NAG CL Interface
e01znc (dimn_​scat_​shep_​eval)

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1 Purpose

e01znc evaluates the multidimensional interpolating function generated by e01zmc and its first partial derivatives.

2 Specification

#include <nag.h>
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) , xd of a set of scattered data points (xr,fr) , for r=1,2,,m, as computed by e01zmc, and evaluates the interpolant and its first partial derivatives at the set of points xi, for i=1,2,,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: d2.
2: m Integer Input
On entry: must be the same value supplied for argument m in the preceding call to e01zmc.
Constraint: m (d+1) × (d+2) /2+2 .
3: x[d×m] const double Input
Note: the ith ordinate of the point xj is stored in x[(j-1)×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×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)×(d+2)/2)×m+2×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: n1.
8: xe[d×n] const double Input
Note: the ith ordinate of the point xj is stored in xe[(j-1)×d+i-1].
On entry: xe[(j-1)×d] ,, xe[(j-1)×d+d-1] must be set to the evaluation point xj, for j=1,2,,n.
9: q[n] double Output
On exit: q[i-1] contains the value of the interpolant, at xi, for i=1,2,,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×n] double Output
Note: the (i,j)th element of the matrix is stored in qx[(j-1)×d+i-1].
On exit: qx[(j-1)×d+i-1] contains the value of the partial derivatives with respect to the ith independent variable (dimension) of the interpolant Q (x) at xj, for j=1,2,,n, and for each of the partial derivatives i=1,2,,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: d2.
On entry, n=value.
Constraint: n1.
NE_INT_2
On entry, ((d+1)×(d+2)/2)×m+2×d+1 exceeds the largest machine integer.
d=value and m=value.
On entry, m=value and d=value.
Constraint: m (d+1) × (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(mn) time will be required.

10 Example

This program evaluates the function (in six variables)
f(x) = x1 x2 x3 1 + 2 x4 x5 x6  
at a set of randomly generated data points and calls e01zmc to construct an interpolating function Qx. It then calls e01znc to evaluate the interpolant at a set of points on the line xi=x, for i=1,2,,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.
See also e01zmc.

10.1 Program Text

Program Text (e01znce.c)

10.2 Program Data

Program Data (e01znce.d)

10.3 Program Results

Program Results (e01znce.r)