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Re: Re: Problems with differentiating Piecewise functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87032] Re: [mg86992] Re: Problems with differentiating Piecewise functions
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sat, 29 Mar 2008 04:25:32 -0500 (EST)
  • References: <200803260955.EAA09634@smc.vnet.net> <fsg6r6$j9q$1@smc.vnet.net> <200803280816.DAA04765@smc.vnet.net> <26A932C4-A122-434F-898B-BBEF5FD453F4@mimuw.edu.pl>

On 28 Mar 2008, at 12:26, Andrzej Kozlowski wrote:
>
> On 28 Mar 2008, at 09:16, 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.
>>
>
> You are violating one of the most fundamental principles of computer  
> algebra: do not mix approximate numbers with symbolic computation.  
> More correctly, such "mixing" requires a great deal of care and  
> symbolic algebraic techniques that can deal with approximate numbers  
> (and even more so with machine precision input) are still in their  
> infancy. (In fact Mathematica is one of the few systems that make it  
> to some extent possible). In your case the symbolic technique you  
> are using is differentiation of a piecewise function, and this  
> cannot be reliably combined with numerical precision input.
>
> The answer to your problem is simple. Switch from using a mixture of  
> symbolic methods and approximate input to a purely numeric setting.  
> The best way to do this, I think, is by using FunctionInterpolation.  
> So what you need to do is this:
>
> 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}}]
>
> f = FunctionInterpolation[pw[x], {x, 0, 80}];
>
> Plot[f[x], {x, 40, 60}]
>
> shows a continuous curve
>
> Plot[f'[x], {x, 40, 60}]
>
> shows an almost smooth curve, with a slight "kink" at the break  
> point. One could probably play with the options to  
> FunctionInterpolation to "smoothen" it.
>
> Andrzej Kozlowski


Actually, you can get a smooth looking derivative by using a higher  
interpolation order:

f = FunctionInterpolation[pw[x], {x, 0, 80}, InterpolationOrder -> 10];

Plot[f'[x], {x, 40, 60}]

looks very smooth and so does:

Plot[f''[x], {x, 40, 60}]

Andrzej Kozlowski


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