NAG Library Function Document

nag_5d_shep_interp (e01tmc)

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}‖}_{2}^{2}

The basic method is global in that the interpolated value at any point depends on all the data, but nag_5d_shep_interp (e01tmc) 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 nag_5d_shep_interp (e01tmc) 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, and advice on alternatives is given in Section 9.2.

nag_5d_shep_interp (e01tmc) is derived from the new implementation of QSHEP3 described by Renka (1988b). It uses the modification for five-dimensional interpolation described by Berry and Minser (1999).

Values of the interpolant

Q (x)

generated by nag_5d_shep_interp (e01tmc), and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to nag_5d_shep_eval (e01tnc).

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: m – IntegerInput: On entry: $m$ , the number of data points.
Note: on the basis of experimental results reported in Berry and Minser (1999), it is recommended to use $m \geq 4000$ .

Constraint: $m \geq 23$ .
2: x[ $5 \times m$ ] – const doubleInput: Note: the $(i, j)$ th element of the matrix $X$ is stored in $x [(j - 1) \times 5 + i - 1]$ .
On entry: $x [(r - 1) \times 5], \dots, x [(r - 1) \times 5 + 4]$ must be set to the Cartesian coordinates of the data point $x_{r}$ , for $r = 1, 2, \dots, m$ .
Constraint: these coordinates must be distinct, and must not all lie on the same four-dimensional hypersurface.
3: f[m] – const doubleInput: On entry: $f [r - 1]$ must be set to the data value $f_{r}$ , for $r = 1, 2, \dots, m$ .
4: nw – IntegerInput: 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 $w_{r}$ , for $r = 1, 2, \dots, m$ (see Section 3). Note that $R_{w}$ is different for each weight. If $nw \leq 0$ the default value $nw = \min (32, m - 1)$ is used instead.
Constraint: $nw \leq \min (50, m - 1)$ .
5: nq – IntegerInput: On entry: the number $N_{q}$ of data points to be used in the least squares fit for coefficients defining the quadratic functions $q_{r} (x)$ (see Section 3). If $nq \leq 0$ the default value $nq = \min (50, m - 1)$ is used instead.
Constraint: $nq \leq 0$ or $20 \leq nq \leq \min (70, m - 1)$ .
6: iq[ $2 \times m + 1$ ] – IntegerOutput: On exit: integer data defining the interpolant $Q (x)$ .
7: rq[ $21 \times m + 11$ ] – doubleOutput: On exit: real data defining the interpolant $Q (x)$ .
8: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_DATA_HYPERSURFACE: On entry, all the data points lie on the same four-dimensional hypersurface. No unique solution exists.
NE_DUPLICATE_NODE: There are duplicate nodes in the dataset. $x [(k - 1) \times 5 + i - 1] = x [(k - 1) \times 5 + j - 1]$ , for $i = ⟨value⟩$ , $j = ⟨value⟩$ and $k = 1, 2, \dots, 5$ . The interpolant cannot be derived.
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 23$ .
On entry, $nq = ⟨value⟩$ .
Constraint: $nq \leq 0$ or $nq \geq 20$ .
NE_INT_2: On entry, $nq = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $nq \leq \min (70, m - 1)$ .
On entry, $nw = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $nw \leq \min (50, m - 1)$ .
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

On successful exit, the function generated interpolates the input data exactly and has quadratic precision. Overall accuracy of the interpolant is affected by the choice of arguments nw and nq as well as the smoothness of the function represented by the input data. Berry and Minser (1999) report on the results obtained for a set of test functions.

8 Parallelism and Performance

nag_5d_shep_interp (e01tmc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_5d_shep_interp (e01tmc) 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.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

9.1 Timing

The time taken for a call to nag_5d_shep_interp (e01tmc) 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

O (m)

. At worst

O (m^{2})

time will be required.

9.2 Choice of $N_{w}$ and $N_{q}$

Default values of the arguments

N_{w}

and

N_{q}

may be selected by calling nag_5d_shep_interp (e01tmc) with

nw \leq 0

and

nq \leq 0

. These default values may well be satisfactory for many applications.

If non-default values are required they must be supplied to nag_5d_shep_interp (e01tmc) 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

nw = \min (32, m - 1)

and

nq = \min (50, m - 1)

have been chosen on the basis of experimental results reported in Berry and Minser (1999). In these experiments the error norm was found to increase with the decrease of

N_{q}

, but to be little affected by the choice of

N_{w}

. The choice of both, directly affected the time taken by the function. For further advice on the choice of these arguments see Berry and Minser (1999).

10 Example

This program reads in a set of

30

data points and calls nag_5d_shep_interp (e01tmc) to construct an interpolating function

Q (x)

. It then calls nag_5d_shep_eval (e01tnc) 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 larger.

See also Section 10 in nag_5d_shep_eval (e01tnc).

NAG Library Function Documentnag_5d_shep_interp (e01tmc)

+− Contents

1 Purpose

2 Specification

3 Description