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 <[email protected]> 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
>
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