naginterfaces.library.interp.dim3_scat_shep¶

naginterfaces.library.interp.dim3_scat_shep(x, y, z, f, nw, nq)[source]¶

dim3_scat_shep generates a three-dimensional interpolant to a set of scattered data points, using a modified Shepard method.

For full information please refer to the NAG Library document for e01tg

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/e01/e01tgf.html

Parameters

xfloat, array-like, shape $(m)$: $x [r - 1]$ , $y [r - 1]$ , $z [r - 1]$ must be set to the Cartesian coordinates of the data point $(x_{r}, y_{r}, z_{r})$ , for $r = 1, 2, \dots, m$ .
yfloat, array-like, shape $(m)$: $x [r - 1]$ , $y [r - 1]$ , $z [r - 1]$ must be set to the Cartesian coordinates of the data point $(x_{r}, y_{r}, z_{r})$ , for $r = 1, 2, \dots, m$ .
zfloat, array-like, shape $(m)$: $x [r - 1]$ , $y [r - 1]$ , $z [r - 1]$ must be set to the Cartesian coordinates of the data point $(x_{r}, y_{r}, z_{r})$ , for $r = 1, 2, \dots, m$ .
ffloat, array-like, shape $(m)$: $f [r - 1]$ must be set to the data value $f_{r}$ , for $r = 1, 2, \dots, m$ .
nwint: The number $N_{w}$ of data points that determines each radius of influence $R_{w}$ , appearing in the definition of each of the weights $w_{r}$ , for $r = 1, 2, \dots, m$ (see Notes). Note that $R_{w}$ is different for each weight. If $n w \leq 0$ the default value $n w = m i n (32, m - 1)$ is used instead.
nqint: The number $N_{q}$ of data points to be used in the least squares fit for coefficients defining the nodal functions $q_{r} (x, y, z)$ (see Notes). If $n q \leq 0$ the default value $n q = m i n (17, m - 1)$ is used instead.

Returns

iqint, ndarray, shape $(2 \times m + 1)$: Integer data defining the interpolant $Q (x, y, z)$ .
rqfloat, ndarray, shape $(10 \times m + 7)$: Real data defining the interpolant $Q (x, y, z)$ .

Raises

NagValueError

(errno $1$ )

On entry, $lrq$ is too small: $lrq = ⟨ v a l u e ⟩$ .

(errno $1$ )

On entry, $liq$ is too small: $liq = ⟨ v a l u e ⟩$ .

(errno $1$ )

On entry, $n w = ⟨ v a l u e ⟩$ and $m = ⟨ v a l u e ⟩$ .

Constraint: $n w \leq m i n (40, m - 1)$ .

(errno $1$ )

On entry, $n q = ⟨ v a l u e ⟩$ and $m = ⟨ v a l u e ⟩$ .

Constraint: $n q \leq m i n (40, m - 1)$ .

(errno $1$ )

On entry, $n q = ⟨ v a l u e ⟩$ .

Constraint: $n q \leq 0$ or $n q \geq 9$ .

(errno $1$ )

On entry, $m = ⟨ v a l u e ⟩$ .

Constraint: $m \geq 10$ .

(errno $2$ )

There are duplicate nodes in the dataset. $(x [I - 1], y [I - 1], z [I - 1]) = (x [J - 1], y [J - 1], z [J - 1])$ for: $I = ⟨ v a l u e ⟩$ and $J = ⟨ v a l u e ⟩$ . The interpolant cannot be derived.

(errno $3$ )

All nodes are coplanar. There is no unique solution.

Notes

dim3_scat_shep constructs a smooth function $Q (x, y, z)$ which interpolates a set of $m$ scattered data points $(x_{r}, y_{r}, z_{r}, f_{r})$ , for $r = 1, 2, \dots, m$ , using a modification of Shepard’s method. The surface is continuous and has continuous first partial derivatives.

The basic Shepard method, which is a generalization of the two-dimensional method described in Shepard (1968), interpolates the input data with the weighted mean

Q (x, y, z) = \frac{\sum_{r = 1}^{m} w_{r} (x, y, z) q_{r}}{\sum_{r = 1}^{m} w_{r} (x, y, z)},

where

q_{r} = f_{r} and w_{r} (x, y, z) = \frac{1}{d_{r}^{2}} and d_{r}^{2} = {(x - x_{r})}_{r}^{2} + {(y - y_{r})}_{r}^{2} + {(z - z_{r})}_{r}^{2} .

The basic method is global in that the interpolated value at any point depends on all the data, but this function uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each $w_{r} (x, y, z)$ to be zero outside a sphere with centre $(x_{r}, y_{r}, z_{r})$ and some radius $R_{w}$ . Also, to improve the performance of the basic method, each $q_{r}$ above is replaced by a function $q_{r} (x, y, z)$ , which is a quadratic fitted by weighted least squares to data local to $(x_{r}, y_{r}, z_{r})$ and forced to interpolate $(x_{r}, y_{r}, z_{r}, f_{r})$ . In this context, a point $(x, y, z)$ is defined to be local to another point if it lies within some distance $R_{q}$ of it. Computation of these quadratics constitutes the main work done by this function.

The efficiency of the function is further enhanced by using a cell method for nearest neighbour searching due to Bentley and Friedman (1979).

The radii $R_{w}$ and $R_{q}$ are chosen to be just large enough to include $N_{w}$ and $N_{q}$ data points, respectively, for user-supplied constants $N_{w}$ and $N_{q}$ . Default values of these arguments are provided by the function, and advice on alternatives is given in Further Comments.

This function is derived from the function QSHEP3 described by Renka (1988b).

Values of the interpolant $Q (x, y, z)$ generated by this function, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to dim3_scat_shep_eval().

References

Bentley, J L and Friedman, J H, 1979, Data structures for range searching, ACM Comput. Surv. (11), 397–409

Franke, R and Nielson, G, 1980, Smooth interpolation of large sets of scattered data, Internat. J. Num. Methods Engrg. (15), 1691–1704

Renka, R J, 1988, Multivariate interpolation of large sets of scattered data, ACM Trans. Math. Software (14), 139–148

Renka, R J, 1988, Algorithm 661: QSHEP3D: Quadratic Shepard method for trivariate interpolation of scattered data, ACM Trans. Math. Software (14), 151–152

Shepard, D, 1968, A two-dimensional interpolation function for irregularly spaced data, Proc. 23rd Nat. Conf. ACM, 517–523, Brandon/Systems Press Inc., Princeton

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naginterfaces.library.interp.dim3_scat_shep¶

naginterfaces.library.interp.dim3_​scat_​shep¶

naginterfaces.library.interp.dim3_scat_shep¶