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

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

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

**Re: Re: Problems with differentiating Piecewise functions**

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**Re: Re: Problems with differentiating Piecewise functions**

**Re: Problems with differentiating Piecewise functions**