NAG FL Interface
e01znf (dimn_​scat_​shep_​eval)

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1 Purpose

e01znf evaluates the multidimensional interpolating function generated by e01zmf and its first partial derivatives.

2 Specification

Fortran Interface
Subroutine e01znf ( d, m, x, f, iq, rq, n, xe, q, qx, ifail)
Integer, Intent (In) :: d, m, iq(2*m+1), n
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x(d,m), f(m), rq(*), xe(d,n)
Real (Kind=nag_wp), Intent (Out) :: q(n), qx(d,n)
C Header Interface
#include <nag.h>
void  e01znf_ (const Integer *d, const Integer *m, const double x[], const double f[], const Integer iq[], const double rq[], const Integer *n, const double xe[], double q[], double qx[], Integer *ifail)
The routine may be called by the names e01znf or nagf_interp_dimn_scat_shep_eval.

3 Description

e01znf 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 e01zmf, and evaluates the interpolant and its first partial derivatives at the set of points xi, for i=1,2,,n.
e01znf must only be called after a call to e01zmf.
e01znf 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 e01zmf.
Constraint: d2.
2: m Integer Input
On entry: must be the same value supplied for argument m in the preceding call to e01zmf.
Constraint: m (d+1) × (d+2) /2+2 .
3: x(d,m) Real (Kind=nag_wp) array Input
Note: the ith ordinate of the point xj is stored in x(i,j).
On entry: must be the same array supplied as argument x in the preceding call to e01zmf. It must remain unchanged between calls.
4: f(m) Real (Kind=nag_wp) array Input
On entry: must be the same array supplied as argument f in the preceding call to e01zmf. It must remain unchanged between calls.
5: iq(2×m+1) Integer array Input
On entry: must be the same array returned as argument iq in the preceding call to e01zmf. It must remain unchanged between calls.
6: rq(*) Real (Kind=nag_wp) array Input
Note: the dimension 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 e01zmf. It must remain unchanged between calls.
7: n Integer Input
On entry: n, the number of evaluation points.
Constraint: n1.
8: xe(d,n) Real (Kind=nag_wp) array Input
Note: the ith ordinate of the point xj is stored in xe(i,j).
On entry: xe(1:d,j) must be set to the evaluation point xj, for j=1,2,,n.
9: q(n) Real (Kind=nag_wp) array Output
On exit: q(i) 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 e01znf returns with ifail=3.
10: qx(d,n) Real (Kind=nag_wp) array Output
On exit: qx(i,j) 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 e01znf returns with ifail=3.
11: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, d=value.
Constraint: d2.
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.
On entry, n=value.
Constraint: n1.
ifail=2
On entry, values in iq appear to be invalid. Check that iq has not been corrupted between calls to e01zmf and e01znf.
On entry, values in rq appear to be invalid. Check that rq has not been corrupted between calls to e01zmf and e01znf.
ifail=3
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.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Computational errors should be negligible in most practical situations.

8 Parallelism and Performance

e01znf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e01znf 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 routine. 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 e01znf 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.

9.1 Internal Changes

Internal changes have been made to this routine as follows:
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

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 e01zmf to construct an interpolating function Qx. It then calls e01znf 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 e01zmf.

10.1 Program Text

Program Text (e01znfe.f90)

10.2 Program Data

Program Data (e01znfe.d)

10.3 Program Results

Program Results (e01znfe.r)