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_4d_scat_shep (e01tk)

Purpose

nag_interp_4d_scat_shep (e01tk) generates a four-dimensional interpolant to a set of scattered data points, using a modified Shepard method.

Syntax

[iq, rq, ifail] = e01tk(x, f, nw, nq, 'm', m)
[iq, rq, ifail] = nag_interp_4d_scat_shep(x, f, nw, nq, 'm', m)

Description

nag_interp_4d_scat_shep (e01tk) constructs a smooth function Q (x) Q (x) , x4x4 which interpolates a set of mm scattered data points (xr,fr) (xr,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 method, which is a generalization of the two-dimensional method described in Shepard (1968), interpolates the input data with the weighted mean
Q (x) = ( r = 1m wr (x) qr )/( r = 1m wr (x) ) ,
Q (x) = r=1 m wr (x) qr r=1 m wr (x) ,
where qr = fr qr = fr , wr (x) = 1/(dr2) wr (x) = 1dr2  and dr2 = xxr22 dr2 = x-xr2 2 .
The basic method is global in that the interpolated value at any point depends on all the data, but nag_interp_4d_scat_shep (e01tk) uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each wr (x) wr (x)  to be zero outside a hypersphere with centre xr xr  and some radius RwRw. Also, to improve the performance of the basic method, each qrqr above is replaced by a function qr (x) qr (x) , which is a quadratic fitted by weighted least squares to data local to xr xr  and forced to interpolate (xr,fr) (xr,fr) . In this context, a point x x  is defined to be local to another point if it lies within some distance RqRq of it.
The efficiency of nag_interp_4d_scat_shep (e01tk) is enhanced by using a cell method for nearest neighbour searching due to Bentley and Friedman (1979) with a cell density of 33.
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 ].
nag_interp_4d_scat_shep (e01tk) 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 (x) Q (x)  generated by nag_interp_4d_scat_shep (e01tk), and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to nag_interp_4d_scat_shep_eval (e01tl).

References

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. Software 25 353–366
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. Software 14 139–148
Renka R J (1988b) Algorithm 661: QSHEP3D: Quadratic Shepard method for trivariate interpolation of scattered data ACM Trans. Math. Software 14 151–152
Shepard D (1968) A two-dimensional interpolation function for irregularly spaced data Proc. 23rd Nat. Conf. ACM 517–523 Brandon/Systems Press Inc., Princeton

Parameters

Compulsory Input Parameters

1:     x(44,m) – double array
x(1 : 4,r)x1:4r must be set to the Cartesian coordinates of the data point xrxr, for r = 1,2,,mr=1,2,,m.
Constraint: these coordinates must be distinct, and must not all lie on the same three-dimensional hypersurface.
2:     f(m) – double array
m, the dimension of the array, must satisfy the constraint m16m16.
f(r)fr must be set to the data value frfr, for r = 1,2,,mr=1,2,,m.
3:     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 (32,m1)nw=min(32,m-1) is used instead.
Constraint: nwmin (50,m1)nwmin(50,m-1).
4:     nq – int64int32nag_int scalar
The number NqNq of data points to be used in the least squares fit for coefficients defining the quadratic functions qr (x) qr (x)  (see Section [Description]). If nq0nq0 the default value nq = min (38,m1)nq=min(38,m-1) is used instead.
Constraint: nq0nq0 or 14nqmin (50,m1)14nqmin(50,m-1).

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the array f and the second dimension of the array x. (An error is raised if these dimensions are not equal.)
mm, the number of data points.
Constraint: m16m16.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     iq(2 × m + 12×m+1) – int64int32nag_int array
Integer data defining the interpolant Q (x) Q (x) .
2:     rq(15 × m + 915×m+9) – double array
Real data defining the interpolant Q (x) Q (x) .
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
Constraint: m16m16.
Constraint: nq0nq0 or nq14nq14.
Constraint: nqmin (50,m1)nqmin(50,m-1).
Constraint: nwmin (50,m1)nwmin(50,m-1).
  ifail = 2ifail=2
There are duplicate nodes in the dataset.
  ifail = 3ifail=3
On entry, all the data points lie on the same three-dimensional hypersurface. No unique solution exists.

Accuracy

On successful exit, the function generated interpolates the input data exactly and has quadratic precision. Overall accuracy of the interpolant is affected by the choice of parameters nw and nq as well as the smoothness of the function represented by the input data.

Further Comments

Timing

The time taken for a call to nag_interp_4d_scat_shep (e01tk) will depend in general on the distribution of the data points and on the choice of NwNw and NqNq parameters. If the data points 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_4d_scat_shep (e01tk) 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_4d_scat_shep (e01tk) through positive values of nw and nq. Increasing these parameter values makes the method less local. This may increase the accuracy of the resulting interpolant at the expense of increased computational cost.

Example

function nag_interp_4d_scat_shep_example
x = [0.81, 0.91, 0.13, 0.91, 0.63, 0.10, 0.28, 0.55, 0.96, 0.96, ...
     0.16, 0.97, 0.96, 0.49, 0.80, 0.14, 0.42, 0.92, 0.79, 0.96, ...
     0.66, 0.04, 0.85, 0.93, 0.68, 0.76, 0.74, 0.39, 0.66, 0.17;
     0.15, 0.96, 0.88, 0.49, 0.41, 0.13, 0.93, 0.01, 0.19, 0.32, ...
     0.05, 0.14, 0.73, 0.48, 0.34, 0.24, 0.45, 0.19, 0.32, 0.26, ...
     0.83, 0.70, 0.33, 0.58, 0.29, 0.26, 0.26, 0.68, 0.52, 0.08;
     0.44, 0.00, 0.22, 0.39, 0.72, 0.77, 0.24, 0.04, 0.95, 0.53, ...
     0.16, 0.36, 0.28, 0.58, 0.64, 0.12, 0.03, 0.48, 0.15, 0.93, ...
     0.41, 0.40, 0.15, 0.88, 0.88, 0.09, 0.33, 0.69, 0.17, 0.35;
     0.83, 0.09, 0.21, 0.79, 0.68, 0.47, 0.90, 0.41, 0.66, 0.96, ...
     0.30, 0.72, 0.75, 0.19, 0.57, 0.06, 0.68, 0.67, 0.13, 0.89, ...
     0.17, 0.54, 0.03, 0.81, 0.60, 0.41, 0.64, 0.37, 1.00, 0.71];

f = [6.3900;
    2.5000;
    9.3400;
    7.5200;
    6.9100;
    4.6800;
   45.4000;
    5.4800;
    2.7500;
    7.4300;
    6.0500;
    5.7700;
    8.6800;
    2.3800;
    3.7000;
    1.3400;
   15.1800;
    4.3500;
    1.5000;
    3.4300;
    3.1000;
   14.3300;
    0.3500;
    4.3000;
    3.7700;
    4.1600;
    6.7500;
    5.2200;
   16.2300,
   10.6200];

xe = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9;
      0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9;
      0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9;
      0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9];

% Generate the interpolant
nq = int64(0);
nw = int64(0);
[iq, rq, ifail] = nag_interp_4d_scat_shep(x, f, nw, nq);

% Evaluate the interpolant using nag_interp_4d_scat_shep_eval
[q, qx, ifail] = nag_interp_4d_scat_shep_eval(x, f, iq, rq, xe);

fprintf('\n   |  Interpolated Evaluation Points         |  Values\n');
fprintf('---|-----------------------------------------+--------\n');
fprintf('i  |  xe(i,1)   xe(i,2)   xe(i,3)   xe(i,4)  | q(i)\n');
fprintf('---|-----------------------------------------+--------\n');
for i=1:9
  fprintf(' %d |%8.4f  %8.4f  %8.4f  %8.4f  %8.4f \n', i, xe(:, i), q(i));
end
 

   |  Interpolated Evaluation Points         |  Values
---|-----------------------------------------+--------
i  |  xe(i,1)   xe(i,2)   xe(i,3)   xe(i,4)  | q(i)
---|-----------------------------------------+--------
 1 |  0.1000    0.1000    0.1000    0.1000    2.7195 
 2 |  0.2000    0.2000    0.2000    0.2000    4.3110 
 3 |  0.3000    0.3000    0.3000    0.3000    5.5380 
 4 |  0.4000    0.4000    0.4000    0.4000    6.5540 
 5 |  0.5000    0.5000    0.5000    0.5000    7.5910 
 6 |  0.6000    0.6000    0.6000    0.6000    8.7447 
 7 |  0.7000    0.7000    0.7000    0.7000   10.0457 
 8 |  0.8000    0.8000    0.8000    0.8000   11.5797 
 9 |  0.9000    0.9000    0.9000    0.9000   13.1997 

function e01tk_example
x = [0.81, 0.91, 0.13, 0.91, 0.63, 0.10, 0.28, 0.55, 0.96, 0.96, ...
     0.16, 0.97, 0.96, 0.49, 0.80, 0.14, 0.42, 0.92, 0.79, 0.96, ...
     0.66, 0.04, 0.85, 0.93, 0.68, 0.76, 0.74, 0.39, 0.66, 0.17;
     0.15, 0.96, 0.88, 0.49, 0.41, 0.13, 0.93, 0.01, 0.19, 0.32, ...
     0.05, 0.14, 0.73, 0.48, 0.34, 0.24, 0.45, 0.19, 0.32, 0.26, ...
     0.83, 0.70, 0.33, 0.58, 0.29, 0.26, 0.26, 0.68, 0.52, 0.08;
     0.44, 0.00, 0.22, 0.39, 0.72, 0.77, 0.24, 0.04, 0.95, 0.53, ...
     0.16, 0.36, 0.28, 0.58, 0.64, 0.12, 0.03, 0.48, 0.15, 0.93, ...
     0.41, 0.40, 0.15, 0.88, 0.88, 0.09, 0.33, 0.69, 0.17, 0.35;
     0.83, 0.09, 0.21, 0.79, 0.68, 0.47, 0.90, 0.41, 0.66, 0.96, ...
     0.30, 0.72, 0.75, 0.19, 0.57, 0.06, 0.68, 0.67, 0.13, 0.89, ...
     0.17, 0.54, 0.03, 0.81, 0.60, 0.41, 0.64, 0.37, 1.00, 0.71];

f = [6.3900;
    2.5000;
    9.3400;
    7.5200;
    6.9100;
    4.6800;
   45.4000;
    5.4800;
    2.7500;
    7.4300;
    6.0500;
    5.7700;
    8.6800;
    2.3800;
    3.7000;
    1.3400;
   15.1800;
    4.3500;
    1.5000;
    3.4300;
    3.1000;
   14.3300;
    0.3500;
    4.3000;
    3.7700;
    4.1600;
    6.7500;
    5.2200;
   16.2300,
   10.6200];

xe = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9;
      0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9;
      0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9;
      0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9];

% Generate the interpolant
nq = int64(0);
nw = int64(0);
[iq, rq, ifail] = e01tk(x, f, nw, nq);

% Evaluate the interpolant using e01tl
[q, qx, ifail] = e01tl(x, f, iq, rq, xe);

fprintf('\n   |  Interpolated Evaluation Points         |  Values\n');
fprintf('---|-----------------------------------------+--------\n');
fprintf('i  |  xe(i,1)   xe(i,2)   xe(i,3)   xe(i,4)  | q(i)\n');
fprintf('---|-----------------------------------------+--------\n');
for i=1:9
  fprintf(' %d |%8.4f  %8.4f  %8.4f  %8.4f  %8.4f \n', i, xe(:, i), q(i));
end
 

   |  Interpolated Evaluation Points         |  Values
---|-----------------------------------------+--------
i  |  xe(i,1)   xe(i,2)   xe(i,3)   xe(i,4)  | q(i)
---|-----------------------------------------+--------
 1 |  0.1000    0.1000    0.1000    0.1000    2.7195 
 2 |  0.2000    0.2000    0.2000    0.2000    4.3110 
 3 |  0.3000    0.3000    0.3000    0.3000    5.5380 
 4 |  0.4000    0.4000    0.4000    0.4000    6.5540 
 5 |  0.5000    0.5000    0.5000    0.5000    7.5910 
 6 |  0.6000    0.6000    0.6000    0.6000    8.7447 
 7 |  0.7000    0.7000    0.7000    0.7000   10.0457 
 8 |  0.8000    0.8000    0.8000    0.8000   11.5797 
 9 |  0.9000    0.9000    0.9000    0.9000   13.1997 


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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013