Derivative at a point is not translated into Maxima

[robert.marik]

Alex reported ​http://groups.google.cz/group/sage-support/browse_thread/thread/81b96a7731600ec2 this bug (see the link for short discussion related to the problem)

Hi all:

I found some strange behavior in solve that's related to function
composition.  Check out this short example.

----------------------------------------------------------------------
| Sage Version 4.2, Release Date: 2009-10-24                         |
| Type notebook() for the GUI, and license() for information.        |
----------------------------------------------------------------------
sage: var('x,t')
(x, t)
sage: f= function('f',x)
sage: e= {x:exp(t)}
sage: ft= f.subs(e); ft
f(e^t)
sage: Ft = t^2 + ft^2; Ft
t^2 + f(e^t)^2
sage: a= diff(Ft,t); a
2*e^t*f(e^t)*D[0](f)(e^t) + 2*t
sage: solve(a==0,diff(f,x).subs(e))
[D[0](f)(t) == -t*e^(-t)/f(e^t)]

Did you spot the strangeness?  Somehow diff(f,x).subs(e) became diff
(f,x).subs({x:t}) after solving.  Does anybody know how to fix this?

Alex 

[robert.marik]

apply together with the patch for #385 (if necessary)

[robert.marik]

Submitted patch solves the problem described above

limitation and disadvantage of this patch are:

• it works for univariable functions only

• introduces new dummy variable into Maxima session

All comments are welcomed and any help appretiated

[kcrisman]

Replying to robert.marik:

Submitted patch solves the problem described above

Yes, this should do so. It would probably be best to add one more doctest, just to show that the original one works (i.e. without showing the Maxima internals).

limitation and disadvantage of this patch are:

• it works for univariable functions only

That's fine - that's still improving the previous situation, correct? Looking at the code, it would be helpful to know what happens if params has more than one element - particularly in

return "at(diff('%s(dummy_var_der), %s),dummy_var_der=%s)"%(f.name(), ", ".join(params), opnds[0]) 

it seems like one could have more items in the substitution tuple than there are %s to fill up, even if for some reason len(args)==1. What is the relation between args and params - could there be more params than args?

Also, make sure to add a doctest showing that the NotImplementedError? comes into play. Search for 'Traceback' in the file to see how the doctests for errors look.

• introduces new dummy variable into Maxima session

This is unfortunate, but probably unavoidable currently. You should test whether this breaks anything in Maxima, but I don't see why it would.

[jason]

Also, there needs to be a blank line after "TESTS::" for documentation formatting.

[robert.marik]

Replying to kcrisman:

This is unfortunate, but probably unavoidable currently. You should test whether this breaks anything in Maxima, but I don't see why it would.

Perhaps we can use variable from f.variables() instead of introducing dummy variable.

[kcrisman]

Why not see if that works, then update the patch? One might have to be careful in testing for how many variables/args there are. I am never clear on the difference between those two things, anyway, in Sage.

[robert.marik]

Replying to kcrisman:

Why not see if that works, then update the patch? One might have to be careful in testing for how many variables/args there are. I am never clear on the difference between those two things, anyway, in Sage.

My suggestion related to f.variables() does not work, since I cannot acces the variables inside the function derivative.

The original patch does not work for something like diff(f(x,y),x).subs(x=3) so I updated it and tested for various combination how many variables are in the function, how many variables appear at the resulting expression, how many variables are substituted etc.

Perhaps could be improved for multivariable functions, but Maxima expression "at (expr, [eqn_1, ..., eqn_n])" cannot be translated to current Sage and both issues should be fixed together. Perhaps in this ticket? (leaving as needs_work, despite the fact that the original problem has been fixed and doctested)

[robert.marik]

Apply only this patch.

[kcrisman]

Perhaps could be improved for multivariable functions, but Maxima expression "at (expr, [eqn_1, ..., eqn_n])" cannot be translated to current Sage and both issues should be fixed together. Perhaps in this ticket? (leaving as needs_work, despite the fact that the original problem has been fixed and doctested)

Can you give an example of Sage command -> Maxima command -> Maxima output that would give rise to such a situation? That should make it fairly straightforward to change the at function appropriately.

[robert.marik]

Replying to kcrisman:

Can you give an example of Sage command -> Maxima command -> Maxima output that would give rise to such a situation? That should make it fairly straightforward to change the at function appropriately.

I thought that this happens for taylor polynomial in more variables. However, it is not the case (btw. taylor polynomial in more variables seem to be not supported in Sage and I submited patch for #7472).

Hence I think that the problem is patched and if someone finds situation where she/he needs to enhace the functionallity, this can be solved in another ticket.

[robert.marik]

The patch fixes also #6376.

[kcrisman]

Okay, positive review, and all relevant tests pass. It is not optimal because of the dummy variable, one variable, etc., but it fixes things and there are open tickets (some with patches!) for making it even better. Great. I will also put a comment on #6376.

[mhansen]

This causes a failure in sage/interfaces/maxima.py with this example

maxima(derivative(ceil(x*y*d), d,x,x,y))

since multivariate derivatives are not supported.

[mhansen]

I've backed this out for now.

[nbruin]

On sage 6.5beta1 I get

sage: solve(a==0,diff(f,x).subs(e))
[D[0](f)(e^t) == -t*e^(-t)/f(e^t)]

so I think this problem has been resolved. Do we close it as (now) invalid?

[pbruin]

All doctests in the patch now work without the patch, even the ones that were expected to fail.

[kcrisman]

Awesome! I tested this last night but it must have been too late, because I didn't notice it worked. :P

In which case we should document that it is fixed so we don't get bitten if (say) we switch symbolics or something again.

[kcrisman]

As you say, e.g.

sage: x,y,=var('x y') 
sage: a = function('f', x, y).diff(x).subs(x=4).subs(y=8) 
sage: b=maxima(a); b         
?%at('diff('f(t0,t1),t0,1),[t0=4,t1=8])
sage: b.sage()
D[0](f)(4, 8)

Huh. Maybe this is a side effect of making symbolic variables unique in Maxima? Or the library interface? I bet no one has looked at this for a long time.

[pbruin]

Replying to kcrisman:

Awesome! I tested this last night but it must have been too late, because I didn't notice it worked. :P

In which case we should document that it is fixed so we don't get bitten if (say) we switch symbolics or something again.

Good point, I guess we can just add the doctests from the patch.

[kcrisman]

New commits:

​476d8c4

Add doctests for #7401 about derivatives at a point

[nbruin]

Replying to kcrisman:

New commits:

​476d8c4

Add doctests for #7401 about derivatives at a point

Two comments:

• use a = function('f')(x,y).diff(x) to avoid the confusing (and hopefully to deprecated) function('f',x,y).
• perhaps include an example that also tests the behaviour when the auxiliary variables aren't necessary:

  sage: a = function('f')(x,y).diff(x)
  sage: a
  D[0](f)(x, y)
  sage: maxima(a)
  'diff('f(_SAGE_VAR_x,_SAGE_VAR_y),_SAGE_VAR_x,1)
  

since that also illustrates a template how to arrive at more user-friendly printing.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​cc7fe0a

More doctests for #7401

[kcrisman]

Since you basically wrote all the other tests, adding to authors.

[pbruin]

The reviewer patch (typesetting, logical ordering of doctests, checking a few more intermediate results) adds/changes nothing essential.