Add `simplify_real` method to symbolic expressions

[orlitzky]

Symbolic expressions in sage are by default assumed complex. There is a maxima variable, called the "simplification domain," which affects whether or not it simplifies sqrt(x^2) to abs(x). Since our expressions are complex, we set the simplification domain to complex, but provide no easy way to change it.

By adding a simplify_real() method to Expression, we give the user a way to perform the aforementioned simplification by declaring his expression real.

This might provide a quick fix for #14305. See also:

https://groups.google.com/forum/?fromgroups=#!topic/sage-support/jhCJujRtNA4/discussion

Depends on #11912

CC: @egourgoulhon

Component: symbolics

Author: Michael Orlitzky

Branch/Commit: 649e3b3

Reviewer: Karl-Dieter Crisman, Ralf Stephan

Issue created by migration from https://trac.sagemath.org/ticket/14630

[orlitzky]

Author: Michael Orlitzky

[orlitzky]

comment:1

MathJAX is throwing a "Math Processing Error" -- I'm not sure whether it's busted or I screwed up but the docs definitely need a review.

[kcrisman]

comment:2

You need to do \left|x\\right|.

---

I'm not sure about all the stuff in the doc and the code, but I think that some of the doc is doing the code and vice versa? For instance,

# Forget all assumptions

but you don't do that in the code. And the corresponding part in the doc seems to imply that in order to use simplify_real you have to assume variables are real... I may be misunderstanding something here. I think this could be a good way to solve the issue at hand.

My question is how this will interact with the other simplifications. I seem to recall that in some of the more controversial (e.g. radcan-related) simplifications we do, part of the issue is whether the variable is real... that recollection may be outdated. Anyway, I could imagine that simplify_foo simplified differently whether one was real or complex, though I hesitate to add simplify_foo_real for all foo!

[orlitzky]

Attachment: sage-trac_14630.patch.gz

Add simplify_real() method to Expression

[orlitzky]

comment:3

Replying to @kcrisman:

You need to do \left|x\\right|.

Fixed in the new patch, but my MathJAX still doesn't work, so please give it a look.

I'm not sure about all the stuff in the doc and the code, but I think that some of the doc is doing the code and vice versa? For instance,

# Forget all assumptions

but you don't do that in the code.

Whoops, I forgot to call forget(); the comment was correct. I fixed it and added a doctest to ensure that no new assumptions remain after the call.

And the corresponding part in the doc seems to imply that in order to use simplify_real you have to assume variables are real... I may be misunderstanding something here. I think this could be a good way to solve the issue at hand.

I didn't mean that the user has to assume() anything, only that we have to assume things are real in the traditional sense. You can call simplify_real() on an expression containing complex variables, but the answer might not make sense.

My question is how this will interact with the other simplifications. I seem to recall that in some of the more controversial (e.g. radcan-related) simplifications we do, part of the issue is whether the variable is real... that recollection may be outdated. Anyway, I could imagine that simplify_foo simplified differently whether one was real or complex, though I hesitate to add simplify_foo_real for all foo!

Expressions should be complex everywhere. I believe there are one or two functions which treat them as real, but they're documented to do that, so it's fine. Radcan via simplify_radical() will still produce crazy answers that are invalid for complex numbers, but there isn't much that can be done about that now except revoke the name "simplify" from "simplify_radical."

Not many expressions are actually affected by the domain and 'real' assumptions. We can get most of the benefit of simplify_foo_real() by doing simplify_foo().simplify_real(). Simplifications can be combined/repeated to produce better answers, and we already have a situation where calling e.g. simplify_full() twice might give you a better answer than calling it once. So you're kind of on your own regarding how many simplifications to try and in what order.

[sagetrac-sluther]

comment:4

What about adding a parameter to all the simplify_* functions instead of adding more functions?
Like "def simplify(real_domain=False):..."

[orlitzky]

comment:5

Replying to @sagetrac-sluther:

What about adding a parameter to all the simplify_* functions instead of adding more functions?
Like "def simplify(real_domain=False):..."

If instead of simplify_rational, simplify_log, etc. we had simplify(rational=True, log=True,...) that's what I would have done. As it is, if you want to do both rational and log simplifications, you just have to do expr.simplify_rational().simplify_log().

Particularly with simplify_real(), it only does one specific thing that isn't done by other methods so you can be somewhat confident that simplify_foo().simplify_real() will give you both simplifications. In other words I don't think you'd get a nicer expression back by setting the simplification domain to real before simplify_rational() than you would by calling the two in succession.

[sagetrac-sluther]

comment:6

Well I'd say this feels like mixing different concepts. The log, radical, etc. are about the form of the expression that's going to be transformed, whereas the domain is about the values the variables in the expression may take.

You're right that sometimes the application of simplifications in different order yields different results. But I find this rather annoying (not that I know how to fix it). By adding even more functions to the mix we make this situation even worse.

I usually use simplify_full(). So as I understand it I'd always need to call simplify_full() and simplify_real() in some order to benefit from the knowledge about the domain.

And then I'm still left wondering if there are simplifications in simplify_full() hat could benefit from this knowledge too, but didn't get it because there's only simplify_real() and not simplify_full_real().

Is there a list of which simplify_* function could use this domain parameter?

[burcin]

comment:7

Instead of adding a new function simplify_real() or a domain parameter to the existing functions, can we use a context manager instead?

Example use would be:

sage: t = sqrt(x^2)
sage: t.simplify()
sqrt(x^2)
sage: with maxima_domain(RR):
....:    u = t.simplify()
....:
sage: u
abs(x)
sage: t.simplify()
sqrt(x^2)