I think assumptions only work with variables and not functions of
variables. You can specify x>0 so that |x| will simplify to x, but you
cannot specify that cosh(x) > 0 for all real x. The cosh case is simple,
consider the norm of a vector in a Minkowski space. It would be great if
one could specify ahead of time that x_0**2-(x_1**2+x_2**2+x_3**2) was
greater than zero or less than zero or zero as three separate cases.
On Tue, Feb 23, 2016 at 11:19 AM, Nolan Dyck <nolan.dyc...@gmail.com> wrote:
> I understand that symbols are assumed complex by default. But when I tell
> sympy the symbol is real, it does not make the appropriate simplification.
>
> Here is what the dev branch produces for me right now:
>
> >>> from sympy import *
> >>> x = Symbol('x',real=True)
> >>> simplify(abs(cosh(x)))
> Abs(cosh(x))
>
> Is it not possible to assess the 'realness' of the variable within the
> simplify function?
>
> Nolan
>
> On Tuesday, February 23, 2016 at 7:43:34 AM UTC-5, brombo wrote:
>>
>> Sympy assumes symbols to be complex. For a real symbol you need
>>
>> x = Symbol('x',real=True)
>>
>> On Mon, Feb 22, 2016 at 9:44 PM, Nolan Dyck <nolan....@gmail.com> wrote:
>>
>>> Hi Everyone,
>>>
>>> I've been poking around in the sympy source, and I've noticed that the `
>>> simplify` command does not deal with expressions like the following:
>>>
>>> >>> from sympy import *
>>> >>> from sympy.abc import x
>>> >>> simplify(abs(cosh(x)))
>>> Abs(cosh(x))
>>>
>>> A simple glance at the graph of cosh(x) reveals that Abs(cosh(x)) ==
>>> cosh(x). So, in case it's not obvious the above expression should return:
>>>
>>> cosh(x)
>>>
>>> I'd like to take a stab at implementing this but I need some direction.
>>> I hope this isn't duplicating something I have stupidly missed (the
>>> inequality solvers don't seem to identify cases where absolute value
>>> brackets are unnecessary. There are three key cases (as far as I can see):
>>>
>>> 1. The argument within the abs brackets is non-negative over all
>>> combinations of all independent variables. Therefore, the absolute value
>>> brackets redundant / unnecessary and may be removed.
>>> 2. The argument within the abs brackets is non-positive over all
>>> combinations of all independent variables. Therefore, the absolute value
>>> brackets may be removed and the expression may be multiplied by -1 for
>>> the
>>> same effect.
>>> 3. The argument within the abs brackets contains both positive and
>>> negative values depending on the values of the independent variables.
>>>
>>> Ok, so to implement the above rules in the general sense it makes sense
>>> to me to perform the following steps given the expression
>>> Abs(f(x_1,x_2,...x_n)):
>>>
>>> 1. Determine the number of real roots of f(x).
>>> 1. If there are one or more real roots then for each root:
>>> 1. Determine whether the gradient of f(x) is zero:
>>> 1. If non-zero slope at root, then argument expression
>>> obtains opposite signed value at some point, so return Abs(f(x))
>>> 2. Slope of f(x) at root is 0.
>>> 3. Determine if the root is an inflection point (need to
>>> figure out exactly how to test for this over multiple variables in
>>> the
>>> expression)
>>> 1. If at inflection point then the expression will still
>>> become opposite signed on either side of the root, so return
>>> Abs(f(x))
>>> 2. Any and all roots coincide with extrema values of f(x).
>>> Therefore f(x) may be represented without absolute value brackets.
>>> 2. If f(x) >= 0 remove the absolute value brackets and return the
>>> argument expression.
>>> 3. If f(x) < 0 remove the absolute value brackets, multiply the
>>> expression by -1 and return it.
>>>
>>> There are a few things which I'm not sure how it will work out:
>>>
>>> - Imaginary numbers. Does anyone know if I will need to write
>>> special code for this, or should the above procedure work out anyway?
>>> - The case where a symbol in the expression has been defined with
>>> the positive flag:
>>>
>>> >>> y = Symbol('y')
>>> >>> simplify(abs(sinh(y)))
>>> Abs(sinh(y))
>>>
>>> >>> y = Symbol('y',positive=True)
>>> >>> simplify(abs(sinh(y)))
>>> sinh(y)
>>>
>>> - Are there sneaky ways of determining in a precise manner whether a
>>> function which cannot be reduced (e.g. cosh(x)+cos(x)) has real roots,
>>> even
>>> if finding those roots would only be possible numerically? Is there
>>> another
>>> sympy module which can help with this?
>>> - What about variables which produce no real roots over a given
>>> range? Is there a way to handle those? E.g.
>>>
>>> >>> y = Symbol('y',range=[0,pi])
>>> >>> simplify(abs(sin(y)))
>>> sin(y)
>>>
>>> - Redundant absolute value brackets are removed somewhere. Can
>>> anyone tell me where exactly in the code this happens in the simplify
>>> function? I can't seem to find it:
>>>
>>> >>> from sympy.abc import x
>>> >>> simplify(abs(abs(x)+1))
>>> Abs(x) + 1
>>> >>> simplify(abs(x+1))
>>> Abs(x + 1)
>>>
>>> So, right now I have forked the sympy repo (see here
>>> <https://github.com/NauticalMile64/sympy>) and set up my own little
>>> function in sympy.symplify called abssimp.py (just copied combsimp.py and
>>> started from there), and added an appropriate if-absolute check in the main
>>> simplify function. Is this the right way to go about adding such a feature?
>>> Would the code that I write here also be used in solve or something?
>>>
>>> Any guidance / advice would be appreciated.
>>>
>>> Thanks!
>>> Nolan Dyck
>>>
>>> --
>>> You received this message because you are subscribed to the Google
>>> Groups "sympy" group.
>>> To unsubscribe from this group and stop receiving emails from it, send
>>> an email to sympy+un...@googlegroups.com.
>>> To post to this group, send email to sy...@googlegroups.com.
>>> Visit this group at https://groups.google.com/group/sympy.
>>> To view this discussion on the web visit
>>> https://groups.google.com/d/msgid/sympy/aeb34566-d4b8-46f4-8eb5-f7acabbd12f4%40googlegroups.com
>>> <https://groups.google.com/d/msgid/sympy/aeb34566-d4b8-46f4-8eb5-f7acabbd12f4%40googlegroups.com?utm_medium=email&utm_source=footer>
>>> .
>>> For more options, visit https://groups.google.com/d/optout.
>>>
>>
>> --
> You received this message because you are subscribed to the Google Groups
> "sympy" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to sympy+unsubscr...@googlegroups.com.
> To post to this group, send email to sympy@googlegroups.com.
> Visit this group at https://groups.google.com/group/sympy.
> To view this discussion on the web visit
> https://groups.google.com/d/msgid/sympy/9128b7db-cb5b-4551-81fb-d7eb3b8195f4%40googlegroups.com
> <https://groups.google.com/d/msgid/sympy/9128b7db-cb5b-4551-81fb-d7eb3b8195f4%40googlegroups.com?utm_medium=email&utm_source=footer>
> .
>
> For more options, visit https://groups.google.com/d/optout.
>
--
You received this message because you are subscribed to the Google Groups
"sympy" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to sympy+unsubscr...@googlegroups.com.
To post to this group, send email to sympy@googlegroups.com.
Visit this group at https://groups.google.com/group/sympy.
To view this discussion on the web visit
https://groups.google.com/d/msgid/sympy/CALOxT-%3DNgzQNij8r83nMhG6WUvyyDSVjC9FvvYKCHHLvuD7WFg%40mail.gmail.com.
For more options, visit https://groups.google.com/d/optout.
