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MathGroup Archive 1996

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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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