e01zmf generates a multidimensional interpolant to a set of scattered data points, using a modified Shepard method. When the number of dimensions is no more than five, there are corresponding routines in Chapter E01 which are specific to the given dimensionality. e01sgf generates the two-dimensional interpolant, while e01tgf,e01tkfande01tmf generate the three-, four- and five-dimensional interpolants respectively.
The routine may be called by the names e01zmf or nagf_interp_dimn_scat_shep.
3Description
e01zmf constructs a smooth function $Q\left(\mathbf{x}\right)$, $\mathbf{x}\in {\mathbb{R}}^{d}$ which interpolates a set of $m$ scattered data points $({\mathbf{x}}_{r},{f}_{r})$, for $r=1,2,\dots ,m$, using a modification of Shepard's method. The surface is continuous and has continuous first partial derivatives.
The basic Shepard method, which is a generalization of the two-dimensional method described in Shepard (1968), interpolates the input data with the weighted mean
where ${q}_{r}={f}_{r}$, ${w}_{r}\left(\mathbf{x}\right)=\frac{1}{{{\Vert \mathbf{x}-{\mathbf{x}}_{r}\Vert}_{2}}^{2}}$.
The basic method is global in that the interpolated value at any point depends on all the data, but e01zmf uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each ${w}_{r}\left(\mathbf{x}\right)$ to be zero outside a hypersphere with centre ${\mathbf{x}}_{r}$ and some radius ${R}_{w}$. Also, to improve the performance of the basic method, each ${q}_{r}$ above is replaced by a function ${q}_{r}\left(\mathbf{x}\right)$, which is a quadratic fitted by weighted least squares to data local to ${\mathbf{x}}_{r}$ and forced to interpolate $({\mathbf{x}}_{r},{f}_{r})$. In this context, a point $\mathbf{x}$ is defined to be local to another point if it lies within some distance ${R}_{q}$ of it.
The efficiency of e01zmf is enhanced by using a cell method for nearest neighbour searching due to Bentley and Friedman (1979) with a cell density of $3$.
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 parameters are provided, and advice on alternatives is given in Section 9.2.
e01zmf is derived from the new implementation of QSHEP3 described by Renka (1988b). It uses the modification for high-dimensional interpolation described by Berry and Minser (1999).
Values of the interpolant $Q\left(\mathbf{x}\right)$ generated by e01zmf, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to e01znf.
4References
Bentley J L and Friedman J H (1979) Data structures for range searching ACM Comput. Surv.11 397–409
Berry M W, Minser K S (1999) Algorithm 798: high-dimensional interpolation using the modified Shepard method ACM Trans. Math. Software25 353–366
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. Software14 139–148
Renka R J (1988b) Algorithm 661: QSHEP3D: Quadratic Shepard method for trivariate interpolation of scattered data ACM Trans. Math. Software14 151–152
Shepard D (1968) A two-dimensional interpolation function for irregularly spaced data Proc. 23rd Nat. Conf. ACM 517–523 Brandon/Systems Press Inc., Princeton
5Arguments
1: $\mathbf{d}$ – IntegerInput
On entry: $d$, the number of dimensions.
Constraint:
${\mathbf{d}}\ge 2$.
2: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of data points.
Note: on the basis of experimental results reported in Berry and Minser (1999), when ${\mathbf{d}}\ge 5$ it is recommended to use ${\mathbf{m}}\ge 4000$.
3: $\mathbf{x}({\mathbf{d}},{\mathbf{m}})$ – Real (Kind=nag_wp) arrayInput
On entry:
${\mathbf{x}}(1:{\mathbf{d}},\mathit{r})$ must be set to the Cartesian coordinates of the data point ${\mathbf{x}}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.
Constraint:
these coordinates must be distinct, and must not all lie on the same $(d-1)$-dimensional hypersurface.
4: $\mathbf{f}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{f}}\left(r\right)$ must be set to the data value
${f}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.
5: $\mathbf{nw}$ – IntegerInput
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
${w}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$ (see Section 3). Note that ${R}_{w}$ is different for each weight. If ${\mathbf{nw}}\le 0$ the default value ${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(2\times ({\mathbf{d}}+1)\times ({\mathbf{d}}+2),{\mathbf{m}}-1)$ is used instead.
Suggested value:
${\mathbf{nw}}=\mathrm{-1}$.
Constraint:
${\mathbf{nw}}\le {\mathbf{m}}-1$.
6: $\mathbf{nq}$ – IntegerInput
On entry: the number ${N}_{q}$ of data points to be used in the least squares fit for coefficients defining the quadratic functions ${q}_{r}\left(\mathbf{x}\right)$ (see Section 3). If ${\mathbf{nq}}\le 0$ the default value ${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(({\mathbf{d}}+1)\times ({\mathbf{d}}+2)\times 6/5,{\mathbf{m}}-1)$ is used instead.
Suggested value:
${\mathbf{nq}}=\mathrm{-1}$.
Constraint:
${\mathbf{nq}}\le 0$ or $({\mathbf{d}}+1)\times ({\mathbf{d}}+2)/2-1\le {\mathbf{nq}}\le {\mathbf{m}}-1$.
On exit: integer data defining the interpolant $Q\left(\mathbf{x}\right)$.
8: $\mathbf{rq}(*)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array rq
must be at least
$(({\mathbf{d}}+1)\times ({\mathbf{d}}+2)/2)\times {\mathbf{m}}+2\times {\mathbf{d}}+1$.
On exit: real data defining the interpolant $Q\left(\mathbf{x}\right)$.
9: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ 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 $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{d}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{d}}\ge 2$.
On entry, $(({\mathbf{d}}+1)\times ({\mathbf{d}}+2)/2)\times {\mathbf{m}}+2\times {\mathbf{d}}+1$ exceeds the largest machine integer. ${\mathbf{d}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{d}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge ({\mathbf{d}}+1)\times ({\mathbf{d}}+2)/2+2$.
On entry, ${\mathbf{nq}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{d}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nq}}\le 0$ or ${\mathbf{nq}}\ge ({\mathbf{d}}+1)\times ({\mathbf{d}}+2)/2-1$.
On entry, ${\mathbf{nq}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nq}}\le {\mathbf{m}}-1$.
On entry, ${\mathbf{nw}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nw}}\le {\mathbf{m}}-1$.
${\mathbf{ifail}}=2$
There are duplicate nodes in the dataset. ${\mathbf{x}}(k,i)={\mathbf{x}}(k,j)$, for $i=\u27e8\mathit{\text{value}}\u27e9$, $j=\u27e8\mathit{\text{value}}\u27e9$ and $k=1,2,\dots ,{\mathbf{d}}$. The interpolant cannot be derived.
${\mathbf{ifail}}=3$
On entry, all the data points lie on the same hypersurface. No unique solution exists.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
In experiments undertaken by Berry and Minser (1999), the accuracies obtained for a conditional function resulting in sharp functional transitions were of the order of ${10}^{\mathrm{-1}}$ at best. In other cases in these experiments, the function generated interpolates the input data with maximum absolute error of the order of ${10}^{\mathrm{-2}}$.
8Parallelism and Performance
e01zmf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e01zmf 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.
9Further Comments
9.1Timing
The time taken for a call to e01zmf will depend in general on the distribution of the data points and on the choice of ${N}_{w}$ and ${N}_{q}$ parameters. If the data points are uniformly randomly distributed, then the time taken should be $\mathit{O}\left(m\right)$. At worst $\mathit{O}\left({m}^{2}\right)$ time will be required.
9.2Choice of ${\mathit{N}}_{\mathit{w}}$ and ${\mathit{N}}_{\mathit{q}}$
Default values of the parameters ${N}_{w}$ and ${N}_{q}$ may be selected by calling e01zmf with ${\mathbf{nw}}\le 0$ and ${\mathbf{nq}}\le 0$. These default values may well be satisfactory for many applications.
If non-default values are required they must be supplied to e01zmf through positive values of nw and nq. Increasing these argument values makes the method less local. This may increase the accuracy of the resulting interpolant at the expense of increased computational cost. The default values ${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(2\times ({\mathbf{d}}+1)\times ({\mathbf{d}}+2),{\mathbf{m}}-1)$ and ${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(({\mathbf{d}}+1)\times ({\mathbf{d}}+2)\times 6/5,{\mathbf{m}}-1)$ have been chosen on the basis of experimental results reported in Renka (1988a) and Berry and Minser (1999). For further advice on the choice of these arguments see Renka (1988a) and Berry and Minser (1999).
9.3Internal Changes
Internal changes have been made to this routine as follows:
At Mark 26.0: The algorithm used by this routine, based on a Modified Shepard method, has been changed to produce more reliable results for some data sets which were previously not well handled. In addition, handling of evaluation points which are far away from the original data points has been improved by use of an extrapolation method which returns useful results rather than just an error message as was done at earlier Marks.
At Mark 26.1: The algorithm has undergone further changes which enable it to work better on certain data sets, for example data presented on a regular grid. The results returned when evaluating the function at points which are not in the original data set used to construct the interpolating function are now likely to be slightly different from those returned at previous Marks of the Library, but the function still interpolates the original data.
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.
10Example
This program reads in a set of $30$ data points and calls e01zmf to construct an interpolating function $Q\left(\mathbf{x}\right)$. It then calls e01znf 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 very much larger.