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Re: Re: Mathematica can't win against Tiger Woods
*To*: mathgroup at smc.vnet.net
*Subject*: [mg19735] Re: [mg19687] Re: Mathematica can't win against Tiger Woods
*From*: Klamser at t-online.de (Klamser)
*Date*: Mon, 13 Sep 1999 02:41:02 -0400
*References*: <7r7jvo$ck4@smc.vnet.net> <199909112035.QAA04387@smc.vnet.net.>
*Sender*: owner-wri-mathgroup at wolfram.com
I think it is very amusing, talking about the time Mathematica needs to
solve your problem.
What have you done 10 - 12 years ago? If you are faster by hand, use
your hands.
I think it is a real wonder, modern CAM (Computer aides Mathematics)
programs solve such complex problems at all. And not only Mathematica
does it, also others. Some say, that artificial
intelligence never comes. I think CAM is a good start.
If I read Roman E. Maders Mathematicas Programmer II and see the extreme
simple sort algorithms on page 16, I think it is not the problem of some
seconds for solving a problem. I think the problem is to understand the
pattern matching algorithm of Mathematica. Or look at Visual DSolve by
Dan Schwalbe & Stan Wagon. It is astonishing to see the object
orientated methods of mathematica makes it possible to use the graphic
output of Mathematica to solve a problem with other numeric methods.
That does not work with any other object oriented language (I know).
Nothing is perfect, Mathematica is not perfect. Wolfram should do more
to eliminate errors and give the error free code to everyone (not only
premier service members or good friends like Jens-Peer Kuska).
Mathematica 3.0 was a good value for the money. Mathematica 4.0 has
corrected many errors. The new features: Was that a good value for the
money?
But discussing about the time is not useful. Try to discuss about
Jens-Peer Kuska wrote:
>
> Hi William,
>
> first of all the result is not very complicated. It is less than 10
> printed pages and it has no special function like a hypergeometric one
> in it.
>
> Typical a computer algebra uses the most general result or the most
> general algorithm. It must do so to fit into all possible cases.
> Since Mathematica knows nothing about your problem it can't know that
> g is real and positive.
>
> For your problem it seems to be better to calculate the velocities x'[t]
> and y'[t]
> and get
> {{vx[t] -> (b*g)/(a^2 + b^2) +
> (C[1]*Cos[b*t] - C[2]*Sin[b*t])/E^(a*t),
> vy[t] -> -((a*g)/(a^2 + b^2)) +
> (C[2]*Cos[b*t] + C[1]*Sin[b*t])/E^(a*t)}}
>
> a very simple one. It is the second integration (to obtain x[t] and
> y[t])
> that causes Mathematica to split Sin[] and Cos[] int exponentials and it
> must do so due to the Exp[-a*t] term.
> If you integrate the vx[t] and vy[t] to obtain the positions and run
> FullSimplify[] again you get
>
> {(E^(a*t1)*(b*g*t1 + a*C[1] - b*C[2]) +
> (-(a*C[1]) + b*C[2])*Cos[b*t1] + (b*C[1] + a*C[2])*
> Sin[b*t1])/((a^2 + b^2)*E^(a*t1)),
> (E^(a*t1)*(-(a*g*t1) + b*C[1] + a*C[2]) -
> (b*C[1] + a*C[2])*Cos[b*t1] + (-(a*C[1]) + b*C[2])*
> Sin[b*t1])/((a^2 + b^2)*E^(a*t1))}
>
> and the answer can't be more more simple.
>
> Hope that helps
> Jens
>
> "William M. MacDonald" wrote:
> >
> > I want to use the study of golf drives in teaching theoretical methods. An
> > approximate pair of equations to get insight assumes that the drag force is
> > linearly proportional to velocity, instead of the actual quadratic
> > dependence. The equations for a ball with backspin to provide lift are
> > x''[t]== - (a x'[t]+b y'[t]),
> > y''[t]== - g - (a y'[t]- b x'[t])
> > Mathematica returns a very complicated and apparently complex expression in
> > about 9 seconds on my 250 MHz G3 Powerbook. Simplify takes 1min and 20
> > seconds and still returns an apparently complex expression. If I apply
> > FullSimplify on the solution for say x[t], I get no answer in 6 minutes.
> >
> > I have a PC version of another system that I can run on my Powerbook using
> > Virtual PC. It requires 6 seconds to deliver a lengthy but obviously real,
> > no Exp[(a+ I b)t] terms or (a + I b)(a - I b) terms.
> >
> > I have never been able to learn why Mathematica is so slow in solving
> > coupled equations and returns (as USUAL unless you use Simplify) such
> > inelegant results. Is there any computer algebra NERD out
> > there who knows the answer. (Don't tell me to use AlgebraicManipulation; I
> > am trying to sell Mathematica to users who don't want to spend time
> > learning
> > fancy tricks.)
> >
> > --
> > William M. MacDonald
> > Professor of Physics
> > University of Maryland
> >
> > Internet: wm2 at umail.umd.edu
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