On Dec 17, 2008, at 11:49 AM, David Joyner wrote:
>
> On Wed, Dec 17, 2008 at 2:41 PM, Simon King <k...@mathematik.uni-
> jena.de> wrote:
>>
>> Dear Sage supporters,
>>
>> inspired by http://groups.google.com/group/sage-support/
>> browse_thread/thread/729f4a557d970195?hl=en,
>> I was playing around with the solve command.
>>
>> I found two questions:
>> 1. How can one add assumptions (such as x>0 or x is an integer)?
>> The following does not work:
>> sage: assume(x>0)
>> sage: solve(x^2-1,x)
>> [x == -1, x == 1]
>> The Sage Tutorial doesn't mention assumptions in the section about
>> solving equations.
>
>
> I don't know the answer
The problem is that maxima ignores the assume command in solving. I
don't know if this is intentional behavior or a bug.
(%i1) assume(x>0);
(%o1) [x > 0]
(%i2) solve(x^2-1=0, x);
(%o2) [x = - 1, x = 1]
Assumptions that something is an integer while solving would be
highly nontrivial. Imagine
sage: var('a,b,c,n')
sage: assume(a, 'integer')
sage: assume(b, 'integer')
sage: assume(c, 'integer')
sage: assume(d, 'integer')
sage: assume(n > 2)
sage: solve(a^n + b^n == c^n, n)
> but if assume doesn't work then I would vote to
> deprecate it and replace it by an option such as
>
> solve(x^2-1,x,assume="x>0")
> or
> solve(x^2-1,x,options="assume(x>0)")
> or something. Then just pass the assume statement to
> maxima before the solve statement.
Usually I like to avoid using strings to carry more structured
mathematical information. I do think solve(x^2-1, x, assume=x>0)
wouldn't be a bad thing to support. This all depends on how
assumptions are going to be done in the new symbolics.
>> 2. Why does it take so long to return the solution dictionary?
>> From the solve-docstring:
>> sage: ct=cputime()
>> sage: time solve([x^2+y^2 == 1, y^2 == x^3 + x + 1], x, y,
>> solution_dict=True)
>> CPU times: user 0.07 s, sys: 0.00 s, total: 0.07 s
>> Wall time: 0.18 s
>> [{y: -sqrt(3 - sqrt(3)*I)/sqrt(2), x: (-sqrt(3)*I - 1)/2},
>> {y: sqrt(3 - sqrt(3)*I)/sqrt(2), x: (-sqrt(3)*I - 1)/2},
>> {y: -sqrt(sqrt(3)*I + 3)/sqrt(2), x: (sqrt(3)*I - 1)/2},
>> {y: sqrt(sqrt(3)*I + 3)/sqrt(2), x: (sqrt(3)*I - 1)/2},
>> {y: -1, x: 0},
>> {y: 1, x: 0}]
>> sage: cputime(ct)
>> 2.6121629999999989
>>
>> So, for computing the solutions 0.07 CPU-seconds suffice, but for
>> printing them 2.6 CPU-seconds are needed??
cputime only measures the time of the Sage process (formatting the
equations, feeding them to maxima, and parsing and printing the
result). The forked maxima process, which is actually doing all the
work in this example, takes the rest. As for printing, I'm not sure
what's taking so long (simplifying each result individually?), but it
shouldn't.
sage: time soln = solve([x^2+y^2 == 1, y^2 == x^3 + x + 1], x, y,
solution_dict=True)
CPU times: user 0.02 s, sys: 0.01 s, total: 0.03 s
Wall time: 0.14 s
sage: print soln # fast
sage: str(soln) # fast
sage: repr(soln) # fast
sage: soln # slow !?
- Robert
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