void	e01znc (Integer d, Integer m, const double x[], const double f[], const Integer iq[], const double rq[], Integer n, const double xe[], double q[], double qx[], NagError *fail)

The function may be called by the names: e01znc, nag_interp_dimn_scat_shep_eval or nag_nd_shep_eval.

3 Description

e01znc takes as input the interpolant

Q (x)

x \in ℝ^{d}

of a set of scattered data points

(x_{r}, f_{r})

, for

r = 1, 2, \dots, m

, as computed by e01zmc, and evaluates the interpolant and its first partial derivatives at the set of points

x_{i}

, for

i = 1, 2, \dots, n

e01znc must only be called after a call to e01zmc.

e01znc is derived from the new implementation of QS3GRD described by Renka (1988). It uses the modification for high-dimensional interpolation described by Berry and Minser (1999).

4 References

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

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

5 Arguments

1: $d$ – Integer Input: On entry: must be the same value supplied for argument d in the preceding call to e01zmc.

Constraint: $d \geq 2$ .
2: $m$ – Integer Input: On entry: must be the same value supplied for argument m in the preceding call to e01zmc.

Constraint: $m \geq (d + 1) \times (d + 2) / 2 + 2$ .
3: $x [d \times m]$ – const double Input: Note: the $i$ th ordinate of the point $x_{j}$ is stored in $x [(j - 1) \times d + i - 1]$ .

On entry: must be the same array supplied as argument x in the preceding call to e01zmc. It must remain unchanged between calls.
4: $f [m]$ – const double Input: On entry: must be the same array supplied as argument f in the preceding call to e01zmc. It must remain unchanged between calls.
5: $iq [2 \times m + 1]$ – const Integer Input: On entry: must be the same array returned as argument iq in the preceding call to e01zmc. It must remain unchanged between calls.
6: $rq [\dim]$ – const double Input: Note: the dimension, dim, of the array rq must be at least $((d + 1) \times (d + 2) / 2) \times m + 2 \times d + 1$ .

On entry: must be the same array returned as argument rq in the preceding call to e01zmc. It must remain unchanged between calls.
7: $n$ – Integer Input: On entry: $n$ , the number of evaluation points.

Constraint: $n \geq 1$ .
8: $xe [d \times n]$ – const double Input: Note: the $i$ th ordinate of the point $x_{j}$ is stored in $xe [(j - 1) \times d + i - 1]$ .

On entry: $xe [(j - 1) \times d], \dots, xe [(j - 1) \times d + d - 1]$ must be set to the evaluation point $x_{j}$ , for $j = 1, 2, \dots, n$ .
9: $q [n]$ – double Output: On exit: $q [i - 1]$ contains the value of the interpolant, at $x_{i}$ , for $i = 1, 2, \dots, n$ . If any of these evaluation points lie outside the region of definition of the interpolant the corresponding entries in q are set to an extrapolated approximation, and e01znc returns with $fail . code =$ NE_BAD_POINT.
10: $qx [d \times n]$ – double Output: Note: the $(i, j)$ th element of the matrix is stored in $qx [(j - 1) \times d + i - 1]$ .

On exit: $qx [(j - 1) \times d + i - 1]$ contains the value of the partial derivatives with respect to the $i$ th independent variable (dimension) of the interpolant $Q (x)$ at $x_{j}$ , for $j = 1, 2, \dots, n$ , and for each of the partial derivatives $i = 1, 2, \dots, d$ . If any of these evaluation points lie outside the region of definition of the interpolant, the corresponding entries in qx are set to extrapolated approximations to the partial derivatives, and e01znc returns with $fail . code =$ NE_BAD_POINT.
11: $fail$ – 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_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_BAD_POINT: On entry, at least one evaluation point lies outside the region of definition of the interpolant. At such points the corresponding values in q and qx contain extrapolated approximations. Points should be evaluated one by one to identify extrapolated values.
NE_INT: On entry, $d = ⟨ value ⟩$ .
Constraint: $d \geq 2$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .
NE_INT_2: On entry, $((d + 1) \times (d + 2) / 2) \times m + 2 \times d + 1$ exceeds the largest machine integer.
$d = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .

On entry, $m = ⟨ value ⟩$ and $d = ⟨ value ⟩$ .
Constraint: $m \geq (d + 1) \times (d + 2) / 2 + 2$ .
NE_INT_ARRAY: On entry, values in iq appear to be invalid. Check that iq has not been corrupted between calls to e01zmc and e01znc.
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.
NE_REAL_ARRAY: On entry, values in rq appear to be invalid. Check that rq has not been corrupted between calls to e01zmc and e01znc.

7 Accuracy

Computational errors should be negligible in most practical situations.

8 Parallelism and Performance

e01znc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

e01znc 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

The time taken for a call to e01znc will depend in general on the distribution of the data points. If the data points are approximately uniformly distributed, then the time taken should be only

O (n)

. At worst

O (m n)

time will be required.

10 Example

This program evaluates the function (in six variables)

f (x) = \frac{x_{1} x_{2} x_{3}}{1 + 2 x_{4} x_{5} x_{6}}

at a set of randomly generated data points and calls e01zmc to construct an interpolating function

Q_{x}

. It then calls e01znc to evaluate the interpolant at a set of points on the line

x_{i} = x

, for

i = 1, 2, \dots, 6

. To reduce the time taken by this example, the number of data points is limited. Increasing this value to the suggested minimum of

4000

improves the interpolation accuracy at the expense of more time.

e01zn: FL CL CPP AD

NAG CL Interfacee01znc (dimn_​scat_​shep_​eval)

▸▿ 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

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
e01znc (dimn_scat_shep_eval)