e01zmc 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 functions in Chapter E01 which are specific to the given dimensionality. e01sgc generates the two-dimensional interpolant, while e01tgc,e01tkcande01tmc generate the three-, four- and five-dimensional interpolants respectively.
The function may be called by the names: e01zmc, nag_interp_dimn_scat_shep or nag_nd_shep_interp.
e01zmc constructs a smooth function , which interpolates a set of scattered data points , for , 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 , .
The basic method is global in that the interpolated value at any point depends on all the data, but e01zmc uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each to be zero outside a hypersphere with centre and some radius . Also, to improve the performance of the basic method, each above is replaced by a function , which is a quadratic fitted by weighted least squares to data local to and forced to interpolate . In this context, a point is defined to be local to another point if it lies within some distance of it.
The efficiency of e01zmc is enhanced by using a cell method for nearest neighbour searching due to Bentley and Friedman (1979) with a cell density of .
The radii and are chosen to be just large enough to include and data points, respectively, for user-supplied constants and . Default values of these parameters are provided, and advice on alternatives is given in Section 9.2.
e01zmc 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 generated by e01zmc, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to e01znc.
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
On entry: the d components of the first data point must be stored in elements of x. The second data point must be stored in elements of x, and so on. In general, the m data points must be stored in
, for and .
these coordinates must be distinct, and must not all lie on the same -dimensional hypersurface.
4: – const doubleInput
On entry: must be set to the data value
, for .
5: – IntegerInput
On entry: the number of data points that determines each radius of influence , appearing in the definition of each of the weights
, for (see Section 3). Note that is different for each weight. If the default value is used instead.
6: – IntegerInput
On entry: the number of data points to be used in the least squares fit for coefficients defining the quadratic functions (see Section 3). If the default value is used instead.
7: – IntegerOutput
On exit: integer data defining the interpolant .
8: – doubleOutput
Note: the dimension, dim, of the array rq
must be at least
On exit: real data defining the interpolant .
9: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, all the data points lie on the same hypersurface. No unique solution exists.
There are duplicate nodes in the dataset. , for , and . The interpolant cannot be derived.
On entry, .
On entry, exceeds the largest machine integer.
On entry, and .
On entry, and .
Constraint: or .
On entry, and .
On entry, and .
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.
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.
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 at best. In other cases in these experiments, the function generated interpolates the input data with maximum absolute error of the order of .
8Parallelism and Performance
e01zmc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e01zmc 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 e01zmc will depend in general on the distribution of the data points and on the choice of and parameters. If the data points are uniformly randomly distributed, then the time taken should be . At worst time will be required.
9.2Choice of and
Default values of the parameters and may be selected by calling e01zmc with and . These default values may well be satisfactory for many applications.
If non-default values are required they must be supplied to e01zmc 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 and 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).
This program reads in a set of data points and calls e01zmc to construct an interpolating function . It then calls e01znc 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.