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MathGroup Archive 2006

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Re: Errors in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg69258] Re: Errors in Mathematica
  • From: Peter Pein <petsie at dordos.net>
  • Date: Sun, 3 Sep 2006 23:47:13 -0400 (EDT)
  • References: <ed3qkr$rnh$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Chris schrieb:
> NDSolve[{P'[x] ==   Q[x], Q'[x]  == 10^(-10) ,
> P[10^(-6)]==0,  Q[10^(-6)] == 0,
> n[x] == Exp[-A[x]]  ,
> A'[x] == -A[x]/236,
> A[10^(-6)]== -.0001},
> {P,Q,A,n}, {x, 10^(-6),1000},
> MaxSteps rightarrow Infinity]
> Plot[Evaluate[P[x]/. %], {x,10^(-6), 10}]
> 
> Here 2 equations link the functions P and Q.
> Two other equations link the functions n and A.
> There is no coupling between the two sets, and yet:
> 
> Note the denominator 236 that  appears in the second set.
> As this number is changed the solution for P[x], and in particular the
> value P[6] changes.
> Of course this should not happen.
> There is a discontinuity between 236 and 237 and another between 274
> and 275.
> After that, further increase has no effect on P[6].
> 
> The problem goes away if the small numbers are increased.
> It also goes away if the exponential function is replaced by  unity or
> some elementary function.
> 
> If the small numbers are made much smaller it causes Mathematica to
> shut down.
> 
> Can anyone tell me how to handle this?
> 

Hi Chris,

  I guess the change from 236 to 237 changes the stepsize of the 
algorithm at some point. You solve for (approx.) the range 0..1000 but 
use the value at 6. Using a different StartingStepSize eliminates this 
problem:

P6 gives P[6] as a function of the 'magic' denominator and the 
StartingStepSize:

P6[denom_, sss_:Automatic] := P[6] /. First[NDSolve[{P'[x] == Q[x], 
Q'[x] == 10^(-10), P[10^(-6)] == 0, Q[10^(-6)] == 0, n[x] == Exp[-A[x]], 
A'[x] == -A[x]/denom, A[10^(-6)] == -10^(-4)}, {P, Q, A, n}, {x, 
10^(-6), 1000}, MaxSteps -> Infinity, StartingStepSize -> sss]]

P6[237] - P6[236]
--> 3.896888607874218*^-11

this is the effect, you have observed.
Let's try a smaller sss:

P6[237, 10^(-6)] - P6[236, 10^(-6)]
--> 0.

To see how small the stepsize has to be:

Plot[Log[10, $MachineEpsilon + Abs[P6[237, Re[10^(-x)]] -
   P6[236, Re[10^(-x)]]]], {x, 0, 1},
   AxesOrigin -> {0, Log[10, $MachineEpsilon]}];

The plot shows that for sss ~10^0.7 (ca. 5) and smaller values the 
effect vanishes.

HTH,
Peter


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