solve() can return undefined points as "solutions"

[sagetrac-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)
[x == 0]
sage: solve(sin(x^2)/x^2 == 0, x)
[x == 0]
sage: solve(sin(x^2)/x^3 == 0, x)
[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.)

Component: calculus

Stopgaps: todo

Author: Matt Torrence

Branch/Commit: u/gh-Torrencem/2617_solve_check_domain @ 2cc6fd5

Reviewer: Vincent Delecroix

Issue created by migration from https://trac.sagemath.org/ticket/2617

[sagetrac-mabshoff]

comment:1

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]

comment:2

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]

comment:3

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]

comment:4

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]

comment:5

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]

comment:6

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]

comment:7

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()))$