Re: Derivative of InterpolatingFunction
- To: mathgroup at smc.vnet.net
- Subject: [mg4374] Re: Derivative of InterpolatingFunction
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Mon, 15 Jul 1996 07:11:10 -0400
- Organization: University of Western Australia
- Sender: owner-wri-mathgroup at wolfram.com
Andrei Constantinescu wrote:
> I have the following problem:
>
> fct = InterpolatingFunction[{{0., 3.14159}, {0., 3.14159}}, <>]
>
> ... so its an InterpolatingFunction of 2 variables .
>
> What I want now to do is the derivative of this function.
> But I fail !
>
> I get for example:
>
> In[69]:= Derivative[1,0][fct]
>
> (1,0)
> Out[69]= (InterpolatingFunction[{{25., 210.}, {0., 17.}}, <>])
> In[71]:= %69[100., 4.]
In The Mathematica Journal 4(2):31 the following appears:
Partial Derivatives
Presently, Mathematica cannot handle partial derivatives of
InterpolatingFunctions. The package DInterpolatingFunction.m, provided
by Hon Wah Tam (tam at wri.com) and included in the electronic
supplement, computes partial derivatives of two-dimensional
InterpolatingFunctions. Enhancements for higher dimensions will
eventually be incorporated into Mathematica.
After placing DInterpolatingFunction.m in your home directory, it can
be loaded using the command:
<< DInterpolatingFunction`
Define an array of data points:
g[x_, y_] := Sin[3 x] Cos[4 y]
tb = Table[{x, y, g[x, y]}, {x, 0, 1, .1}, {y, 0, 1, .05}];
and then interpolate it:
ifunc = Interpolation[Flatten[tb, 1]];
You can evaluate the interpolated function directly:
ifunc[0.23, 0.41]
-0.0440066
and plot its graph:
Plot3D[ifunc[x, y], {x, 0, 1}, {y, 0, 1}];
With the DInterpolatingFunction.m package, you can also compute partial
derivatives. Here is the derivative with respect to the first variable:
dx = Derivative[1, 0][ifunc];
dx[0.23, 0.41]
-0.160227
which can be compared with the expected value:
Derivative[1, 0][g][0.23, 0.41]
-0.159991
Similarly:
dy = Derivative[0, 1][ifunc];
dy[0.23, 0.41]
-2.53594
Derivative[0, 1][g][0.23, 0.41]
-2.54005
Here is the package:
(* DInterpolatingFunction.m *)
Unprotect[ InterpolatingFunction ];
BeginPackage[ "DInterpolatingFunction`", "Global`" ]
Begin[ "`Private`" ]
DiffElem[ element_, zerolist_ ] :=
Module[ { coeffs, dvdfs, order, var,
polynomial, j, newelement },
dvdfs = element[[1]];
coeffs = element[[2]];
order = Length[ coeffs ];
polynomial = dvdfs[[ order + 1 ]];
For[ j = order, j >= 1, j--,
polynomial = ( var + coeffs[[j]] ) * polynomial + dvdfs[[j]]
];
polynomial = D[ Expand[ polynomial ], var ];
dvdfs = CoefficientList[ polynomial, var ];
If[ Length[ dvdfs ] < order,
dvdfs = Join[ dvdfs, zerolist[[ order - Length[ dvdfs ] ]] ]
];
newelement = { dvdfs, zerolist[[order-1]] };
Return[ newelement ];
];
Interpolate[ x_, dvdfs_, coeffs_, tn_, order_ ] :=
Module[ { answer, xmtn, j },
answer = dvdfs[[order+1]];
xmtn = x - tn;
For[ j = order, j >= 1, j--,
answer = ( xmtn + coeffs[[j]] ) * answer + dvdfs[[j]]
];
Return[ answer ];
]
DiffLine[ line_, zerolist_, lastdimpts_ ] :=
Module[ { i, newline, dvdfs, coeffs, order, tn, x, df, element },
newline = Table[ DiffElem[ line[[i]], zerolist ],
{ i, 2, Length[ line ] } ];
{ dvdfs, coeffs } = newline[[1]];
tn = lastdimpts[[2]];
x = lastdimpts[[1]];
order = Length[ coeffs ];
df = Interpolate[ x, dvdfs, coeffs, tn, order ];
element = { { df, 0 }, { 0 } };
PrependTo[ newline, element ];
Return[ newline ];
]
InterpolatingFunction /:
Derivative[0,1][
InterpolatingFunction[ range_, table_ ] ] :=
Module[ { gridpoints, tbl, i, j, order,
zerolist, lastdimpts, newtable,answer },
gridpoints = table[[1]];
lastdimpts = Last[ gridpoints ];
tbl = table[[2]];
order = Max[ Table[ Length[ tbl[[1,i,2]] ],
{ i, Length[ tbl[[1]] ] } ] ];
zerolist = Table[ Table[ 0, { j, i } ], { i, order } ];
newtable = Table[ DiffLine[ tbl[[i]], zerolist, lastdimpts ],
{ i, Length[ tbl ] } ];
answer = InterpolatingFunction[ range, { gridpoints, newtable }
];
Return[ answer ];
];
InterpolatingFunction /:
Derivative[1,0][InterpolatingFunction[ range_, table_ ] ] :=
Module[ { gridpoints, ifunc, data, i, j, x, y },
gridpoints = table[[1]];
ifunc = InterpolatingFunction[ range, table ];
data = Table[ { y = gridpoints[[2,j]], x = gridpoints[[1,i]],
ifunc[x,y] },
{ j, Length[ gridpoints[[2]] ] },
{ i, Length[ gridpoints[[1]] ] } ];
data = Flatten[ data, 1 ];
ifunc = Interpolation[ data ];
ifunc = Derivative[0,1][ifunc];
data = Table[ { x = gridpoints[[1,i]], y = gridpoints[[2,j]],
ifunc[y,x] },
{ i, Length[ gridpoints[[1]] ] },
{ j, Length[ gridpoints[[2]] ] } ];
data = Flatten[ data, 1 ];
ifunc = Interpolation[ data ];
Return[ ifunc ];
];
End[]
EndPackage[]
Cheers,
Paul
_________________________________________________________________
Paul Abbott
Department of Physics Phone: +61-9-380-2734
The University of Western Australia Fax: +61-9-380-1014
Nedlands WA 6907 paul at physics.uwa.edu.au
AUSTRALIA http://www.pd.uwa.edu.au/Paul
_________________________________________________________________
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