[with patch, positive review] cannot differentiate cotangent

[burcin]

From sage-support:

On Fri, 18 Sep 2009 13:15:46 -0500
Jason Grout <jason-sage@creativetrax.com> wrote:

> On alpha.sagenb.org, I get the following:
> 
> sage: t=var('t')
> sage: diff(cot(t),t)
> D[0](cot)(t)
> sage: diff(cos(t)/sin(t),t)
> -cos(t)^2/sin(t)^2 - 1
> 
> 
> Does Sage not know that cot(t) is cos(t)/sin(t)? 

Unfortunately it doesn't.

GiNaC doesn't define the function cot. Sage defines it in the file sage/functions/trig.py starting at line 184. I suppose it was written quickly by Mike during the symbolics switch.

Defining a custom derivative function (named _derivative_) in that class should fix this.

Here is the thread:

 http://groups.google.com/group/sage-support/browse_thread/thread/752de34c876720cc

[timdumol]

Added _derivative() functions to all functions in trig.py and hyperbolic.py

[timdumol]

This should do the trick.

[burcin]

Thanks for the quick patch.

Here is my review:

• the keyword argument diff_param is only useful for multivariate functions. In this case since all these functions are univariate, so we know the argument is args[0]. You can safely drop the first two lines of the _derivative_() methods and replace the third with x = args[0].
• continuing the previous item, we should tell these functions they are univariate. ATM, they silently drop the second argument:

  sage: arccsc(a,b)
  arccsc(a)
  

  You can do this by giving nargs=1 as a parameter to the base class constructor.
• It is better to give the variable as an argument to diff in the doctests, for example diff(asech(x), x). I actually prefer asech(x).derivative(x), but this is your patch. :)
• The formula for the derivative of asech(x) you use is only true for x real. You need -1/(x * (x+1) * sqrt( (1-x)/(1+x) )).
• Similarly, the derivative of acsch(x) is -1/(x^2 * sqrt(1 + 1/x^2) )

Can someone else check the derivatives to make sure there is no mistake?

[timdumol]

Added _derivative_() functions to all functions in trig.py and hyperbolic.py. Rev 2. Apply this patch only.

[timdumol]

Functions that inherit from PrimitiveFunction are automatically given nargs = 1 on L800 of symbolic/function.pyx. So the silent dropping of arguments is more of a usability problem on the Symbolic side.

I've made the changes for the derivatives of asech(x) and acsch(x). I've also generalized the derivatives for asec(x) and acsc(x).

Anyone who wants to review the derivatives can check:  http://mathworld.wolfram.com/Derivative.html,  http://mathworld.wolfram.com/InverseHyperbolicFunctions.html and  http://mathworld.wolfram.com/InverseTrigonometricFunctions.html

[burcin]

Replying to timdumol:

Functions that inherit from PrimitiveFunction are automatically given nargs = 1 on L800 of symbolic/function.pyx. So the silent dropping of arguments is more of a usability problem on the Symbolic side.

You're right. I fixed this and many other things about symbolic functions last weekend, by rewriting sage/symbolic/function.pyx. Unfortunately I don't think I'll be able to clean up my changes and submit them any time soon... oh, well...

I'm giving your patch a positive review. It applies cleanly, and passes all tests on my 4.1.1.alpha. Great job! Many thanks.

[jason]

burcin: it seems that there are several people interested in helping with symbolics, so maybe if you just posted what you had, people could clean it up for you, if that would be easier for your time situation?

[mvngu]

Merged trac_6956-derivatives.2.patch.

[mvngu]

There is no 4.1.2.alpha3. Sage 4.1.2.alpha3 was William Stein's release for working on making the notebook a standalone package.