On Apr 6, 12:03 pm, Michael Orlitzky <mich...@orlitzky.com> wrote:
> I sort of buy that, but if,
>
> sage: f = function('f', x)
>
> doesn't make `f` a function, that's a user-interface WTF =)
I would tend to agree, but it's a fact of life that function('f',x)
returns the symbolic function f evaluated at x. The docstring of
"function" doesn't help either. The confusion between the nature of f
and f(x) is a usual one in calculus. The vagueness actually lies more
in the nature of 'x' than the nature of 'f': Is 'x' a polynomial
variable? an arbitrary real number? an arbitrary complex number? The
identity element in the ring of differentiable functions on R?
> After #12801, I'm having trouble reconciling these two examples:
>
> sage: f = function('f', x)
> sage: g = function('g')
> sage: f.diff(x).substitute_function(f,g)
> D[0](f)(x)
>
> versus,
>
> sage: f = function('f')
> sage: g = function('g')
> sage: f(x).diff(x).substitute_function(f,g)
> D[0](g)(x)
>
> It would be slightly better I think if the first example worked, but I
> can live with using the second (although I never would have discovered
> it on my own).
Again, the result of "function('f',x)" is the expression "f(x)", which
does not occur in "D[0](f)(x)". Really, substitute_function should
throw an error if it is asked to substitute something that cannot be
viewed as a "function". Note that function('f',x)(3) throws a
deprecation warning. Life would be much clearer if that were to
finally become a proper error.
> What I would /really/ like to be able to do is,
>
> midpoint = (1/2)*( f(a) + f(b) )
>
> and then approximate multiple functions by swapping out the symbolic `f`
> for a real function like sine.
which you *can* provided you avoid the pitfall of "function('f',x)":
sage: f=function('f')
sage: var('a,b')
(a, b)
sage: midpoint = (1/2)*( f(a) + f(b) )
sage: midpoint
1/2*f(a) + 1/2*f(b)
sage: midpoint.substitute_function(f,sin)
1/2*sin(a) + 1/2*sin(b)
You can even make that into a function of a,b
sage: mp
(a, b) |--> 1/2*f(a) + 1/2*f(b)
although substitute_function silently changes the nature of the object
[I think that's a bug]:
sage: mp.substitute_function(f,sin)
1/2*sin(a) + 1/2*sin(b)
[now we just have an expression in a,b again]
That said, I think your confusion shows that it's probably better if
function('f',x) were to be deprecated in favour of function('f')(x).
The relevant bit of information to glean from the notation is that f
takes one argument. You can already indicate the valid number of
arguments:
sage: f= function('f',nargs=3)
sage: f(1,2)
TypeError: Symbolic function f takes exactly 3 arguments (2 given)
The attribute f.number_of_arguments() doesn't seem to get much use
though:
sage: sage.symbolic.operators.FDerivativeOperator(f,[3])
D[3](f)
is happily produced but can never be successfully evaluated.
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