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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_interp_2d_scat_shep (e01sg)

Purpose

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

Syntax

[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)

Description

nag_interp_2d_scat_shep (e01sg) constructs a smooth function Q(x,y)Q(x,y) which interpolates a set of mm scattered data points (xr,yr,fr)(xr,yr,fr), for r = 1,2,,mr=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
Q(x,y) = (r = 1mwr(x,y)qr)/(r = 1mwr(x,y)),
Q(x,y)=r=1mwr(x,y)qr r=1mwr(x,y) ,
where qr = fr qr = fr , wr (x,y) = 1/(dr2) wr (x,y) = 1dr2  and dr2 = (xxr)2 + (yyr)2 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 wr(x,y)wr(x,y) to be zero outside a circle with centre (xr,yr)(xr,yr) and some radius RwRw. Also, to improve the performance of the basic method, each qrqr above is replaced by a function qr(x,y)qr(x,y), which is a quadratic fitted by weighted least squares to data local to (xr,yr)(xr,yr) and forced to interpolate (xr,yr,fr)(xr,yr,fr). In this context, a point (x,y)(x,y) is defined to be local to another point if it lies within some distance RqRq 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 RwRw and RqRq are chosen to be just large enough to include NwNw and NqNq data points, respectively, for user-supplied constants NwNw and NqNq. Default values of these parameters are provided by the function, and advice on alternatives is given in Section [Choice of and ].
This function is derived from the function QSHEP2 described by Renka (1988b).
Values of the interpolant Q(x,y)Q(x,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).

References

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

Parameters

Compulsory Input Parameters

1:     x(m) – double array
2:     y(m) – double array
m, the dimension of the array, must satisfy the constraint m6m6.
The Cartesian coordinates of the data points (xr,yr)(xr,yr), for r = 1,2,,mr=1,2,,m.
Constraint: these coordinates must be distinct, and must not all be collinear.
3:     f(m) – double array
m, the dimension of the array, must satisfy the constraint m6m6.
f(r)fr must be set to the data value frfr, for r = 1,2,,mr=1,2,,m.
4:     nw – int64int32nag_int scalar
The number NwNw of data points that determines each radius of influence RwRw, appearing in the definition of each of the weights wrwr, for r = 1,2,,mr=1,2,,m (see Section [Description]). Note that RwRw is different for each weight. If nw0nw0 the default value nw = min (19,m1)nw=min(19,m-1) is used instead.
Constraint: nwmin (40,m1)nwmin(40,m-1).
5:     nq – int64int32nag_int scalar
The number NqNq of data points to be used in the least squares fit for coefficients defining the nodal functions qr(x,y)qr(x,y) (see Section [Description]). If nq0nq0 the default value nq = min (13,m1)nq=min(13,m-1) is used instead.
Constraint: nq0nq0 or 5nqmin (40,m1)5nqmin(40,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.)
mm, the number of data points.
Constraint: m6m6.

Input Parameters Omitted from the MATLAB Interface

liq lrq

Output Parameters

1:     iq(liq) – int64int32nag_int array
liq2 × m + 1liq2×m+1.
Integer data defining the interpolant Q(x,y)Q(x,y).
2:     rq(lrq) – double array
lrq6 × m + 5lrq6×m+5.
Real data defining the interpolant Q(x,y)Q(x,y).
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry,m < 6m<6,
or0 < nq < 50<nq<5,
or nq > min (40,m1) nq > min(40,m-1) ,
or nw > min (40,m1) nw > min(40,m-1) ,
orliq < 2 × m + 1liq<2×m+1,
orlrq < 6 × m + 5lrq<6×m+5.
  ifail = 2ifail=2
On entry,(x(i),y(i)) = (x(j),y(j))(xi,yi)=(xj,yj) for some ijij.
  ifail = 3ifail=3
On entry,all the data points are collinear. No unique solution exists.

Accuracy

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

Further Comments

Timing

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 O(m)O(m). At worst O(m2)O(m2) time will be required.

Choice of Nw and Nq

Default values of the parameters NwNw and NqNq may be selected by calling nag_interp_2d_scat_shep (e01sg) with nw0nw0 and nq0nq0. 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 parameters 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 = min (19,m1) nw = min(19,m-1)  and nq = min (13,m1) nq = min(13,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 NwNw and NqNq, generally increasing monotonically and slowly with distance from the optimal pair. The method is not therefore thought to be particularly sensitive to the parameter values. For further advice on the choice of these parameters see Renka (1988a).

Example

function nag_interp_2d_scat_shep_example
x = [11.16;
     12.85;
     19.85;
     19.72;
     15.91;
     0;
     20.87;
     3.45;
     14.26;
     17.43;
     22.8;
     7.58;
     25;
     0;
     9.66;
     5.22;
     17.25;
     25;
     12.13;
     22.23;
     11.52;
     15.2;
     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;
     20;
     12.78;
     17.87;
     3.46;
     12.39;
     1.98;
     11.87;
     0;
     20;
     14.66;
     19.57;
     3.87;
     10.79;
     6.21;
     8.53;
     0;
     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;
     15.3;
     34.6;
     5.74;
     41.24;
     10.74;
     18.6;
     5.47;
     29.87;
     4.4;
     58.2;
     4.73;
     40.36;
     6.43;
     8.74;
     13.71;
     10.25;
     15.74;
     21.6;
     19.31;
     12.11;
     53.1;
     49.43;
     3.25;
     28.63;
     5.52;
     44.08];
nw = int64(0);
nq = int64(0);
[iq, rq, ifail] = nag_interp_2d_scat_shep(x, y, f, nw, nq)
 

iq =

                    3
                   12
                    1
                    4
                    8
                    5
                    3
                    6
                    9
                    7
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    2
                   22
                   11
                   10
                   19
                   16
                   17
                   23
                   15
                   18
                   13
                   14
                   13
                   30
                   15
                   25
                   24
                   20
                   21
                   20
                   21
                   22
                   26
                   27
                   28
                   26
                   29
                   28
                   29
                   30


rq =

         0
         0
    8.3333
    6.6667
   22.5597
  271.5245
  205.4824
  189.9545
  341.5185
  174.9092
  480.1409
  351.2840
  282.3028
  217.6970
  274.5241
  259.8121
  297.1145
  306.2440
  508.9409
  301.4125
  270.6925
  291.9401
  351.3640
  125.8490
  235.8665
  135.0205
  301.3909
  172.9517
  159.2721
  367.6490
  379.5581
  371.3905
  286.9056
  270.1129
  397.8650
    0.1878
   -0.1651
   -0.2132
   -1.1301
    1.0468
    0.0625
   -0.0885
   -0.1691
   -0.8079
   -0.1882
    0.0832
   -0.0586
    0.1087
   -1.0392
   -0.3381
   -0.0486
    0.0172
   -0.0871
   -1.2037
   -0.1929
   -0.0898
    0.0049
    0.0372
   -0.6652
   -0.8723
    0.2572
    0.1824
   -0.7009
   -5.7655
   -9.6106
   -0.1071
    0.0604
    0.0720
   -0.9609
    0.1964
    0.4051
    0.1409
   -0.3944
   -5.0685
    1.9018
    0.0383
    0.1960
   -0.3135
   -0.3698
   -2.2983
   -0.0608
    0.0197
   -0.0858
   -0.9400
   -0.5989
    0.0549
   -0.0392
    0.0193
   -0.6920
   -0.4476
    0.1788
    0.0060
   -0.1424
   -2.6019
    0.0566
    0.0575
    0.0023
    0.0253
   -0.4455
   -0.4517
    0.1591
   -0.0533
    0.1236
   -4.5812
   -1.7523
    0.1713
    0.1845
   -0.5129
   -1.4284
   -6.1995
    0.3180
    0.0419
   -1.0292
   -3.0937
   -0.8993
   -0.0205
    0.1689
   -0.1461
   -0.1984
   -1.8325
   -0.0044
    0.0448
    0.0002
   -1.1667
   -0.5838
    0.1173
   -0.1318
    0.1124
   -0.9391
   -0.3562
    0.0017
    0.0532
    0.0019
   -1.0258
   -0.7284
    0.1174
   -0.0572
    0.0674
   -0.8996
   -0.7621
   -0.0128
   -0.0210
   -0.0592
   -0.6935
   -0.2959
    0.3389
   -0.3648
    0.0668
   -2.8368
    0.8061
   -0.0989
    0.0636
   -0.0704
   -0.8872
   -0.3389
    0.5183
    0.2246
   -0.8528
   -6.8360
   -1.9763
   -0.0216
   -0.0341
    0.0905
   -3.4386
    0.1377
   -0.0919
    0.0075
    0.0564
   -1.4055
    0.0492
    0.2750
    0.1535
   -0.6945
   -3.4433
   -5.0134
    0.0801
    0.0351
    0.0076
   -0.7771
   -0.3231
    0.1475
   -0.0108
   -0.0167
   -3.7187
   -0.6231


ifail =

                    0


function e01sg_example
x = [11.16;
     12.85;
     19.85;
     19.72;
     15.91;
     0;
     20.87;
     3.45;
     14.26;
     17.43;
     22.8;
     7.58;
     25;
     0;
     9.66;
     5.22;
     17.25;
     25;
     12.13;
     22.23;
     11.52;
     15.2;
     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;
     20;
     12.78;
     17.87;
     3.46;
     12.39;
     1.98;
     11.87;
     0;
     20;
     14.66;
     19.57;
     3.87;
     10.79;
     6.21;
     8.53;
     0;
     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;
     15.3;
     34.6;
     5.74;
     41.24;
     10.74;
     18.6;
     5.47;
     29.87;
     4.4;
     58.2;
     4.73;
     40.36;
     6.43;
     8.74;
     13.71;
     10.25;
     15.74;
     21.6;
     19.31;
     12.11;
     53.1;
     49.43;
     3.25;
     28.63;
     5.52;
     44.08];
nw = int64(0);
nq = int64(0);
[iq, rq, ifail] = e01sg(x, y, f, nw, nq)
 

iq =

                    3
                   12
                    1
                    4
                    8
                    5
                    3
                    6
                    9
                    7
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    2
                   22
                   11
                   10
                   19
                   16
                   17
                   23
                   15
                   18
                   13
                   14
                   13
                   30
                   15
                   25
                   24
                   20
                   21
                   20
                   21
                   22
                   26
                   27
                   28
                   26
                   29
                   28
                   29
                   30


rq =

         0
         0
    8.3333
    6.6667
   22.5597
  271.5245
  205.4824
  189.9545
  341.5185
  174.9092
  480.1409
  351.2840
  282.3028
  217.6970
  274.5241
  259.8121
  297.1145
  306.2440
  508.9409
  301.4125
  270.6925
  291.9401
  351.3640
  125.8490
  235.8665
  135.0205
  301.3909
  172.9517
  159.2721
  367.6490
  379.5581
  371.3905
  286.9056
  270.1129
  397.8650
    0.1878
   -0.1651
   -0.2132
   -1.1301
    1.0468
    0.0625
   -0.0885
   -0.1691
   -0.8079
   -0.1882
    0.0832
   -0.0586
    0.1087
   -1.0392
   -0.3381
   -0.0486
    0.0172
   -0.0871
   -1.2037
   -0.1929
   -0.0898
    0.0049
    0.0372
   -0.6652
   -0.8723
    0.2572
    0.1824
   -0.7009
   -5.7655
   -9.6106
   -0.1071
    0.0604
    0.0720
   -0.9609
    0.1964
    0.4051
    0.1409
   -0.3944
   -5.0685
    1.9018
    0.0383
    0.1960
   -0.3135
   -0.3698
   -2.2983
   -0.0608
    0.0197
   -0.0858
   -0.9400
   -0.5989
    0.0549
   -0.0392
    0.0193
   -0.6920
   -0.4476
    0.1788
    0.0060
   -0.1424
   -2.6019
    0.0566
    0.0575
    0.0023
    0.0253
   -0.4455
   -0.4517
    0.1591
   -0.0533
    0.1236
   -4.5812
   -1.7523
    0.1713
    0.1845
   -0.5129
   -1.4284
   -6.1995
    0.3180
    0.0419
   -1.0292
   -3.0937
   -0.8993
   -0.0205
    0.1689
   -0.1461
   -0.1984
   -1.8325
   -0.0044
    0.0448
    0.0002
   -1.1667
   -0.5838
    0.1173
   -0.1318
    0.1124
   -0.9391
   -0.3562
    0.0017
    0.0532
    0.0019
   -1.0258
   -0.7284
    0.1174
   -0.0572
    0.0674
   -0.8996
   -0.7621
   -0.0128
   -0.0210
   -0.0592
   -0.6935
   -0.2959
    0.3389
   -0.3648
    0.0668
   -2.8368
    0.8061
   -0.0989
    0.0636
   -0.0704
   -0.8872
   -0.3389
    0.5183
    0.2246
   -0.8528
   -6.8360
   -1.9763
   -0.0216
   -0.0341
    0.0905
   -3.4386
    0.1377
   -0.0919
    0.0075
    0.0564
   -1.4055
    0.0492
    0.2750
    0.1535
   -0.6945
   -3.4433
   -5.0134
    0.0801
    0.0351
    0.0076
   -0.7771
   -0.3231
    0.1475
   -0.0108
   -0.0167
   -3.7187
   -0.6231


ifail =

                    0



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Chapter Introduction
NAG Toolbox

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