hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_interp_2d_scat_shep (e01sg)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_interp_2d_scat_shep (e01sg) generates a two-dimensional interpolant to a set of scattered data points, using a modified Shepard method.


[iq, rq, ifail] = e01sg(x, y, f, nw, nq, 'm', m)
[iq, rq, ifail] = nag_interp_2d_scat_shep(x, y, f, nw, nq, 'm', m)


nag_interp_2d_scat_shep (e01sg) constructs a smooth function Qx,y which interpolates a set of m scattered data points xr,yr,fr, for r=1,2,,m, using a modification of Shepard's method. The surface is continuous and has continuous first partial derivatives.
The basic Shepard (1968) method interpolates the input data with the weighted mean
Qx,y=r=1mwrx,yqr r=1mwrx,y ,  
where qr = fr , wr x,y = 1dr2  and dr2 = x-xr 2 + y-yr 2 .
The basic method is global in that the interpolated value at any point depends on all the data, but this function uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each wrx,y to be zero outside a circle with centre xr,yr and some radius Rw. Also, to improve the performance of the basic method, each qr above is replaced by a function qrx,y, which is a quadratic fitted by weighted least squares to data local to xr,yr and forced to interpolate xr,yr,fr. In this context, a point x,y is defined to be local to another point if it lies within some distance Rq of it. Computation of these quadratics constitutes the main work done by this function.
The efficiency of the function is further enhanced by using a cell method for nearest neighbour searching due to Bentley and Friedman (1979).
The radii Rw and Rq are chosen to be just large enough to include Nw and Nq data points, respectively, for user-supplied constants Nw and Nq. Default values of these arguments are provided by the function, and advice on alternatives is given in Choice of and .
This function is derived from the function QSHEP2 described by Renka (1988b).
Values of the interpolant Qx,y generated by this function, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to nag_interp_2d_scat_shep_eval (e01sh).


Bentley J L and Friedman J H (1979) Data structures for range searching ACM Comput. Surv. 11 397–409
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. Software 14 139–148
Renka R J (1988b) Algorithm 660: QSHEP2D: Quadratic Shepard method for bivariate interpolation of scattered data ACM Trans. Math. Software 14 149–150
Shepard D (1968) A two-dimensional interpolation function for irregularly spaced data Proc. 23rd Nat. Conf. ACM 517–523 Brandon/Systems Press Inc., Princeton


Compulsory Input Parameters

1:     xm – double array
2:     ym – double array
The Cartesian coordinates of the data points xr,yr, for r=1,2,,m.
Constraint: these coordinates must be distinct, and must not all be collinear.
3:     fm – double array
fr must be set to the data value fr, for r=1,2,,m.
4:     nw int64int32nag_int scalar
The number Nw of data points that determines each radius of influence Rw, appearing in the definition of each of the weights wr, for r=1,2,,m (see Description). Note that Rw is different for each weight. If nw0 the default value nw=min19,m-1 is used instead.
Constraint: nwmin40,m-1.
5:     nq int64int32nag_int scalar
The number Nq of data points to be used in the least squares fit for coefficients defining the nodal functions qrx,y (see Description). If nq0 the default value nq=min13,m-1 is used instead.
Constraint: nq0 or 5nqmin40,m-1.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the dimension of the arrays x, y, f. (An error is raised if these dimensions are not equal.)
m, the number of data points.
Constraint: m6.

Output Parameters

1:     iqliq int64int32nag_int array
Integer data defining the interpolant Qx,y.
2:     rqlrq – double array
Real data defining the interpolant Qx,y.
3:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
On entry,m<6,
or nq > min40,m-1 ,
or nw > min40,m-1 ,
On entry,xi,yi=xj,yj for some ij.
On entry,all the data points are collinear. No unique solution exists.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


On successful exit, the function generated interpolates the input data exactly and has quadratic accuracy.

Further Comments


The time taken for a call to nag_interp_2d_scat_shep (e01sg) will depend in general on the distribution of the data points. If x and y are uniformly randomly distributed, then the time taken should be Om. At worst Om2 time will be required.

Choice of Nw and Nq

Default values of the arguments Nw and Nq may be selected by calling nag_interp_2d_scat_shep (e01sg) with nw0 and nq0. These default values may well be satisfactory for many applications.
If non-default values are required they must be supplied to nag_interp_2d_scat_shep (e01sg) through positive values of nw and nq. Increasing these arguments makes the method less local. This may increase the accuracy of the resulting interpolant at the expense of increased computational cost. The default values nw = min19,m-1  and nq = min13,m-1  have been chosen on the basis of experimental results reported in Renka (1988a). In these experiments the error norm was found to vary smoothly with Nw and Nq, generally increasing monotonically and slowly with distance from the optimal pair. The method is not therefore thought to be particularly sensitive to the argument values. For further advice on the choice of these arguments see Renka (1988a).


This program reads in a set of 30 data points and calls nag_interp_2d_scat_shep (e01sg) to construct an interpolating function Qx,y. It then calls nag_interp_2d_scat_shep_eval (e01sh) 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 larger.
function e01sg_example

fprintf('e01sg example results\n\n');

% Scattered Grid Data
x = [11.16; 12.85; 19.85; 19.72; 15.91;  0.00; 20.87;  3.45; 14.26; ...
     17.43; 22.80;  7.58; 25.00;  0.00;  9.66;  5.22; 17.25; 25.00; ...
     12.13; 22.23; 11.52; 15.20;  7.54; 17.32;  2.14;  0.51; 22.69; ...
      5.47; 21.67;  3.31];
y = [ 1.24;  3.06; 10.72;  1.39;  7.74; 20.00; 20.00; 12.78; 17.87; ...
      3.46; 12.39;  1.98; 11.87;  0.00; 20.00; 14.66; 19.57;  3.87; ...
     10.79;  6.21;  8.53;  0.00; 10.69; 13.78; 15.03;  8.37; 19.63; ...
     17.13; 14.36; 0.33];
f = [22.15; 22.11;  7.97; 16.83; 15.30; 34.60;  5.74; 41.24; 10.74; ...
     18.60;  5.47; 29.87;  4.40; 58.20;  4.73; 40.36;  6.43;  8.74; ...
     13.71; 10.25; 15.74; 21.60; 19.31; 12.11; 53.10; 49.43;  3.25; ...
     28.63;  5.52; 44.08];

% Generate interpolant
nw = int64(0);
nq = int64(0);
[iq, rq, ifail] = e01sg(x, y, f, nw, nq);

% Interpolation points
u = [20.00;  6.41;  7.54;  9.91; 12.30];
v = [ 3.14; 15.44; 10.69; 18.27;  9.22];

% Interpolate at interpolation points 
[q, qx, qy, ifail] = e01sh(x, y, f, iq, rq, u, v);

fprintf('Interpolated values Q and its derivatives at (u,v)\n'); 
fprintf('     u      v      q      qx     qy\n');
for i = 1:size(u,1)
  fprintf('%7.2f%7.2f%7.2f%7.2f%7.2f\n', u(i), v(i), q(i), qx(i), qy(i));

e01sg example results

Interpolated values Q and its derivatives at (u,v)
     u      v      q      qx     qy
  20.00   3.14  15.89  -1.28  -0.63
   6.41  15.44  34.05  -3.62  -3.56
   7.54  10.69  19.31  -2.84   0.81
   9.91  18.27  13.68  -1.59  -4.71
  12.30   9.22  14.56  -0.68  -0.78

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015