add tolerance to is_hermitian() for sparse "double" matrices

[orlitzky]

Discovered in #33023, the implementation of is_hermitian() for sparse double matrices is the generic one that tests equality of entries, and can fail due to numerical imprecision.

We already have an is_hermitian() for DENSE double matrices, but it is not used for the sparse class, which is its sibling and not a child. A quick way to fix this is to implement the "naive" version from the sense class in the sparse class. But a better solution might be to move the "naive" algorithm for dense matrices up into the superclass, and just allow the generic method to take a tolerance. If that's stable, it would allow us to avoid reimplementing is_hermitian() three times.

When this has been fixed, the # known bug tags need to be removed from the following tests in src/sage/matrix/matrix_double_sparse.pyx:

sage: L = A.cholesky()                    # known bug, 33031        
sage: (A - L*L.T).norm(1) < 1e-10         # known bug, 33031        
True                                                                
sage: B = A.change_ring(RR)               # known bug, 33031        
sage: (B.cholesky() - L).norm(1) < 1e-10  # known bug, 33031

sage: L = A.cholesky()                    # known bug, 33031        
sage: (A - L*L.H).norm(1) < 1e-10         # known bug, 33031        
True                                                                
sage: B = A.change_ring(CC)               # known bug, 33031        
sage: (B.cholesky() - L).norm(1) < 1e-10  # known bug, 33031

Depends on #33023

CC: @collares @kliem

Component: linear algebra

Author: Michael Orlitzky

Branch/Commit: 4c14a9e

Reviewer: Dima Pasechnik

Issue created by migration from https://trac.sagemath.org/ticket/33031

[orlitzky]

Description changed:

--- 
+++ 
@@ -2,3 +2,20 @@
 
 We already have an `is_hermitian()` for DENSE double matrices, but it is not used for the sparse class, which is its sibling and not a child. A quick way to fix this is to implement the "naive" version from the sense class in the sparse class. But a better solution might be to move the "naive" algorithm for dense matrices up into the superclass, and just allow the generic method to take a tolerance. If that's stable, it would allow us to avoid reimplementing `is_hermitian()` three times.
 
+When this has been fixed, the `# known bug` tags need to be removed from the following tests in `src/sage/matrix/matrix_double_sparse.pyx`:
+
+```
+sage: L = A.cholesky()                    # known bug, 33031        
+sage: (A - L*L.T).norm(1) < 1e-10         # known bug, 33031        
+True                                                                
+sage: B = A.change_ring(RR)               # known bug, 33031        
+sage: (B.cholesky() - L).norm(1) < 1e-10  # known bug, 33031
+```
+
+```
+sage: L = A.cholesky()                    # known bug, 33031        
+sage: (A - L*L.H).norm(1) < 1e-10         # known bug, 33031        
+True                                                                
+sage: B = A.change_ring(CC)               # known bug, 33031        
+sage: (B.cholesky() - L).norm(1) < 1e-10  # known bug, 33031
+```

[dimpase]

comment:2

This also needs to take into account cases with entries such as RBF and CBF, RIF amd CIF.

[orlitzky]

comment:3

Replying to @dimpase:

This also needs to take into account cases with entries such as RBF and CBF, RIF amd CIF.

is_hermitian() already fails on RBF and RIF matrices for want of a conjugate() method...

[orlitzky]

comment:4

Here's a proof-of-concept that attempts to fix sparse RDF/CDF without breaking anything else.

---

New commits:

295508e	Trac #33023: disable flaky cholesky() tests.
a2359e7	Trac #33031: add tolerance to _is_hermitian().
7ff6363	Trac #33031: use superclass for naive RBF/CDF is_hermitian().
dd5b12d	Trac #33031: add tolerance for sparse RDF/CDF is_hermitian().
d202b67	Trac #33031: speed up is_hermitian() for sparse matrices.
1dea31b	Trac #33031: fix is_hermitian() over finite fields.

[orlitzky]

Branch: u/mjo/ticket/33031

[orlitzky]

Author: Michael Orlitzky

[orlitzky]

Commit: 1dea31b

[orlitzky]

Dependencies: #33023

[orlitzky]

Work Issues: actually fix is_hermitian() and not just cholesky()

[sagetrac-git]

Changed commit from 1dea31b to 0f04c34

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

0752485	Trac #33031: add tolerance for sparse RDF/CDF is_hermitian().
95714d1	Trac #33031: speed up is_hermitian() for sparse matrices.
9339252	Trac #33031: fix is_hermitian() over finite fields.
0f04c34	Trac #33031: add tolerance for sparse RDF/CDF is_skew_hermitian().