e01 Chapter Contents
e01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_nd_shep_interp (e01zmc)

## 1  Purpose

nag_nd_shep_interp (e01zmc) generates a multi-dimensional interpolant to a set of scattered data points, using a modified Shepard method. When the number of dimensions is no more than five, there are corresponding functions in Chapter e01 which are specific to the given dimensionality. nag_2d_shep_interp (e01sgc) generates the two-dimensional interpolant, while nag_3d_shep_interp (e01tgc)nag_4d_shep_interp (e01tkc) and nag_5d_shep_interp (e01tmc) generate the three-, four- and five-dimensional interpolants respectively.

## 2  Specification

 #include #include
 void nag_nd_shep_interp (Integer d, Integer m, const double x[], const double f[], Integer nw, Integer nq, Integer iq[], double rq[], NagError *fail)

## 3  Description

nag_nd_shep_interp (e01zmc) constructs a smooth function $Q\left(\mathbf{x}\right)$, $\mathbf{x}\in {ℝ}^{d}$ which interpolates a set of $m$ scattered data points $\left({\mathbf{x}}_{r},{f}_{r}\right)$, 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
 $Q x = ∑ r=1 m wr x qr ∑ r=1 m wr x ,$
where ${q}_{r}={f}_{r}$, ${w}_{r}\left(\mathbf{x}\right)=\frac{1}{{{‖\mathbf{x}-{\mathbf{x}}_{r}‖}_{2}}^{2}}$.
The basic method is global in that the interpolated value at any point depends on all the data, but nag_nd_shep_interp (e01zmc) uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each ${w}_{r}\left(\mathbf{x}\right)$ to be zero outside a hypersphere with centre ${\mathbf{x}}_{r}$ and some radius ${R}_{w}$. Also, to improve the performance of the basic method, each ${q}_{r}$ above is replaced by a function ${q}_{r}\left(\mathbf{x}\right)$, which is a quadratic fitted by weighted least squares to data local to ${\mathbf{x}}_{r}$ and forced to interpolate $\left({\mathbf{x}}_{r},{f}_{r}\right)$. In this context, a point $\mathbf{x}$ is defined to be local to another point if it lies within some distance ${R}_{q}$ of it.
The efficiency of nag_nd_shep_interp (e01zmc) 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 parameters are provided, and advice on alternatives is given in Section 9.2.
nag_nd_shep_interp (e01zmc) 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 $Q\left(\mathbf{x}\right)$ generated by nag_nd_shep_interp (e01zmc), and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to nag_nd_shep_eval (e01znc).

## 4  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 (1988a) Multivariate interpolation of large sets of scattered data ACM Trans. Math. Software 14 139–148
Renka R J (1988b) 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

## 5  Arguments

1:     dIntegerInput
On entry: $d$, the number of dimensions.
Constraint: ${\mathbf{d}}\ge 2$.
2:     mIntegerInput
On entry: $m$, the number of data points.
Note: on the basis of experimental results reported in Berry and Minser (1999), when ${\mathbf{d}}\ge 5$ it is recommended to use ${\mathbf{m}}\ge 4000$.
Constraint: ${\mathbf{m}}\ge \left({\mathbf{d}}+1\right)×\left({\mathbf{d}}+2\right)/2+2$.
3:     x[${\mathbf{d}}×{\mathbf{m}}$]const doubleInput
Note: the $\left(i,j\right)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{d}}+i-1\right]$.
On entry: the d components of the first data point must be stored in elements $0,1,\dots ,{\mathbf{d}}-1$ of x. The second data point must be stored in elements ${\mathbf{d}},{\mathbf{d}}+1,\dots ,2×{\mathbf{d}}-1$ of x, and so on. In general, the m data points must be stored in ${\mathbf{x}}\left[\mathit{i}×{\mathbf{d}}+\mathit{j}\right]$, for $\mathit{i}=0,1,\dots ,{\mathbf{m}}-1$ and $\mathit{j}=0,1,\dots ,{\mathbf{d}}-1$.
Constraint: these coordinates must be distinct, and must not all lie on the same $\left(d-1\right)$-dimensional hypersurface.
4:     f[m]const doubleInput
On entry: ${\mathbf{f}}\left[r-1\right]$ must be set to the data value ${f}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.
5:     nwIntegerInput
On entry: the number ${N}_{w}$ of data points that determines each radius of influence ${R}_{w}$, appearing in the definition of each of the weights ${w}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$ (see Section 3). Note that ${R}_{w}$ is different for each weight. If ${\mathbf{nw}}\le 0$ the default value ${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(2×\left({\mathbf{d}}+1\right)×\left({\mathbf{d}}+2\right),{\mathbf{m}}-1\right)$ is used instead.
Suggested value: ${\mathbf{nw}}=-1$
Constraint: ${\mathbf{nw}}\le {\mathbf{m}}-1$.
6:     nqIntegerInput
On entry: the number ${N}_{q}$ of data points to be used in the least squares fit for coefficients defining the quadratic functions ${q}_{r}\left(\mathbf{x}\right)$ (see Section 3). If ${\mathbf{nq}}\le 0$ the default value ${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\left({\mathbf{d}}+1\right)×\left({\mathbf{d}}+2\right)×6/5,{\mathbf{m}}-1\right)$ is used instead.
Suggested value: ${\mathbf{nq}}=-1$
Constraint: ${\mathbf{nq}}\le 0$ or $\left({\mathbf{d}}+1\right)×\left({\mathbf{d}}+2\right)/2-1\le {\mathbf{nq}}\le {\mathbf{m}}-1$.
7:     iq[$2×{\mathbf{m}}+1$]IntegerOutput
On exit: integer data defining the interpolant $Q\left(\mathbf{x}\right)$.
8:     rq[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array rq must be at least $\left(\left({\mathbf{d}}+1\right)×\left({\mathbf{d}}+2\right)/2\right)×{\mathbf{m}}+2×{\mathbf{d}}+1$.
On exit: real data defining the interpolant $Q\left(\mathbf{x}\right)$.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_DATA_HYPERSURFACE
On entry, all the data points lie on the same hypersurface. No unique solution exists.
NE_DUPLICATE_NODE
There are duplicate nodes in the dataset. ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{d}}+k-1\right]={\mathbf{x}}\left[\left(j-1\right)×{\mathbf{d}}+k-1\right]$, for $i=⟨\mathit{\text{value}}⟩$, $j=⟨\mathit{\text{value}}⟩$ and $k=1,2,\dots ,{\mathbf{d}}$. The interpolant cannot be derived.
NE_INT
On entry, ${\mathbf{d}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{d}}\ge 2$.
NE_INT_2
On entry, $\left(\left({\mathbf{d}}+1\right)×\left({\mathbf{d}}+2\right)/2\right)×{\mathbf{m}}+2×{\mathbf{d}}+1$ exceeds the largest machine integer.
${\mathbf{d}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{d}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge \left({\mathbf{d}}+1\right)×\left({\mathbf{d}}+2\right)/2+2$.
On entry, ${\mathbf{nq}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{d}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nq}}\le 0$ or ${\mathbf{nq}}\ge \left({\mathbf{d}}+1\right)×\left({\mathbf{d}}+2\right)/2-1$.
On entry, ${\mathbf{nq}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nq}}\le {\mathbf{m}}-1$.
On entry, ${\mathbf{nw}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nw}}\le {\mathbf{m}}-1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

## 7  Accuracy

In experiments undertaken by Berry and Minser (1999), the accuracies obtained for a conditional function resulting in sharp functional transitions were of the order of ${10}^{-1}$ at best. In other cases in these experiments, the function generated interpolates the input data with maximum absolute error of the order of ${10}^{-2}$.

## 8  Parallelism and Performance

nag_nd_shep_interp (e01zmc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nd_shep_interp (e01zmc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

### 9.1  Timing

The time taken for a call to nag_nd_shep_interp (e01zmc) will depend in general on the distribution of the data points and on the choice of ${N}_{w}$ and ${N}_{q}$ parameters. If the data points are uniformly randomly distributed, then the time taken should be $\mathit{O}\left(m\right)$. At worst $\mathit{O}\left({m}^{2}\right)$ time will be required.

### 9.2  Choice of ${N}_{w}$ and ${N}_{q}$

Default values of the parameters ${N}_{w}$ and ${N}_{q}$ may be selected by calling nag_nd_shep_interp (e01zmc) with ${\mathbf{nw}}\le 0$ and ${\mathbf{nq}}\le 0$. These default values may well be satisfactory for many applications.
If non-default values are required they must be supplied to nag_nd_shep_interp (e01zmc) through positive values of nw and nq. Increasing these argument values makes the method less local. This may increase the accuracy of the resulting interpolant at the expense of increased computational cost. The default values ${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(2×\left({\mathbf{d}}+1\right)×\left({\mathbf{d}}+2\right),{\mathbf{m}}-1\right)$ and ${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\left({\mathbf{d}}+1\right)×\left({\mathbf{d}}+2\right)×6/5,{\mathbf{m}}-1\right)$ have been chosen on the basis of experimental results reported in Renka (1988a) and Berry and Minser (1999). For further advice on the choice of these arguments see Renka (1988a) and Berry and Minser (1999).

## 10  Example

This program reads in a set of $30$ data points and calls nag_nd_shep_interp (e01zmc) to construct an interpolating function $Q\left(\mathbf{x}\right)$. It then calls nag_nd_shep_eval (e01znc) to evaluate the interpolant at a set of points.
Note that this example is not typical of a realistic problem: the number of data points would normally be very much larger.

### 10.1  Program Text

Program Text (e01zmce.c)

### 10.2  Program Data

Program Data (e01zmce.d)

### 10.3  Program Results

Program Results (e01zmce.r)