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Re: Re: Problems with differentiating Piecewise functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87036] Re: [mg86992] Re: Problems with differentiating Piecewise functions
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Sat, 29 Mar 2008 04:26:17 -0500 (EST)
  • Reply-to: hanlonr at cox.net

You can eliminate the gap in the Plot by excluding 50

Plot[pw[x], {x, 0, 100}, Exclusions -> 50]

For numerical analysis, Rationalize the expressions and make them exactly equal, i.e., slight correction to one or both expressions to make them exactly equal at the join.


Bob Hanlon

---- hlovatt <howard.lovatt at gmail.com> wrote: 
> On Mar 28, 12:18 am, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> > On 26 Mar 2008, at 10:55, hlovatt wrote:
> >
> > > If I set up a piecewise function and differentiate it:
> >
> > > In[112]:= pw1 = Piecewise[{{x^2, x <= 0}, {x, x > 0}}]
> >
> > > Out[112]= \[Piecewise] {
> > >  {x^2, x <= 0},
> > >  {x, x > 0}
> > > }
> >
> > > In[113]:= pw1 /. x -> 0
> >
> > > Out[113]= 0
> >
> > > In[114]:= pw1d = D[pw1, x]
> >
> > > Out[114]= \[Piecewise] {
> > >  {2 x, x < 0},
> > >  {1, x > 0},
> > >  {Indeterminate, \!\(\*
> > >     TagBox["True",
> > >      "PiecewiseDefault",
> > >      AutoDelete->False,
> > >      DeletionWarning->True]\)}
> > > }
> >
> > > In[115]:= pw1d /. x -> 0
> >
> > > Out[115]= Indeterminate
> >
> > > Then at the joins between the pieces I get Indeterminate values,
> > > because the limit x <= 0 has become x < 0 after differentiation. Does
> > > anyone know a solution to this problem?
> >
> > > Thanks,
> >
> > > Howard.
> >
> >   What do you mean by "a solution to this problem"? You have a
> > function that is not differentiable at 0 and you would like it's
> > derivative to have a value there? You can't expect a "solution to a
> > problem" when you do not tell us what is the problem (except the fact
> > that not all functions are differentiable - but that's life).
> > Note that is your pieceise function is actually differentiable than
> > the derivative is defined everywhere:
> >
> > pw2 = Piecewise[{{x^2, x <= 0}, {x^3, x > 0}}];
> >
> > pw2d = D[pw2, x]
> > Piecewise[{{2*x, x < 0}, {0, x == 0}}, 3*x^2]
> >
> > This is also as it should be. What else would you expect?
> >
> > Andrzej Kozlowski
> 
> Thanks to everyone who replied. I have to apologise for the bad
> example that I posted (I simplified my problem by cutting and pasting
> an example from the help file to keep the post short). I am actually
> fitting cubic splines and the functions are continuous up to the
> second derivative (at least to the accuracy of machine precision). A
> better example is:
> 
> In[54]:= pw[x_] :=
>  Piecewise[{{0.+ 0.007508378277320685 x + 7.561460342471517*10^-7 x^3,
>      x < 50}, {-4.8729206849430454*10^-6 (-125.76959597633721 +
>        x) (1148.1044516606876- 47.50636365246156 x + x^2), 50 <= x}}]
> 
> In[55]:= pw[x]
> 
> Out[55]= \[Piecewise] {
>   {0.+ 0.00750838 x + 7.56146*10^-7 x^3, x < 50},
>   {-4.87292*10^-6 (-125.77 + x) (1148.1- 47.5064 x + x^2), 50 <= x}
>  }
> 
> In[56]:= pw[50]
> 
> Out[56]= 0.469937
> 
> In[57]:= pw[50 + 10^-30]
> 
> Out[57]= 0.469937
> 
> In[58]:= pw[50 - 10^-30]
> 
> Out[58]= 0.469937
> 
> In[60]:= pw'[x]
> 
> Out[60]= \[Piecewise] {
>   {0.00750838+ 2.26844*10^-6 x^2, x < 50},
>   {-4.87292*10^-6 (-125.77 + x) (-47.5064 + 2 x) -
>     4.87292*10^-6 (1148.1- 47.5064 x + x^2), x > 50},
>   {Indeterminate, \!\(\*
>      TagBox["True",
>       "PiecewiseDefault",
>       AutoDelete->False,
>       DeletionWarning->True]\)}
>  }
> 
> In[61]:= pw'[50]
> 
> Out[61]= Indeterminate
> 
> In[62]:= pw'[50 + 10^-30]
> 
> Out[62]= 0.0131795
> 
> In[63]:= pw'[50 - 10^-30]
> 
> Out[63]= 0.0131795
> 
> Also if you Plot pw or pw' you get an annoying gap in the plot at the
> join (but strangely not pw''). My guess is that Mathematica is too
> pedantic about machine precision and is treating each piece as an
> algebraic equation. However this does not explain why Plot behaves
> funnily and doesn't help if you are trying to do numerical analysis.
> 



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