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)

to be zero outside a circle with centre

(x_{r}, y_{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)

, which is a quadratic fitted by weighted least squares to data local to

(x_{r}, y_{r})

and forced to interpolate

(x_{r}, y_{r}, f_{r})

. In this context, a point

(x, y)

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 Section 9.2.

This function is derived from the function QSHEP2 described by Renka (1988b).

Values of the interpolant

Q (x, y)

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 e01shc.

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

Renka R J (1988b) Algorithm 660: QSHEP2D: Quadratic Shepard method for bivariate interpolation of scattered data ACM Trans. Math. Software 14 149–150

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: $m$ – Integer Input: On entry: $m$ , the number of data points.

Constraint: $m \geq 6$ .
2: $x [m]$ – const double Input
3: $y [m]$ – const double Input: On entry: the Cartesian coordinates of the data points $(x_{r}, y_{r})$ , for $r = 1, 2, \dots, m$ .

Constraint: these coordinates must be distinct, and must not all be collinear.
4: $f [m]$ – const double Input: On entry: $f [r - 1]$ must be set to the data value $f_{r}$ , for $r = 1, 2, \dots, m$ .
5: $nw$ – Integer Input: 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 $nw = \min (19, m - 1)$ is used instead.

Constraint: $nw \leq \min (40, m - 1)$ .
6: $nq$ – Integer Input: On entry: the number $N_{q}$ of data points to be used in the least squares fit for coefficients defining the nodal functions $q_{r} (x, y)$ (see Section 3). If $nq \leq 0$ the default value $nq = \min (13, m - 1)$ is used instead.

Constraint: $nq \leq 0$ or $5 \leq nq \leq \min (40, m - 1)$ .
7: $iq [\dim]$ – Integer Output: On exit: integer data defining the interpolant $Q (x, y)$ .
8: $rq [\dim]$ – double Output: On exit: real data defining the interpolant $Q (x, y)$ .
9: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALL_DATA_COLLINEAR: All nodes are collinear. There is no unique solution.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_DATA_NOT_UNIQUE: There are duplicate nodes in the dataset. $(x [I - 1], y [I - 1]) = (x [J - 1], y [J - 1])$ , for $I = ⟨ value ⟩$ and $J = ⟨ value ⟩$ . The interpolant cannot be derived.
NE_INT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 6$ .

On entry, $nq = ⟨ value ⟩$ .
Constraint: $nq \leq 0$ or $nq \geq 5$ .
NE_INT_2: On entry, $nq = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $nq \leq \min (40, m - 1)$ .

On entry, $nw = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $nw \leq \min (40, m - 1)$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

On successful exit, the function generated interpolates the input data exactly and has quadratic accuracy.

8 Parallelism and Performance

e01sgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

e01sgc 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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also 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 e01sgc will depend in general on the distribution of the data points. If x and y 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 e01sgc 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 e01sgc through positive values of nw and nq. Increasing these arguments 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 (19, m - 1)

and

nq = \min (13, m - 1)

have been chosen on the basis of experimental results reported in Renka (1988a). In these experiments the error norm was found to vary smoothly with

N_{w}

and

N_{q}

, generally increasing monotonically and slowly with distance from the optimal pair. The method is not, therefore, thought to be particularly sensitive to the argument values. For further advice on the choice of these arguments see Renka (1988a).

10 Example

This program reads in a set of

30

data points and calls e01sgc to construct an interpolating function

Q (x, y)

. It then calls e01shc 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.

e01sg: FL CL CPP AD

NAG CL Interfacee01sgc (dim2_​scat_​shep)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Timing

9.2 Choice of Nw and Nq

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
e01sgc (dim2_scat_shep)

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