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Re: Re: Problems with differentiating Piecewise functions


hlovatt 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.
>  
>
 One idea is to use Interpolation instead of Piecewise to creat your 
spline. If you know the derivatives at each of the control points, then 
the syntax to use is:

Interpolation[{{{x0}, y0, dy0}, {{x1}, y1, dy1}, ...}, 
InterpolationOrder->3]

For example, suppose we choose x0, x1, x2 to be 0, 50, 100, and obtain:

spline = Interpolation[{{{0}, pw[0], pw'[0]}, {{50}, pw[50], 
pw'[49.99999]}, {{100}, pw[100], pw'[100]}}, InterpolationOrder->3]

Of course, whatever method you used to create your spline should be able 
to generate derivatives at the control points, and you would use this 
data directly instead of creating pw first. spline[x], spline'[x] both 
plot nicely with no gaps.

Carl Woll
Wolfram Research


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