naginterfaces.library.interp.dim4_scat_shep¶

naginterfaces.library.interp.dim4_scat_shep(x, f, nw, nq)[source]¶

dim4_scat_shep generates a four-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 e01tk

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/e01/e01tkf.html

Parameters

xfloat, array-like, shape $(4, m)$: $x [0, r - 1], \dots, x [3, r - 1]$ must be set to the Cartesian coordinates of the data point $x_{r}$ , 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 quadratic functions $q_{r} (x)$ (see Notes). If $n q \leq 0$ the default value $n q = m i n (38, m - 1)$ is used instead.

Returns

iqint, ndarray, shape $(2 \times m + 1)$: Integer data defining the interpolant $Q (x)$ .
rqfloat, ndarray, shape $(15 \times m + 9)$: Real data defining the interpolant $Q (x)$ .

Raises

NagValueError

(errno $1$ )

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

Constraint: $n w \leq m i n (50, m - 1)$ .

(errno $1$ )

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

Constraint: $n q \leq m i n (50, m - 1)$ .

(errno $1$ )

On entry, $n q = ⟨ v a l u e ⟩$ .

Constraint: $n q \leq 0$ or $n q \geq 14$ .

(errno $1$ )

On entry, $m = ⟨ v a l u e ⟩$ .

Constraint: $m \geq 16$ .

(errno $2$ )

There are duplicate nodes in the dataset. $x [i - 1, k - 1] = x [j - 1, k - 1]$ , for $i = ⟨ v a l u e ⟩$ , $j = ⟨ v a l u e ⟩$ and $k = 1, 2, \dots, 4$ . The interpolant cannot be derived.

(errno $3$ )

On entry, all the data points lie on the same three-dimensional hypersurface. No unique solution exists.

Notes

dim4_scat_shep constructs a smooth function $Q (x)$ , $x \in R^{4}$ which interpolates a set of $m$ scattered data points $(x_{r}, f_{r})$ , 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) = \frac{\sum_{r = 1}^{m} w_{r} (x) q_{r}}{\sum_{r = 1}^{m} w_{r} (x)},

where $q_{r} = f_{r}$ , $w_{r} (x) = \frac{1}{d_{r}^{2}}$ and $d_{r}^{2} = {∥ x - x_{r} ∥}_{r}^{2}$ .

The basic method is global in that the interpolated value at any point depends on all the data, but dim4_scat_shep uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each $w_{r} (x)$ to be zero outside a hypersphere with centre $x_{r}$ and some radius $R_{w}$ . Also, to improve the performance of the basic method, each $q_{r}$ above is replaced by a function $q_{r} (x)$ , which is a quadratic fitted by weighted least squares to data local to $x_{r}$ and forced to interpolate $(x_{r}, f_{r})$ . In this context, a point $x$ is defined to be local to another point if it lies within some distance $R_{q}$ of it.

The efficiency of dim4_scat_shep is enhanced by using a cell method for nearest neighbour searching due to Bentley and Friedman (1979) with a cell density of $3$ .

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.

dim4_scat_shep is derived from the new implementation of QSHEP3 described by Renka (1988b). It uses the modification for high-dimensional interpolation described by Berry and Minser (1999).

Values of the interpolant $Q (x)$ generated by dim4_scat_shep, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to dim4_scat_shep_eval().

References

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

Berry, M W, Minser, K S, 1999, Algorithm 798: high-dimensional interpolation using the modified Shepard method, ACM Trans. Math. Software (25), 353–366

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.dim4_scat_shep¶

naginterfaces.library.interp.dim4_​scat_​shep¶

naginterfaces.library.interp.dim4_scat_shep¶