Although both behaviors may be reasonable, I don't like the fact that each
line of the following code gives a completely different result depending on
whether F = RR (returns a result that may be garbage, without warning) or F
= RDF (gives an error, even though there may be a valid solution). My own
preference for default behavior (for F = RR, at least, which is where I
think a naive user is likely to be working) is intermediate between these
two, and is what I think MATLAB does: return a result but give a warning.
sage: matrix(F, [[0]]).solve_right(vector(F, [0]))
sage: matrix(F, [[0]]).solve_right(vector(F, [1]))
Along these lines, I note that the code for
`Matrix_generic_dense.solve_right` has
try:
X = self._solve_right_nonsingular_square(C, check_rank=True)
except NotFullRankError:
X = self._solve_right_general(C, check=check)
It seems to me that the except clause would be a good place to put a
warning message over inexact rings. However, I know nothing about numerical
analysis (and not much about sage), so I am certainly not giving an expert
opinion on anything in this discussion, just the viewpoint of this sage
user, and I apologize if it is noise.
On Saturday, May 23, 2020 at 12:30:56 PM UTC-6, Markus Wageringel wrote:
>
> Indeed, instead of ignoring the `check` keyword altogether over inexact
> fields, it would be useful to verify the solution when that is possible,
> such as for exact input over SR and for ball and intervall fields. I have
> opened #29729 <https://trac.sagemath.org/ticket/29729> for this. For
> general inexact rings though, the result cannot be verified, so it would
> not make sense to do the check by default.
>
>
> Am Samstag, 23. Mai 2020 07:15:45 UTC+2 schrieb Dave Morris:
>>
>> From the documentation at
>> https://www.mathworks.com/help/matlab/ref/mldivide.html, it appears to
>> me that MATLAB gives a warning: "Warning: Matrix is close to singular or
>> badly scaled. Results may be inaccurate." That seems to me to be a better
>> default behavior than what sage is doing now, but I think it would make
>> sense to let users choose to turn off the warning (and maybe also let them
>> upgrade it to an error).
>>
>
> The generic implementation of `solve_right` does not provide this. I
> would suggest to use a type that has a specialized implementation for this,
> such as RDF, CDF, CBF. For example over RDF and CDF, a warning is issued if
> the input is ill-conditioned and an error is raised if the matrix is
> singular, which is a reasonable behavior:
>
> sage: matrix(RDF, [[2/3, 1], [0.4, 0.6]]).solve_right(vector(RDF, [1, 1]))
> /Applications/SageMath/src/bin/sage-ipython:1: LinAlgWarning: Ill-conditioned
> matrix (rcond=2.77556e-17): result may not be accurate.
> #!/usr/bin/env sage-python
> (5404319552844596.0, -3602879701896396.0)
> sage: matrix(RDF, [[0]]).solve_right(vector(RDF, [1]))
> ...
> LinAlgError: Matrix is singular.
>
>
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