solve() can return undefined points as "solutions"

[cwitty]

Consider the following examples (reported by Dean Moore here: ​http://groups.google.com/group/sage-support/browse_thread/thread/5555e780a76b3343#)

sage: solve(sin(x^2)/x == 0)
[x == 0]
sage: solve(sin(x^2)/x^2 == 0)
[x == 0]
sage: solve(sin(x^2)/x^3 == 0)
[x == 0]

None of these functions are even defined at x=0, so that should not be returned as a solution. (The first two functions can be extended to x=0 by taking limits, in which case x=0 is a solution to the first one but not the second; the third function has a vertical asymptote at x=0.)

[mabshoff]

Is this a bug in Maxima? In that case we should report those to them? This also seams like a fairly serious issue, so I am elevating this to critical.

Cheers,

Michael

[kcrisman]

This is a Maxima bug as of 5.16.3, and has been reported there as 2845005 (see ​http://sourceforge.net/tracker/?func=detail&aid=2845005&group_id=4933&atid=104933).

[robert.marik]

Perhaps related issue is also that the solving acot(x) == 0 ends with error message "The number 0 isn't in the domain of cot"

The online tool Mathatmatical Assistant on Web ( ​http://user.mendelu.cz/marik/maw/index.php?lang=en&form=main ) has a wrapper for maxima's solve ( ​http://mathassistant.cvs.sourceforge.net/viewvc/mathassistant/maw/common/maw_solve.mac?revision=1.14&view=markup )

I hope, it could be used also in Sage. I'll try it, hope within a week.

[kcrisman]

Replying to robert.marik:

Perhaps related issue is also that the solving acot(x) == 0 ends with error message "The number 0 isn't in the domain of cot"

No, this is an appropriate error message (it's from Maxima, not Sage). There are no solutions to acot(x)==0, at least over the reals (and presumably over the complex field as well?). Now that we know about that error, it would be easy to put a catch in for something like that error message and return sage: solve(acot(x),x) [] instead. Feel free to open a ticket for that and put me in the cc: field.

But this is unrelated to the issue in the ticket, which is a genuine Maxima bug, as far as I can tell.

[kcrisman]

Replying to kcrisman:

Replying to robert.marik:

Perhaps related issue is also that the solving acot(x) == 0 ends with error message "The number 0 isn't in the domain of cot"

No, this is an appropriate error message (it's from Maxima, not Sage). There are no solutions to acot(x)==0, at least over the reals (and presumably over the complex field as well?). Now that we know about that error, it would be easy to put a catch in for something like that error message and return sage: solve(acot(x),x) []

This will be addressed (not the main point of this ticket) in the patch for #7745. The main point is still a bug in Maxima 5.20.1.

[robert.marik]

I had an idea to introduce new option to solve, which

1. Takes only explicit solutions

2. Substitutes into equation and if an error appears, removes this "solution" from the list.

The problem in this approach is, that for example ln(0)=-Infinity in Sage and so x=0 will be still reported as a solution of x/ln(x)=0. The problem could be solved by substituting values in Maxima and not in Sage, but I am still thinking on some cleaner solution. And still have no idea what should be returned as solution of x*ln(x-3) == 0. Distinguish in this new option, if the user works in real domain or in complex doman? Something like check_domain = False, True, or 'real'?

Any idea?

[kcrisman]

As it turns out, to_poly_solve can handle this sort of thing (see in Maxima the share/contrib/rtest_to_poly_solver.mac line 1092). But we would have to figure out a way to interpret the if statements properly (for instance, to note that twice an integer plus one is not zero).

/* Sage Ticket 2617; see also Sage mailing list 18 March 2008 */

nicedummies(to_poly_solve(sin(x^2)/x,x));
%union(%if(2*%z0+1 # 0,[x = -sqrt(2*%pi*%z0+%pi)],%union()),
             %if(2*%z0+1 # 0,[x = sqrt(2*%pi*%z0+%pi)],%union()),
             %if(%z1 # 0,[x = -sqrt(2)*sqrt(%pi)*sqrt(%z1)],%union()),
             %if(%z1 # 0,[x = sqrt(2)*sqrt(%pi)*sqrt(%z1)],%union()))$

nicedummies(to_poly_solve(sin(x^2)/x^2,x));
%union(%if(2*%z0+1 # 0,[x = -sqrt(2*%pi*%z0+%pi)],%union()),
             %if(2*%z0+1 # 0,[x = sqrt(2*%pi*%z0+%pi)],%union()),
             %if(%z1 # 0,[x = -sqrt(2)*sqrt(%pi)*sqrt(%z1)],%union()),
             %if(%z1 # 0,[x = sqrt(2)*sqrt(%pi)*sqrt(%z1)],%union()))$

nicedummies(to_poly_solve(sin(x^2)/x^3,x));
%union(%if(2*%z0+1 # 0,[x = -sqrt(2*%pi*%z0+%pi)],%union()),
             %if(2*%z0+1 # 0,[x = sqrt(2*%pi*%z0+%pi)],%union()),
             %if(%z1 # 0,[x = -sqrt(2)*sqrt(%pi)*sqrt(%z1)],%union()),
             %if(%z1 # 0,[x = sqrt(2)*sqrt(%pi)*sqrt(%z1)],%union()))$

[zonova]

SO, has this issue been fixed yet? What of kcrisman and robert.marik's suggestions? Also is there a reason that there is no branch to edit?

[kcrisman]

Presumably not fixed. No branch because no one has posted one yet - if you have a fix you can be the first to post a branch!

[zonova]

kcrisman, in your post (from 7 years ago) you had mentioned to_poly_solve in maxima's share/contrib. It's been a while since then, so it is not located there anymore. I couldn't find it anywhere in Maxima's source on github though, so i wasn't sure if it was still used at all. Does sage/maxima use it at all?

I've been looking at several old tickets, all involving solving equations. One was using find_root, which uses scipy, and the other had to do with solve just like this one. I think it would be best to just have one, no? As far as I can tell, they do about the same thing, and they both have issues. On a similar note, if to_poly_solve resolves this issue, then maybe we should use that for all equation solving?

[kcrisman]

We definitely have that method and it is still in Maxima. Looks like it moved to ​https://sourceforge.net/p/maxima/code/ci/master/tree/share/to_poly_solve/.

However, find_root is explicitly supposed to be a numerical solver, while solve is supposed to be an exact solver. Because to_poly_solve sometimes returns numerical answers in rare situations, there could be some overlap. Also, to_poly_solve is not what we want for all solving, because it changes some other things and of course might take longer for simple ones. It has a specific purpose, but that is not a general purpose.

On the other hand, if sympy can now solve everything Maxima does, one could try to switch the default algorithm to use that instead. I don't know if we're at that point, though.

[gh-Torrencem]

This has still been an issue, and so I've implemented an optional solution

You can pass the optional argument (check_domain) to solve, which will tell it to check each solution it finds with SymPy? to see if it's NaN. I added some documentation and an example.

I ran (on my machine) all the current doc-tests with the new argument to make sure it accepts all their solutions, and it does, but I believe it significantly slows down the function, so probably should not end up defaulting to True without some more considerations / optimizations (especially since solve is widely used)

See the added documentation example:

sage: solve((x^2 - 1)/(sin(x - 1)) == 0, x, check_domain=True)
[x == -1]

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New commits:

​6d5aa9c	2617: Add check_domain argument to solve to remove results outside of the domain
​2cc6fd5	2617: Add small documentation note