Solve function in SageMath produces some strange output while finding roots of a derivative of function (exp(x)*(1-x))^n/x^m. Entering the code below
var('x, m, n')
hh=(exp(x)*(1-x))^n/x^m
solve(hh.diff(x)==0,x)produces output
(x, m, n)
[x^2 == (m*x^2 - m)/n]What does `[x^2 == (m*x^2 - m)/n] is supposed to mean? By contrast Sympy quickly produces a correct easy to understand answer :
[m−√m(m−4n)2n,m+√m(m−4n)2n]
Step by step :
sage: var('x, m, n')
(x, m, n)
sage: hh=(exp(x)*(1-x))^n/x^m
sage: solve(hh.diff(x)==0,x)
[x^2 == (m*x^2 - m)/n]That's an implicit solution. Sage's default solver (Maxima's) can be coaxed to find for a more explocit answer :
sage: solve(hh.diff(x)==0,x, to_poly_solve=True)
[x == -sqrt(m/(m - n)), x == sqrt(m/(m - n))]Or you may use solve recursively) :
sage: solve(solve(hh.diff(x)==0,x),x)
[x == -sqrt(m/(m - n)), x == sqrt(m/(m - n))]or
sage: solve(hh.diff(x)==0,x)[0].solve(x)
[x == -sqrt(m/(m - n)), x == sqrt(m/(m - n))]But beware : you should check these vatious solutions.And be wary of simplify_full which may use simplifications validonly certain assumptions...
Left as an exercise for the reader...
HTH,
EDIT :
the final solution [x == -sqrt(m/(m - n)), x == sqrt(m/(m - n))] that you obtained by repeated application of solve procedure is wrong, so this is the bug of "Maxima" after all.
Indeed :
sage: [Eq.subs(t).simplify_full().is_zero() for t in flatten([s.solve(x) for s in S1])]
[False, False]This is now #35541
Workarounds :
sage: Ex.factor().solve(x)
[x == 1, x == 1/2*(m - sqrt(m^2 - 4*m*n))/n, x == 1/2*(m + sqrt(m^2 - 4*m*n))/n, x == 0]
sage: [Ex.subs(s).is_zero() for s in Ex.factor().solve(x)]
[True, True, True, True]
sage: Ex.solve(x, algorithm="giac")
[1/2*(m - sqrt(m^2 - 4*m*n))/n, 1/2*(m + sqrt(m^2 - 4*m*n))/n]
sage: [Ex.subs(x==s).is_zero() for s in Ex.solve(x, algorithm="giac")]
[True, True]
sage: [{v[1].sage():v[2].sage() for v in s} for s in mathematica.Solve(Ex==0, x)]
[{x: -1/2*(sqrt(m - 4*n)*sqrt(m) - m)/n},
{x: 1/2*(sqrt(m - 4*n)*sqrt(m) + m)/n}]
sage: [Ex.subs(s).is_zero() for s in [{v[1].sage():v[2].sage() for v in s} for s in mathematica.Solve(Ex==0, x)]]
[True, True]HTH,
Thank you for the clarification that the solution provided by "solve()" function uses "Maxima" and is supposed to be understood as an implicit solution! However, if this is the case then the final solution [x == -sqrt(m/(m - n)), x == sqrt(m/(m - n))] that you obtained by repeated application of solve procedure is wrong, so this is the bug of "Maxima" after all.
Simplifying the function may help:
var('x, m, n');
hh=(exp(x)*(1-x))^n/x^m
solve((hh.diff(x)).simplify_full()==0,x)Thank you very much! Applying method "simplify_full()" produces expression that "solve" function can correctly solve. However, as a new user of SageMath I would like to ask a follow up question, as to what was the problem with the initial solution of the equation? Was the original answer [x^2 == (m*x^2 - m)/n] provided by solve() function incorrect and a result of a bug, or could it be interpreted to provide a correct answer?
Both in fact :
This has the form of an implicit answer, which I (incorrectly) tried to explicit.
It turns out that this implicit answer is wrong.
See the Github issue I cite in my answer's EDIT.
HTH,
Asked: 1 year ago
Seen: 254 times
Last updated: Apr 19 '23
Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.
Edited for readability.