Re: Re: Re: Integration of
- To: mathgroup at smc.vnet.net
- Subject: [mg105972] Re: [mg105919] Re: [mg105916] Re: [mg105896] Integration of
- From: Leonid Shifrin <lshifr at gmail.com>
- Date: Mon, 28 Dec 2009 04:58:25 -0500 (EST)
- References: <200912240515.AAA25812@smc.vnet.net>
> The OP still has a big problem, however, if p3 and p4
> are undefined.
>
A good point.
Regards,
Leonid
> Bobby
>
> On Fri, 25 Dec 2009 05:09:41 -0600, Leonid Shifrin <lshifr at gmail.com>
> wrote:
>
> > Hi Bobby,
> >
> > Just a few comments
> >
> > Integrate is a SYMBOLIC integrator; it can't integrate numerical
> >> interpolating functions.
> >>
> >
> > Not always true, although true in Brian's case, because he uses his
> > InterpolatingFunction object on functional argument (not the integration
> > variable directly). Consider:
> >
> > In[1]:=
> > Short[data = {#,#^2}&/@Range[-2,2,0.1]]
> >
> > Out[1]//Short= {{-2.,4.},{-1.9,3.61},<<38>>,{2.,4.}}
> >
> > In[2]:=
> > intF = Interpolation[data]
> >
> > Out[2]=
> > InterpolatingFunction[{{-2.,2.}},<>]
> >
> > In[3]:=
> > intFIntegrated = Integrate[intF[x],x]
> >
> > Out[3]=
> > InterpolatingFunction[{{-2.,2.}},<>][x]
> >
> > In[4]:=
> > Plot[intFIntegrated,{x,-2,2}]
> >
> > (* Plot output skipped, but the point is that it does work *)
> >
> > The main point is that Integrate *does* integrate a direct
> > InterpolatingFunction object (if integration is over the same variable on
> > which this InterpolatingFunction is defined), returning another
> > InterpolationFunction object as its antiderivative (does not need
> > specific
> > integration limits). This is possible because internally interpolation is
> > nothing but a bunch of splines which are polynomials of some finite order
> > and therefore can be integrated analytically. As far as I remember, this
> > can
> > be also done for multi-dimensional InterpolatingFunction objects.
> >
> > Moreover, for direct InterpolatingFunction object using Integrate is in
> > fact
> > often both faster and
> > more accurate than NIntegrate, since the latter often has (or used to
> > have
> > in prior versions anyway) slow convergence on InterpolatingFunction
> > objects.
> >
> > What is also true is that Integrate can not be used directly to integrate
> > say functions of InterpolatingFunction objects:
> >
> > In[5]:=
> > intFIntegratedSq = Integrate[intF[x]^2,x]
> >
> > Out[5]=
> > \[Integral]InterpolatingFunction[{{-2.,2.}},<>][x]^2 \[DifferentialD]x
> >
> > Not even in fixed numerical limits
> >
> > In[6]:=
> > intFIntegratedSq = Integrate[intF[x]^2,{x,-2,2}]
> >
> > (* Output skipped *)
> >
> > However, for numerical limits there is a NIntegrateInterpolatingFunction
> > function in FunctionApproximations` package, which still uses symbolic
> > Integrate internally, and which again often gives better results than
> > NIntegrate (or at least this used to be so in v.5 when I was using this
> > functionality):
> >
> > In[7]:=
> > Needs["FunctionApproximations`"]
> >
> > In[8]:=
> > NIntegrateInterpolatingFunction[intF[x]^2,{x,-2,2}]
> >
> > Out[8]= 12.8
> >
> >
> > Regards,
> > Leonid
> >
> >
> >
> >
> >> You might use NIntegrate or NDSolve instead, but that would require p3
> >> and
> >> p4 to have numerical values.
> >>
> >> If they did, you could try solving for xP via
> >>
> >> Clear[xP]
> >> xP = xP /.
> >> First@NDSolve[{xP'[t] == qFC[t], xP[0] == 0}, xP, {t, 0, tMax}]
> >>
> >> Bobby
> >>
> >> On Wed, 23 Dec 2009 23:15:01 -0600, blamm64 <blamm64 at charter.net>
> wrote:
> >>
> >> > Hi All,
> >> >
> >> > Well, my problem with NDSolve diagnosing delay differential algebraic
> >> > equations where there are no explicit delay arguments is because I
> >> > defined a "function of an InterpolatingFunction"; it never gets
> >> > evaluated apparently because Integrate cannot deal with it unless the
> >> > InterpolatingFunction is not "changed" in any way.
> >> >
> >> > Now, I searched the forums and found some posts regarding essentially
> >> > what my post is titled, but I could only find instances where the user
> >> > was basically trying to get rid of integrations of
> >> > InterpolatingFunctions being tied to a particular argument, and what I
> >> > want is just the precise reverse of that.
> >> >
> >> > I have an dataset I interpolate. The independent argument is pressure
> >> > difference. I have a quantity tied to the integral of that function
> >> > derived from the dataset. The quantity is displacement and it is tied
> >> > to time-dependent pressure difference "sent through" that function
> >> > (volumetric flow rate as a function of pressure difference across it.
> >> >
> >> > I "played around" and came up with the following. Please consider:
> >> >
> >> > Model
> >> >
> >> > fmodel[delp_]=30 (-(1/(delp+1)^3)+1);
> >> >
> >> > Data
> >> >
> >> > qdata = Table[{delp,fmodel[delp]},{delp,0,1000,1}]; (* note this is
> >> > not actual data I have *)
> >> >
> >> > Interpolation
> >> >
> >> > qFConIN = Interpolation[qdata,InterpolationOrder->2];
> >> >
> >> > Interpolation with time dependent variables in the needed form p3[t] -
> >> > p4[t] "pre-loaded"
> >> >
> >> > dp3p4 = p3[t]-p4[t];
> >> >
> >> > qFC[t_]=qFConIN[dp3p4]
> >> > Out[5]= InterpolatingFunction[{{0,1000}},<>][p3[t]-p4[t]]
> >> >
> >> > Interpolation with time dependent variables "pre-loaded", but no
> >> > independent argument (t) on l.h.s.
> >> >
> >> > qFCnoT=qFConIN[dp3p4]
> >> > Out[6]= InterpolatingFunction[{{0,1000}},<>][p3[t]-p4[t]]
> >> >
> >> > Derivatives (zero and higher) work fine (see below after setting
> >> > explicit, but dummy, p3[t] and p4[t])
> >> >
> >> > dfmodeldt=D[fmodel[dp3p4],t];
> >> >
> >> > dqFCdt=D[qFC[t],t]
> >> > Out[8]= InterpolatingFunction[{{0,1000}},<>][p3[t]-p4[t]] (p3'[t]-
> >> > (p4'[t])
> >> >
> >> > dqFCnoTdt=D[qFCnoT,t]
> >> > Out[9]= InterpolatingFunction[{{0,1000}},<>][p3[t]-p4[t]] (p3'[t]-
> >> > (p4'[t])
> >> >
> >> > Integration of Model
> >> >
> >> > xPfmodel[t_]=Integrate[fmodel[dp3p4],{s,0,t}];
> >> >
> >> > Intergration of "pre-loaded" interpolation (will FAIL)
> >> > Next is what I wanted, but this is what is causing NDSolve to diagnose
> >> > the diffeqs as delay (I have several functions of xP[t] as terms in
> >> > the diffeqs, e.g. fDoor[xP[t]]*qFC[t]/areaPh ... where fDoor is a
> >> > force as a function of displacement function ... qFC[t] gets evaluated
> >> > just fine ... and qFC[t]/areaPh is a fluid velocity, and qFC[t] is the
> >> > pre-defined regulation of that flow which in turn constrains the
> >> > displacement xP[t] )
> >> >
> >> >
> >> > xP[t_]=Integrate[qFC[s],{s,0,t}]
> >> > The output here is Integralsign, limits 0 to t, InterpolatingFunction
> >> > [{{0,1000}},<>][p3[s]-p4[s]] ds
> >> >
> >> > Now here's where it gets interestingly strange
> >> > Even with strange leading t, this integration with "pre-loaded" time-
> >> > dependent arguments (but no independent argument t on the
> >> > interpolation function), apparently works (see below)
> >> >
> >> > xP2[t_]=Integrate[qFCnoT,{s,0,t}]
> >> > Out[12]= t InterpolatingFunction[{{0,1000}},<>][p3[t]-p4[t]]
> >> >
> >> > Now define "dummy" pressures, as functions of time. In actuality, p3
> >> > [t] and p4[t] can vary according to the physics modeled by the diffeqs
> >> > (they are two of five time dependent variables in my diffeqs). All we
> >> > need ensure here is their initial value difference is zero
> >> >
> >> > p3[t_]=30+4 t;
> >> > p4[t_]=30;
> >> >
> >> > Derivatives work fine (a bit higher InterpolationOrder would make
> >> > disparity less)
> >> >
> >> > N[{dqFCdt/.t->2,dqFCnoTdt/.t->2,D[fmodel[dp3p4],t]/.t->2},6]
> >> > Out[15]= {0.0571875,0.0571875,0.0548697}
> >> >
> >> > Integral of model at p3[6] - p4[6]
> >> >
> >> > xPfmodel[6]
> >> > Out[16]= 562464/3125
> >> >
> >> > APPARENTLY works
> >> >
> >> > In[17]:= xP2[6]
> >> > Out[17]= 562464/3125
> >> >
> >> > FAILS
> >> >
> >> > xP[2]
> >> > The output is Integralsign, limits 0 to 2, InterpolatingFunction
> >> > [{{0,1000}},<>][4 s] ds
> >> >
> >> > Now, I tried several different models, and several different p3 and p4
> >> > functions, and I always got agreement between xPfmodel and xP2.
> >> >
> >> > But that really does not *prove* anything. My problem (one of them
> >> > anyway) is I don't know enough Mathematica semantics to decide if xP2
> >> > will always do what I need. I never need the time derivative of xP
> >> > since I already have it explicitly (qFC[t]).
> >> >
> >> > Can anyone enlighten me? What I need to know is the *proper* way to
> >> > get what I need done which is integrate and InterpolatingFunction,
> >> > rather, a function of an InterpolatingFunction. What is the *correct*
> >> > way of getting what is represented by xP[t_] = Integrate[qFC[s], {s,
> >> > 0,t}] (but obviously does not work)?
> >> >
> >> > Thanks,
> >> > -Brian L.
> >> >
> >>
> >>
> >> --
> >> DrMajorBob at yahoo.com
> >>
> >>
>
>
> --
> DrMajorBob at yahoo.com
>
>
- References:
- Integration of Interpolation
- From: blamm64 <blamm64@charter.net>
- Integration of Interpolation