On 04/06/12 16:12, Nils Bruin wrote:
>
> 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)
That was a too-simple example. You can't create e.g. a cubic spline
because of the evaluated derivatives. In general the form over [-1,1]
would look like,
s(f;x) = a(x)*f(-1) + b(x)*f'(-1) + c(x)*f(1) + d(x)*f'(1)
Swapping out `f` after evaluating it at the endpoints is what causes the
biggest problems.
If you make `s` a function that takes `f` as an argument, it works, but
I need to be able to swap out `f` later for two reasons:
1) With higher order approximations, optimal splines can take a long
time to compute.
2) To calculate error bounds, I need to know the coeffients of f,
f.diff(),... What's the coefficient of sin(pi)?
>
> 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:
>
This does work better, I'll use it from now on, thanks.
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