use solve_left for division operation of matrices

[mwageringel]

This ticket improves the __truediv__ division operation of matrices and vectors by implementing it using solve_left (the _backslash_ operator is already implemented using solve_right).

Currently, __truediv__ only works for square matrices with the same parent. The implementation naively computes the inverse of a matrix, which should be avoided, especially over inexact rings.

With the changes of this ticket (and #12406), the operation works over differing rings and even with non-square matrices.

sage: a = matrix(ZZ, [[1, 2], [0, 3]])
sage: b = matrix(ZZ, 3, 2, range(6))
sage: b / a == b * ~a
True

sage: a = matrix(ZZ, [[1, 2], [0, 3], [1, 5]])
sage: (b / a) * a == b
True

sage: b = vector(RDF, [1.5, 2.5])
sage: (b / a * a - b).norm() < 1e-14
True

Depends on #12406

Component: linear algebra

Author: Markus Wageringel

Branch/Commit: 653849a

Reviewer: Jonathan Kliem

Issue created by migration from https://trac.sagemath.org/ticket/29257

[mwageringel]

Branch: u/gh-mwageringel/29257

[mwageringel]

Author: Markus Wageringel

[mwageringel]

Dependencies: #12406

[mwageringel]

comment:1

Based on #12406.

---

New commits:

95e4157	Fix minor typo.
8c41acf	Fix hidden bug in RDF/CDF matrix multiplication dead error-checking.
5844bbe	Change RDF/CDF matrix multiplication to accept matrices for B.
fab2bcd	Improve exceptions
bd6e4fc	Merge tag '9.1.beta4' into #17405
7b07a27	17405: fix solve_right and solve_left over RDF and CDF
0ccb65a	12406: implement coercion for generic solve_right
a815ff9	12406: explicitly convert to AA in generalized_permutahedron
828d67a	29257: use solve_left for `__truediv__` operation of matrices and vectors

[mwageringel]

Commit: 828d67a

[sagetrac-git]

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

859900e

29257: use solve_left for `__truediv__` operation of matrices and vectors

[sagetrac-git]

Changed commit from 828d67a to 859900e

[mwageringel]

comment:3

Now this also passes the tests with Python 2.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. Last 10 new commits:

828d9db	12406: implement reviewer suggestions
6d0bbf6	12406: add a doctest to check consistency of coercion in solve_right
d751e78	Trac # 12406: document that "check" means nothing over inexact rings.
71f0c28	Trac #12406: absorb RDF/CDF solve_* into the superclass methods.
9aa2eb3	Trac #12406: unbreak doctests for exact rings.
988e215	12406: move rank checks to _solve_right_nonsingular_square
99c2a7e	12406: test that `check` is ignored over inexact rings
c1178d9	12406: add NotFullRankError for _solve_right_nonsingular_square
93cea85	12406: test that the solution is inexact
be62f78	29257: use solve_left for `__truediv__` operation of matrices and vectors

[sagetrac-git]

Changed commit from 859900e to be62f78

[mwageringel]

comment:5

Rebased on top of #12406.

[mkoeppe]

comment:6

Batch modifying tickets that will likely not be ready for 9.1, based on a review of the ticket title, branch/review status, and last modification date.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

653849a

Merge tag '9.2.beta8' into #29257

[sagetrac-git]

Changed commit from be62f78 to 653849a

[kliem]

Reviewer: Jonathan Kliem

[kliem]

comment:8

LGTM.

[mwageringel]

comment:9

Thank you. There was a related discussion in #29226 which has not been crosslinked here yet. IMHO, this is still an improvement over the current behavior, but it is okay if you would like to reconsider the review.

[kliem]

comment:10

I see. Of course we can remove division of matrices, but even our documentation https://doc.sagemath.org/html/en/tutorial/tour_linalg.html
advertises using the backslash.

For the time being it's an improvement. Even if we want to get rid of it eventually, at this point we could raise a deprecation error more easily and say, "use solve left/right instead".

[mwageringel]

comment:11

Ok, sounds good to me.

[vbraun]

Changed branch from u/gh-mwageringel/29257 to 653849a