[Maxima-bugs] [ maxima-Bugs-873301 ] Gradef doesn't understand diff (workaround)

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Bugs item #873301, was opened at 2004-01-08 20:39
Message generated for change (Comment added) made by sf-robot
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Category: Lisp Core
Group: None
>Status: Closed
Resolution: Works For Me
Priority: 5
Private: No
Submitted By: Stavros Macrakis (macrakis)
Assigned to: Nobody/Anonymous (nobody)
Summary: Gradef doesn't understand diff (workaround)

Initial Comment:
Suppose that f(x)=exp(g(x)).  Then f'(x)=f(x)*g'(x).  So 
how do we define this using gradef?

Let's try the obvious solution:

      gradef(  f(x), f(x) * diff(g(x),x) )

Now

     diff(f(x^2),x) =>  f(x-1)*'DIFF(g(x-1),x-1,1)

Oops!  That isn't right!  It is substituting x-1 for x in the 
*second* argument of diff as well as the first!

So we have to work around this doing something like

    gradef(  f(x), f(x) * g1(x) )
    gradef(  g(x), g1(x) )

   


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>Comment By: SourceForge Robot (sf-robot)
Date: 2010-06-11 02:20

Message:
This Tracker item was closed automatically by the system. It was
previously set to a Pending status, and the original submitter
did not respond within 14 days (the time period specified by
the administrator of this Tracker).

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Comment By: Dieter Kaiser (crategus)
Date: 2010-05-27 20:55

Message:
As explained in the last posting I would like to suggest to close this bug
report as "works for me".
Setting the status to pending.
Dieter Kaiser

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Comment By: Dieter Kaiser (crategus)
Date: 2010-05-11 20:34

Message:
First, again the example of this bug report:

(%i6) gradef(f(x), f(x)*diff(g(x),x));

We get a wrong result with the above definition:

(%i7) diff(f(2*sin(x)),x);
(%o7) 2*cos(x)*f(2*sin(x))*'diff(g(2*sin(x)),2*sin(x),1)

But Maxima can do it correctly. We have to define the problem the
following way:

(%i8) gradef(f(x), f(x)*at(diff(g(u),u),u=x))$

(%i9) diff(f(2*sin(x)),x);
(%o9) 2*('at('diff(g(u),u,1),u = 2*sin(x)))*cos(x)*f(2*sin(x))

The example of the bug report now gives a correct result too:

(%i10) diff(f(x^2),x);
(%o10) 2*('at('diff(g(u),u,1),u = x^2))*x*f(x^2)

>From this we might conclude, that we have not a bug, but we have to
formulate the problem more precisely.

By the way: 

The derivatives of the nine inverse Jacobi functions wrt the parameter m
are defined as a noun form. Furthermore, the derivatives of the Bessel Y
and K functions wrt the order have a noun form of a derivative on the
property list. All this derivatives do not work too, but have the same
error as reported in this bug report, e.g.

(%i11) diff(inverse_jacobi_ns(x,2*m),m);
(%o11) 2*'diff(inverse_jacobi_ns(x,2*m),2*m,1)

The error for the inverse Jacobi functions is easy to correct. We simply
put NIL for the derivative wrt the parameter m on the property list. We had
some time ago an extension to sdiff to return a noun form for this case.
With this change we get a correct noun form:

(%i13) diff(inverse_jacobi_ns(x,2*m),m);
(%o13) 'diff(inverse_jacobi_ns(x,2*m),m,1)

This is an example of a wrong derivative of a Bessel K function:

(%i15) diff(bessel_k(sin(n),x),n);
(%o15) cos(n)*(%pi*csc(%pi*sin(n))
                  *('diff(bessel_i(-sin(n),x),sin(n),1)
                   -'diff(bessel_i(sin(n),x),sin(n),1))
              /2
              -%pi*bessel_k(sin(n),x)*cot(%pi*sin(n)))

To get this derivative in general correct we can put an expression with AT
on the property list.

Dieter Kaiser

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Comment By: Robert Dodier (robert_dodier)
Date: 2006-07-22 15:48

Message:
Logged In: YES 
user_id=501686

In 5.9.3cvs, I'm seeing a different wrong answer.

gradef( f(x), f(x) * diff(g(x),x) );
diff(f(x^2),x) ;
 => 2*x*f(x^2)*'?%diff(g(x^2),x^2,1)

Should be 2*x*f(x^2)*at('?%diff(g(u),u,1),[u=x^2]) or
something like that (i.e., distinguish point of evaluation
x^2 from variable of differentiation).

<rant> Maxima adopts numerous mathematical conventions,
mostly successfully, but this customary sloppiness with
differentials is something I wish we could clean up. </rant>

Not sure how the proposed workaround is supposed to work. It
turns out g1(x) = bessel_i(1,x)*%e^-x -- surprise!


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