a 3.2.2alpha1 build. Otherwise, if Jason would like to do this, I wouldn't object :-)

[rbeezer]

comment:2

Patch causes left_eigenvectors() and right_eigenvectors() for CDF and RDF to return eigenvalues and eigenvectors in the same format as for exact matrices. Then eigenmatrices for these matrices behave as they should. No attempt is made to identify eigenvalues with multiplicity greater than one.

Patch is failing doctests in the modular form code right now, so will need some more work.

[rbeezer]

Attachment: trac_4756_eigenmatrices_double.patch.gz

[rbeezer]

Self-contained patch, apply only this

[rbeezer]

Author: Rob Beezer

[rbeezer]

comment:3

Attachment: trac_4756-double-eigen.patch.gz

This patch fixes the bug and generally brings double-precision eigenvectors up-to-date.

left_eigenvectors and right_eigenvectors now return their results in the same format as for exact matrices, so routines like eigenmatrix_right can digest the results properly.

Consequently, eigenspaces_left was adjusted for this new format. There was no eigenspaces_right, now there is.

The changed code in modular/modform/numerical.py simply adjusts for the new format to allow doctests to pass and is probably sub-optimal. There is a bug nearby, so this will be addressed on #8018.

Doctests: I had some strange behavior for zero eigenvalues (ie very, very small eigenvalues), so the doctests avoid these. For these eigenvalues, the tests would fail on a first run, but would pass on all subsequent runs (or vice-versa). This was observed on my own machine and boxen.math.washington.edu. So I've avoided these eigenvalues by selecting certain ones in the doctests.

Documentation: documentation was tested for these four functions via notebook introspection. The rest of the file needs attention before it can go into the documentation, see #8046.

[jasongrout]

comment:5

Great job!

I think these changes would make the code easier to read:

 	662	        for k in range(len(spectrum)): 
 	663	            evalue = spectrum[k][0] 
 	664	            evector = spectrum[k][1][0] 
 	665	            pairs.append((evalue, evector.parent().span_of_basis([evector],check=False))) 

changed to

for eval,evectors in spectrum:
    evec = evectors[0]
    evector = evec.parent().span_of_basis([evec],check=False)
    pairs.append((evalue, evector))

(similarly on lines 722-725)

Also:

B = matrix(CDF, [spectrum[i][1][0] for i in range(len(spectrum))]).transpose() 

changed to

B = matrix(CDF, [evecs[0] for _,evecs in spectrum]).transpose()