NAG CL Interfacee01tlc (dim4_​scat_​shep_​eval)

Settings help

CL Name Style:

1Purpose

e01tlc evaluates the four-dimensional interpolating function generated by e01tkc and its first partial derivatives.

2Specification

 #include
 void e01tlc (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: e01tlc, nag_interp_dim4_scat_shep_eval or nag_4d_shep_eval.

3Description

e01tlc takes as input the interpolant $Q\left(\mathbf{x}\right)$, $x\in {ℝ}^{4}$ of a set of scattered data points $\left({\mathbf{x}}_{\mathit{r}},{f}_{\mathit{r}}\right)$, for $\mathit{r}=1,2,\dots ,m$, as computed by e01tkc, and evaluates the interpolant and its first partial derivatives at the set of points ${\mathbf{x}}_{i}$, for $\mathit{i}=1,2,\dots ,n$.
e01tlc must only be called after a call to e01tkc.
e01tlc 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).

4References

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

5Arguments

1: $\mathbf{m}$Integer Input
On entry: must be the same value supplied for argument m in the preceding call to e01tkc.
Constraint: ${\mathbf{m}}\ge 16$.
2: $\mathbf{x}\left[4×{\mathbf{m}}\right]$const double Input
Note: the coordinates of ${x}_{r}$ are stored in ${\mathbf{x}}\left[\left(r-1\right)×4\right]\dots {\mathbf{x}}\left[\left(r-1\right)×4+3\right]$.
On entry: must be the same array supplied as argument x in the preceding call to e01tkc. It must remain unchanged between calls.
3: $\mathbf{f}\left[{\mathbf{m}}\right]$const double Input
On entry: must be the same array supplied as argument f in the preceding call to e01tkc. It must remain unchanged between calls.
4: $\mathbf{iq}\left[2×{\mathbf{m}}+1\right]$const Integer Input
On entry: must be the same array returned as argument iq in the preceding call to e01tkc. It must remain unchanged between calls.
5: $\mathbf{rq}\left[15×{\mathbf{m}}+9\right]$const double Input
On entry: must be the same array returned as argument rq in the preceding call to e01tkc. It must remain unchanged between calls.
6: $\mathbf{n}$Integer Input
On entry: $n$, the number of evaluation points.
Constraint: ${\mathbf{n}}\ge 1$.
7: $\mathbf{xe}\left[4×{\mathbf{n}}\right]$const double Input
Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{xe}}\left[\left(j-1\right)×4+i-1\right]$.
On entry: ${\mathbf{xe}}\left[\left(\mathit{r}-1\right)×4\right],\dots ,{\mathbf{xe}}\left[\left(\mathit{r}-1\right)×4+3\right]$ must be set to the evaluation point ${\mathbf{x}}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
8: $\mathbf{q}\left[{\mathbf{n}}\right]$double Output
On exit: ${\mathbf{q}}\left[\mathit{i}-1\right]$ contains the value of the interpolant, at ${\mathbf{x}}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. 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 e01tlc returns with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_BAD_POINT.
9: $\mathbf{qx}\left[4×{\mathbf{n}}\right]$double Output
Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{qx}}\left[\left(j-1\right)×4+i-1\right]$.
On exit: ${\mathbf{qx}}\left[\left(i-1\right)×4+j-1\right]$ contains the value of the partial derivatives with respect to ${\mathbf{x}}_{j}$ of the interpolant $Q\left(\mathbf{x}\right)$ at ${\mathbf{x}}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, and for each of the four partial derivatives $j=1,2,3,4$. 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 e01tlc returns with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_BAD_POINT.
10: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error 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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
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, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 16$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INT_ARRAY
On entry, values in iq appear to be invalid. Check that iq has not been corrupted between calls to e01tkc and e01tlc.
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 e01tkc and e01tlc.

7Accuracy

Computational errors should be negligible in most practical situations.

8Parallelism and Performance

e01tlc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e01tlc 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.

The time taken for a call to e01tlc 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 $\mathit{O}\left(n\right)$. At worst $\mathit{O}\left(mn\right)$ time will be required.

10Example

This program evaluates the function
 $f (x) = (1.25+cos(5.4x4)) cos(6x1) cos(6x2) 6 + 6 (3x3-1) 2$
at a set of $30$ randomly generated data points and calls e01tkc to construct an interpolating function $Q\left(\mathbf{x}\right)$. It then calls e01tlc to evaluate the interpolant at a set of random points.
To reduce the time taken by this example, the number of data points is limited to $30$. Increasing this value improves the interpolation accuracy at the expense of more time.