On 6/19/2012 12:25 PM, Dennis Darland wrote:
...
> C, however, it is not as simple as producing a Taylor series for a given function. If you had the function, you would already have the solution! I.e. you have equations each with a derivative in the LHS (left hand side) and a functions on the RHS's also possibly involving derivatives. How, from this, cam Maxima determine the Taylor series?
Here's how.
Let us say that you have a function f(x) that solves the differential
equation f'(x)=f(x) and also f(0)=1.
Type into Maxima:
gradef(f(x),f(x));
tellsimp(f(0),1);
taylor(f(x),x,0,10);
.... the result is the solution to the DE, namely the taylor series
for exp(x).
higher order DEs can be accomodated by making up names for first,
second, etc derivatives,
and defining their properties.
This neglects many issues that eventually come up like convergence,
singularities, etc.
It should not be expensive to evaluate a taylor series with numerical
coefficients, even in Lisp.
RJF