Solve for a function defined in a file?

[David Ingerman]

 

 How  to solve([f(x)==0],x) for a function "f(x)" defined in a .sage file?

 The error message: TypeError: Cannot evaluate symbolic expression to a numeric value.

Thank you...

[Dima Pasechnik]

what is f(x) ?
solve() won't work for a Python function, it needs a symbolic
expression, e.g.

sage: type(sin(x))
<type 'sage.symbolic.expression.Expression'>
sage:

>
> Thank you...
>

[David Ingerman]

 Thank you, so what to do for Python function? Matlab had general purpose 'optim(f)' if my memory is right... 

[Dima Pasechnik]

On 2014-06-20, David Ingerman <davidd...@gmail.com> wrote:
> Thank you, so what to do for Python function? Matlab had general purpose
> 'optim(f)' if my memory is right...

you can e.g. use find_root(); this is a numerical thing that accepts
Python functions. Here is an example:

sage: def f(x):
return x-cos(x)
....:
sage: find_root(f,0,1)
0.7390851332151559
sage:

>
> On Wednesday, June 11, 2014 1:50:10 AM UTC-7, Dima Pasechnik wrote:
>>

[David Ingerman]

 Thank you, that's helpful. Is there a way to get all roots of a Python function on an interval? 

[Dima Pasechnik]

On 2014-06-21, David Ingerman <davidd...@gmail.com> wrote:
> Thank you, that's helpful. Is there a way to get all roots of a Python
> function on an interval?

I never heard of robust procedures for such a task, and doubt they are
even possible (think about roots of sin(1/x) on [0,1]).
Certainly you can partition the interval into pieces
and try finding root in each subinterval where the function has
different signs on its ends.



scipy has a variety of root-finding methods implemented. You
can do
sage: import scipy.optimize
and then read on
sage: scipy.optimize.brentq?
(the method called by Sage's find_root)

>
> On Friday, June 20, 2014 2:10:38 AM UTC-7, Dima Pasechnik wrote:
>>

[David Ingerman]

 Thank you, that makes sense. My Python function is not continuous though, has poles, so I'll probably have to plot it to find its zeros... 

[Dima Pasechnik]

On 2014-06-22, David Ingerman <davidd...@gmail.com> wrote:
> Thank you, that makes sense. My Python function is not continuous though,
> has poles, so I'll probably have to plot it to find its zeros...

you might perhaps try finding a pole p by solving 1/f(x)=0, and
regularise f by multiplying f with (x-p)i^m, for appropriate m.
Repeat until all the poles in the interval are taken care of.
Similarly you can divide f by (x-r)^k for a zero r. In effect this
amounts to finding a Pade approximation of f.
(no idea how well this behaves in practice)

>
> On Saturday, June 21, 2014 1:21:06 AM UTC-7, Dima Pasechnik wrote:
>>

[David Ingerman]

 Have zeros of order 2 too, so the sign change doesn't help in general, but may work for some zeros. Thank you!...