Add inverse method to symbolic expressions

[sagetrac-ltw]

Mathematica has an InverseFunction function:

http://reference.wolfram.com/mathematica/ref/InverseFunction.html

Sage can accomplish this via the roots method, as described on AskSage: http://ask.sagemath.org/question/502/can-sage-compute-the-inverse-of-a-function

However, there is no "convenience" method to find inverses of symbolic expressions. Since roots itself just calls Maxima's solve, it should be relatively straightforward to implement an inverse method in the same fashion. This should also be able to find the inverse of vector functions (n-dimensional input, n-dimensional output).

sage-devel discussion: http://groups.google.com/group/sage-devel/browse_thread/thread/d7d9af95a67d4481

CC: @kcrisman

Component: symbolics

Keywords: InverseFunction, inverse function

Author: Eviatar Bach, Ralf Stephan

Branch/Commit: u/rws/add_inverse_method_to_symbolic_expressions @ f9bfdf4

Issue created by migration from https://trac.sagemath.org/ticket/11202

[sagetrac-ltw]

Description changed:

--- 
+++ 
@@ -2,10 +2,10 @@
 
 http://reference.wolfram.com/mathematica/ref/InverseFunction.html
 
-Sage needs it, too!
+Sage can accomplish this via the `roots` method, as described on AskSage: http://ask.sagemath.org/question/502/can-sage-compute-the-inverse-of-a-function
+
+However, there is no "convenience" method to find inverses of symbolic expressions, and I am not sure whether the above method can find the inverse of vector functions (`n`-dimensional input, `n`-dimensional output).
 
 sage-devel discussion: http://groups.google.com/group/sage-devel/browse_thread/thread/d7d9af95a67d4481
 
-AskSage question and answer: http://ask.sagemath.org/question/502/can-sage-compute-the-inverse-of-a-function
 
-

[sagetrac-ltw]

Description changed:

--- 
+++ 
@@ -4,7 +4,7 @@
 
 Sage can accomplish this via the `roots` method, as described on AskSage: http://ask.sagemath.org/question/502/can-sage-compute-the-inverse-of-a-function
 
-However, there is no "convenience" method to find inverses of symbolic expressions, and I am not sure whether the above method can find the inverse of vector functions (`n`-dimensional input, `n`-dimensional output).
+However, there is no "convenience" method to find inverses of symbolic expressions. Since `roots` itself just calls Maxima's `solve`, it should be relatively straightforward to implement an `inverse` method in the same fashion. This should also be able to find the inverse of vector functions (`n`-dimensional input, `n`-dimensional output).
 
 sage-devel discussion: http://groups.google.com/group/sage-devel/browse_thread/thread/d7d9af95a67d4481
 

[eviatarbach]

Attachment: 11202.patch.gz

[eviatarbach]

comment:4

Patch added!

[eviatarbach]

comment:5

Note that this method is faster than using roots, as was suggested by Kelvin Li:

sage: timeit('inverse(log(x), y)')
125 loops, best of 3: 2.88 ms per loop
sage: timeit('(log(x) - y).roots(x)')
125 loops, best of 3: 3.39 ms per loop

[kcrisman]

comment:6

Interesting. I hope someone looks at this more carefully, because it would be great to have this more easily accessible. Two comments:

• The documentation that would come up in this is in the functional.py file. Since some people might think this is pretty fully featured, but it probably isn't, that documentation should be bigger, and point to the full doc in the expression file.
• I'd like to see, from whomever tests it, a pretty sizable number of test cases that indicate we won't get "weird" output. Namely, what happens when solve is baffled? Multivariate things? Things that have inverses (like functions defined by integrals or something) but which won't have them shown here?
  In the past we've had a lot of complaints from adding "early stage" things. I'm still in favor of doing so, as that's the only way this project improves, but we want to make sure that we make it really clear to even the most obtuse reader what the real status and abilities of something like this is.

[eviatarbach]

comment:7

Okay, will do.

I don't know what you mean by it not being "fully featured". It should be able to handle any invertible functions, assuming solve supports them. Those are really the only limitations here. If it can't solve it, it just leaves it unevaluated, the same way that solve does. For example,

(abs(x) == y).solve(x)
[abs(x) == y]

Don't think there are many places for this to go wrong, since all it does is switch around two variables and solves for one of them. But I will add some test cases (or did you want someone else to do it, since you said "from whomever tests it"?). I did include a multivariate example, by the way. I don't know what you mean by functions defined by integrals; could you please give an example?

Thank you.

[jdemeyer]

comment:8

Please fill in your real name as Author.

[eviatarbach]

Author: Eviatar Bach