naginterfaces.library.interp.dim2_​scat

naginterfaces.library.interp.dim2_scat(x, y, f)

dim2_scat generates a two-dimensional surface interpolating a set of scattered data points, using the method of Renka and Cline.

For full information please refer to the NAG Library document for e01sa

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/e01/e01saf.html

Parameters

xfloat, array-like, shape $\left(m\right)$

The coordinates of the $\text{r}$th data point, for $\text{r}=1,2,\dots ,m$. The data points are accepted in any order, but see Further Comments.

yfloat, array-like, shape $\left(m\right)$

The coordinates of the $\text{r}$th data point, for $\text{r}=1,2,\dots ,m$. The data points are accepted in any order, but see Further Comments.

ffloat, array-like, shape $\left(m\right)$

The coordinates of the $\text{r}$th data point, for $\text{r}=1,2,\dots ,m$. The data points are accepted in any order, but see Further Comments.

Returns

triangint, ndarray, shape $(7\times m)$

A data structure defining the computed triangulation, in a form suitable for passing to dim2_scat_eval().

gradsfloat, ndarray, shape $(2,m)$

The estimated partial derivatives at the nodes, in a form suitable for passing to dim2_scat_eval(). The derivatives at node $\text{r}$ with respect to $x$ and $y$ are contained in $grads[0,\text{r}-1]$ and $grads[1,\text{r}-1]$ respectively, for $\text{r}=1,2,\dots ,m$.

Raises

NagValueError

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 3$.

(errno $2$)

All nodes are collinear. There is no unique solution.

(errno $3$)

On entry, $(x[\text{I}-1],y[\text{I}-1])=(x[\text{J}-1],y[\text{J}-1])$, for $\text{I},\text{J}=⟨value⟩⟨value⟩$, $x[\text{I}-1]$, $y[\text{I}-1]=⟨value⟩⟨value⟩$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

dim2_scat constructs an interpolating surface $F(x,y)$ through a set of $m$ scattered data points $(x_{\text{r}},y_{\text{r}},f_{\text{r}})$, for $\text{r}=1,2,\dots ,m$, using a method due to Renka and Cline. In the $(x,y)$ plane, the data points must be distinct. The constructed surface is continuous and has continuous first derivatives.

The method involves firstly creating a triangulation with all the $(x,y)$ data points as nodes, the triangulation being as nearly equiangular as possible (see Cline and Renka (1984)). Then gradients in the $x$- and $y$-directions are estimated at node $\text{r}$, for $\text{r}=1,2,\dots ,m$, as the partial derivatives of a quadratic function of $x$ and $y$ which interpolates the data value $f_r$, and which fits the data values at nearby nodes (those within a certain distance chosen by the algorithm) in a weighted least squares sense. The weights are chosen such that closer nodes have more influence than more distant nodes on derivative estimates at node $r$. The computed partial derivatives, with the $f_r$ values, at the three nodes of each triangle define a piecewise polynomial surface of a certain form which is the interpolant on that triangle. See Renka and Cline (1984) for more detailed information on the algorithm, a development of that by Lawson (1977). The code is derived from Renka (1984).

The interpolant $F(x,y)$ can subsequently be evaluated at any point $(x,y)$ inside or outside the domain of the data by a call to dim2_scat_eval(). Points outside the domain are evaluated by extrapolation.

References

Cline, A K and Renka, R L, 1984, A storage-efficient method for construction of a Thiessen triangulation, Rocky Mountain J. Math. (14), 119–139

Lawson, C L, 1977, Software for $C^1$ surface interpolation, Mathematical Software III, (ed J R Rice), 161–194, Academic Press

Renka, R L, 1984, Algorithm 624: triangulation and interpolation of arbitrarily distributed points in the plane, ACM Trans. Math. Software (10), 440–442

Renka, R L and Cline, A K, 1984, A triangle-based $C^1$ interpolation method, Rocky Mountain J. Math. (14), 223–237