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Re: Problems with differentiating Piecewise functions

• To: mathgroup at smc.vnet.net
• Subject: [mg87017] Re: Problems with differentiating Piecewise functions
• From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
• Date: Sat, 29 Mar 2008 04:22:43 -0500 (EST)
• References: <200803260955.EAA09634@smc.vnet.net> <fsg6r6$j9q$1@smc.vnet.net> <fsib7n$609$1@smc.vnet.net>

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

That result is correct.

> In[62]:= pw'[50 + 10^-30]
>
> Out[62]= 0.0131795
>
> In[63]:= pw'[50 - 10^-30]
>
> Out[63]= 0.0131795

Others may have better ideas. My suggestion gives approx 0.0131795 for the
derivative at 50; choose one of the three limits below. Of course, by using
those limits, you are defining types of derivatives yourself:

In[34]:= Limit[(pw[x + h] - pw[x])/h, h -> 0]

Out[34]= Piecewise[{{2.535885030738735*^-20*(2.960851216166295*^17 +
            8.9453507365065*^13*x^2), x < 50.}},
 -2.1278945401599612*^-27*(1.6311721028240681*^25 - 7.93610765893554*^23*x
 + 6.870059478478754*^21*x^2)]

In[35]:= % /. x -> 50

Out[35]= 0.0131795

In[36]:= Limit[(pw[x + h] - pw[x])/h, h -> 0, Direction -> 1]

Out[36]= Piecewise[{{2.535885030738735*^-20*(2.960851216166295*^17 +
            8.9453507365065*^13*x^2), x < 50.},
     {-2.1278945401599612*^-27*(1.6311721028240681*^25 -
            7.93610765893554*^23*x + 6.870059478478754*^21*x^2), x > 50.}},
            0.013179473534174325]

In[37]:= Limit[(pw[x + h] - pw[x - h])/(2 h), h -> 0]

Out[37]= Piecewise[{{2.535885030738735*^-20*(2.960851216166295*^17 +
            8.9453507365065*^13*x^2), x < 50.},
     {-4.2557890803199225*^-27*(8.155860514120341*^24 -
            3.96805382946777*^23*x + 3.435029739239377*^21*x^2), x > 50.}},
            0.01317947353417433]

> Also if you Plot pw or pw' you get an annoying gap in the plot at the
> join (but strangely not pw'').

For example, one can see a gap with Plot[pw[x], {x, 0, 100}].
But try Plot[pw[x], {x, 0, 50, 100}] instead; there should be no gap then.

BTW, it is perhaps not strange that there was no gap when you plotted pw''.
The reason there was no visible gap has to do, I suspect, with the way that
Mathematica chooses points to be plotted.

David

> 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.

• References:
  • Problems with differentiating Piecewise functions
    • From: hlovatt <howard.lovatt@gmail.com>