The InverseDistaneWeightedInterpolation command creates a function $f\left(\mathrm{x1}\,\mathrm{...}\,\mathrm{xn}\right)=\mathrm{value}$ which can then be evaluated at arbitrary points in $R^{\mathrm{n}}$.

This interpolant is defined at the point $x$ as the weighted average of the values ${\mathrm{values}}_i$, where each value is weighted by the inverse of the distance between ${\mathrm{points}}_i$ and $x$, and only points at distance at most r of $x$ are used.

By default, r is infinity, so all sample points will be used to interpolate a query point.

If no points lie within distance r, the interpolated value will be $Float\left(\mathrm{undefined}\right)$.

For sufficiently large r, this interpolation method produces a $C^1$ continuous interpolant.

This interpolation method does not introduce local minima or maxima which are not already present in the input data.

f can be evaluated at every point in $R^{\mathrm{n}}$, but results for points far away from the sample points may not be meaningful.

As with all interpolation methods, the interpolant f always passes through all of the sample values.

Input sample points must not contain duplicates. The presence of duplicate points can lead to unexpected results.

In order to evaluate f at k points, you can put each point in a row of a Matrix M and call f(M) to obtain the k values of f in a k-element Vector. This will be most efficient if M's options are such that its datatype is float[8], its order is C_order, and its storage is rectangular.

$\mathrm{XY}≔\left[\left[0\,0\right]\,\left[1\,0\right]\,\left[2\,0\right]\,\left[0\,1\right]\,\left[1\,1\right]\,\left[2\,1\right]\,\left[0\,2\right]\,\left[1\,2\right]\,\left[2\,2\right]\right]$

$\mathrm{XY}≔\left[\left[0\,0\right]\,\left[1\,0\right]\,\left[2\,0\right]\,\left[0\,1\right]\,\left[1\,1\right]\,\left[2\,1\right]\,\left[0\,2\right]\,\left[1\,2\right]\,\left[2\,2\right]\right]$

$Z≔\left[0\,0\,0\,0\,1\,0\,0\,0\,0\right]$

$Z≔\left[0\,0\,0\,0\,1\,0\,0\,0\,0\right]$

$f≔\mathrm{Interpolation}:-\mathrm{InverseDistanceWeightedInterpolation}\left(\mathrm{XY}\,Z\,1.5\right)$

$f≔\left(\begin{array}{c}\mathrm{Inv\ⅇrs\ⅇ\; Distanc\ⅇ\; W\ⅇight\ⅇ\ⅆ\; int\ⅇrpolation\; obȷ\ⅇct\; with\; 9\; sampl\ⅇ\; points}\\ \mathrm{Ra\ⅆius\; of\; influ\ⅇnc\ⅇ:\; 1.5}\end{array}\right)$

$f\left(0.5\,0.5\right)$

$0.250000000000000000$

$M≔\mathrm{Matrix}\left(\left[\left[1.5\,0.3\right]\,\left[0.7\,1.4\right]\,\left[1.2\,1.8\right]\right]\,\mathrm{datatype}=\mathrm{float}\left[8\right]\,\mathrm{order}=\mathrm{C\_order}\right)$

$M≔\left[\begin{array}{cc}1.50000000000000& 0.300000000000000\\ 0.700000000000000& 1.40000000000000\\ 1.20000000000000& 1.80000000000000\end{array}\right]$

$f\left(M\right)$

$\left[\begin{array}{c}0.0913991699478318\\ 0.599109577735220\\ 0.0335025560440729\end{array}\right]$

$\mathrm{plot3d}\left(\left(x\,y\right)\mapsto f\left(x\,y\right)\,0..2\,0..2\,\mathrm{labels}=\left[x\,y\,z\right]\right)$

The Interpolation[InverseDistanceWeightedInterpolation] command was introduced in Maple 2018.

For more information on Maple 2018 changes, see Updates in Maple 2018.