Timing comparison for diffeq program in c++, Maple and Maxima

Dennis Darland <dennis.darland at yahoo.com> writes:
> How, given the diffeq below, do you generate a Taylor series in 
> Maxima?
>
> diff ( y , x , 1 ) = tan(sqrt(2.0*x + 3.0));

Well a 1d example like this is easy. Don't forget what a Taylor series
is:

  y(x) = y(c) + (x-c)*y'(c) + (x-c)^2/2*y''(c) + ...

You're saying that you have a formula for y'(x). In which case a
(forward-Euler style) Taylor series is easy enough:

(%i1) rhs: tan(sqrt(2*x+3));
(%o1)                         tan(sqrt(2 x + 3))
(%i2) assume(x>c);
(%o2)                               [x > c]
(%i3) taylor ('integrate(rhs, x, c, x), x, c, 2);
(%o3)/T/ tan(sqrt(2 c + 3)) (x - c)
                               2                                      2
                           (sec (sqrt(2 c + 3)) sqrt(2 c + 3)) (x - c)
                         + -------------------------------------------- + . . .
                                             4 c + 6


(You need to quote integrate, otherwise Maxima tries to do the
integration)

> or better, the system:
> diff (x2,t,2) = 3.0 * diff(x2,t,1) - 2.0 * x2 - diff(x1,t,2) - diff (x1,t,1) + x1;
> diff (x1,t,1) = 4.0 * x2 -  2.0 * diff (x2,t ,1) - 2.0 * x1;

Well, if you look at the documentation for Taylor, you'll see that it
can do multivariate Taylor series. You have a weird mixture of a second
order and first order equation here. Presumably you want something that
works for "any" equation you give it, so there's no point in trying to
spot some structure in the example you've posted. As such, what you
should probably do is transform this into a collection of coupled
first-order ODEs.

Let

  u1 = x1
  u2 = x2
  u3 = dx1/dt
  u4 = dx2/dt

Then you end up with

(%i1) eqs: ['diff(x2,t,2)=3*'diff(x2,t)-2*x2-'diff(x1,t,2)-'diff(x1,t)+x1,
            'diff(x1,t)  =4*x2 - 2*'diff(x2,t)-2*x1];
        2                     2
       d x2     dx2          d x1   dx1       dx1       dx2
(%o1) [---- = 3 --- - 2 x2 - ---- - --- + x1, --- = - 2 --- + 4 x2 - 2 x1]
         2      dt             2    dt        dt        dt
       dt                    dt
(%i2) subst(['diff(x2,t,2)='diff(u4,t), 'diff(x1,t,2)='diff(u3,t),
             'diff(x2,t)=u2, 'diff(x1,t)=u1],
             eqs);
         du4                 du3
(%o2)   [--- = - 2 x2 + x1 - --- + 3 u2 - u1, u1 = 4 x2 - 2 x1 - 2 u2]
         dt                  dt

And the system is in fact linear here, so the Taylor series is trivial.


I think that if you want Maxima to do something useful you should
probably tell us what your "sode" code is actually supposed to do. Is it
just a naive numerical integrator? In which case, give up on symbolic
algebra packages and code it in C/C++ or Octave. You should also
probably read some numerical analysis textbooks: just using Taylor
expansions isn't the best approach.

Or maybe you want your code to do some cool maths that automatically
spots special structures in the problems it's given? In which case, a
higher level tool might be fantastic. But it really isn't clear what
you're trying to achieve.


Rupert
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