Great. Thanks Aaron and Brombo for your replies. I have a few follow up
questions:
1. I'd like to add in those checks into the assumptions system for
hyperbolic functions at least. Is that a reasonable entry level task?
2. I noticed that code sample you gave checks the is_real field on the
first argument. Is there an is_positive field I could check so that sinh
and other functions could also be simplified in some cases (i.e. x is
positive, Abs(sinh(x)) -> sinh(x))?
Nolan
On Tuesday, February 23, 2016 at 3:19:09 PM UTC-5, Aaron Meurer wrote:
>
> For cosh, it should just be handled by the assumptions system. That
> is, cosh should have
>
> def _eval_is_nonnegative(self):
> if self.args[0].is_real:
> return True
>
> (and similar for the new assumptions _eval_refine).
>
> On Mon, Feb 22, 2016 at 9:44 PM, Nolan Dyck <[email protected]
> <javascript:>> 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):
>
> To do anything nontrivial in this area the inequality solvers will
> definitely need to be improved.
>
> >
> > 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.
> > 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.
> > 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)):
> >
> > Determine the number of real roots of f(x).
> >
> > If there are one or more real roots then for each root:
> >
> > Determine whether the gradient of f(x) is zero:
> >
> > If non-zero slope at root, then argument expression obtains opposite
> signed
> > value at some point, so return Abs(f(x))
> >
> > Slope of f(x) at root is 0.
> > Determine if the root is an inflection point (need to figure out exactly
> how
> > to test for this over multiple variables in the expression)
> >
> > If at inflection point then the expression will still become opposite
> signed
> > on either side of the root, so return Abs(f(x))
> >
> > Any and all roots coincide with extrema values of f(x). Therefore f(x)
> may
> > be represented without absolute value brackets.
> >
> > If f(x) >= 0 remove the absolute value brackets and return the argument
> > expression.
> > 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:
>
> This algorithm only works if the function is real and differentiable
> over the given domain.
>
> >
> > 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?
>
> solveset aims to give a full set of solutions to an equation over an
> equation (compared with solve(), which makes no guarantees about
> giving all solutions). It's a difficult problem, though, and so far it
> is limited in what it can do.
>
> > 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)
>
> For a continuous function, you only need to check the intervals
> between the zeros. You can find all the zeros of sin(y) in the range
> [0, pi] (0 and pi), so you only have to check a value between them.
>
> I don't know if this is necessarily the best way to implement this in
> SymPy (one needs the ability to determine if a function is continuous,
> and to find all its roots). At any rate, Symbol doesn't have a "range"
> argument currently. You can use solveset(sin(y), y, domain=Interval(0,
> pi)) (note there is a bug with this in master
> https://github.com/sympy/sympy/issues/10671).
>
> >
> > 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:
>
> It happens in the Abs constructor, i.e., when you create the Abs. Abs
> is defined in sympy.functions.elementary.complexes.
>
> >
> >>>> 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) 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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