Re: Re: Odd behavior of InterpolationFunctionderivative

• To: mathgroup at smc.vnet.net
• Subject: [mg29411] Re: [mg29379] Re: [mg29368] Odd behavior of InterpolationFunctionderivative
• From: Tomas Garza <tgarza01 at prodigy.net.mx>
• Date: Mon, 18 Jun 2001 03:39:11 -0400 (EDT)
• References: <200106150623.CAA28862@smc.vnet.net> <200106160647.CAA07738@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Hello Maryvonne,

I'm afraid your explanation doesn't address the problem I was referring to.
Essentially, what I'm saying is that a given set of numbers {f1, f2,...,fn}
(forget that they were generated using  the Sin function; they are just some
numerical values) produces an InterpolationFunction which, in the case where
Interpolation was called giving only those values has a derivative which is
different from that where the Interpolation is called giving also the
abscissa values explicitly, i.e., {{x1, f1}, {x2, f2},..., {xn, fn}}. The
plot of the Interpolation Function is the same in both cases; however the
plots of the corresponding derivatives - which should be equal, too - are
different.

Just look at the plot of what I called functionTwo: The slope of the
InterpolationFunction is obviously very close to 1 when x = 6; however, the
plot of its derivative has a maximum of about 0.314 (at x = 6). Nowhere in
the graph the "derivative" reflects the value of the slope of the
InterpolationFunction.

Finally, I don't understand your remark:

> Please, try at beginning not use the variable x in both cases.

I don't see why you object to using the same symbol for the argument of a
function in two different contexts (?). There is absolutely no way that this
could induce any confusion whatsoever.

Regards,
Tomas

----- Original Message -----
From: "Maryvonne Teissier" <my.teissier at cybercable.fr>
To: mathgroup at smc.vnet.net
Subject: [mg29411] [mg29379] Re: [mg29368] Odd behavior of
InterpolationFunctionderivative

> Hi Tomas,
>
> No bug ! Just Mathematics!
>
> Please, try at beginning not use the variable x in both cases.
>
> If you want derivative of your InterpolationFunction , you must thinck
> at functionTwo as Sin[a*(w-c)] rather than Sin ...
> When w goes from 1 to 11  you want that a*(w -c) goes from
> -Pi/2 to Pi/2, so a = Pi/10 and c = 6 .  In your (good) example,
> the derivative with respect to w is a*Cos[a*(w-c)]. not only Cos ...Try
> the following and you will see back your strange derivative of functionTwo
>
> functionMy[w_]:=N[Sin[(Pi/10)(w-6)]];
>
> Plot[{functionMy[w],(functionMy[#] &)'[w]}, {w, 1., 11.}];
>
> Sincerly,
> Maryvonne Teissier,
> University of Paris 7.
>
>
> Tomas Garza a *crit :
>
> > "Interpolation[data] constructs an InterpolatingFunction object which
> > represents an approximate function that interpolates the data. The data
> > can have the forms {{x1, f1},{x2, f2},...} or {f1, f2,...}, where in the
> > second case, the xi are taken to have values 1, 2, ..." (on-line Help
> > Browser). The following example shows that while in both cases the
> > InterpolationFunction works properly, the first derivatives appear to be
> > different. Take, for example, Sin[x] in the range (-Pi/2, Pi/2), and
> > construct a list of 11 values thereof in order to obtain an
> > interpolation function.
> >
> > In[1]:=
> > points = Table[x, {x, -Pi/2, Pi/2, Pi/10}];
> > vals = Table[Sin[x], {x, -Pi/2, Pi/2, Pi/10}];
> >
> > In[3]:=
> > functionOne = Interpolation[Transpose[{points, vals}]];
> > functionTwo = Interpolation[vals];
> >
> > In[5]:=
> > Plot[{functionOne[x], (functionOne[#] &)'[x]}, {x, -Pi/2, Pi/2}];
> >
> > In[6]:=
> > Plot[{functionTwo[x], (functionTwo[#] &)'[x]}, {x, 1, 11}];
> >
> > In both plots the graph of the approximation of Sin[x] appears
> > correctly. However, in the second plot the graph of the interpolated
> > derivative, Cos[x], is clearly wrong.
> >
> > What is going on? Is this a bug (aka "feature")?
> >
> > Tomas Garza
> > Mexico City
>
>

```

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