On 6/7/07, Randy LeVeque <rjl@amath.washington.edu> wrote:
> By the way, I'm just trying to figure out how sage does Taylor series.
> Maybe you can pass this on to whoever the best person is to chat with about
> this...
>
> In maple I can do things like
>
> > mtaylor(u(x+h,t+k),[h,k],3);
> 2
> u(x, t) + D[1](u)(x, t) h + D[2](u)(x, t) k + 1/2 D[1, 1](u)(x, t) h
>
> 2
> + h D[1, 2](u)(x, t) k + 1/2 D[2, 2](u)(x, t) k
>
>
> which is very convenient for numerical analysis when computing truncation
> errors of finite difference methods (h and k are mesh widths in space and
> time). In sage a general expansion of this sort doesn't seem possible even
> in a single variable, e.g.,
>
> sage: taylor(u(x+h),h,0,4)
> x + h
>
> Apparently an undefined function like u(x) is taken to be the identity map?
To define a formal function, do u = function('u'). Then
sage: u = function('u')
sage: u(x + h)
u(x + h)
sage: diff(u(x+h), x)
diff(u(x + h), x, 1)
To get the Taylor expansion you would do this:
sage: taylor(u(x+h),h,0,4)
-- however -- this currently doesn't work in SAGE since we hadn't considered
doing this yet. What happens is Maxima does the computation and outputs
the following expression:
'u(x)+(?%at('diff('u(x+h),h,1),h=0))*h+(?%at('diff('u(x+h),h,2),h=0))*h^2/2+(?%at('diff('u(x+h),h,3),h=0))*h^3/6+(?%at('diff('u(x+h),h,4),h=0))*h^4/24
SAGE doesn't know yet how to parse the "at" function, so you get
an error -- it will have to be added. [Note that I don't necessarily consider
maxima the ultimate underlying engine for SAGE's symbolic computation
capabilities -- but it does provide a very quick way for SAGE to have
a powerful symbolic system for which a lot of subtle bugs have
already been fixed (over the last 40 years of Maxima development). ]
Definitely point out lots of things like this in your talk at SD4!
-- William
