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NAG Toolbox: nag_interp_4d_scat_shep_eval (e01tl)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_interp_4d_scat_shep_eval (e01tl) evaluates the four-dimensional interpolating function generated by nag_interp_4d_scat_shep (e01tk) and its first partial derivatives.

Syntax

[q, qx, ifail] = e01tl(x, f, iq, rq, xe, 'm', m, 'n', n)
[q, qx, ifail] = nag_interp_4d_scat_shep_eval(x, f, iq, rq, xe, 'm', m, 'n', n)

Description

nag_interp_4d_scat_shep_eval (e01tl) takes as input the interpolant Q x , x4 of a set of scattered data points xr,fr , for r=1,2,,m, as computed by nag_interp_4d_scat_shep (e01tk), and evaluates the interpolant and its first partial derivatives at the set of points xi , for i=1,2,,n.
nag_interp_4d_scat_shep_eval (e01tl) must only be called after a call to nag_interp_4d_scat_shep (e01tk).
nag_interp_4d_scat_shep_eval (e01tl) is derived from the new implementation of QS3GRD described by Renka (1988). It uses the modification for high-dimensional interpolation described by Berry and Minser (1999).

References

Berry M W, Minser K S (1999) Algorithm 798: high-dimensional interpolation using the modified Shepard method ACM Trans. Math. Software 25 353–366
Renka R J (1988) Algorithm 661: QSHEP3D: Quadratic Shepard method for trivariate interpolation of scattered data ACM Trans. Math. Software 14 151–152

Parameters

Compulsory Input Parameters

1:     x4m – double array
Note: the coordinates of xr are stored in x1r x4r .
must be the same array supplied as argument x in the preceding call to nag_interp_4d_scat_shep (e01tk). It must remain unchanged between calls.
2:     fm – double array
must be the same array supplied as argument f in the preceding call to nag_interp_4d_scat_shep (e01tk). It must remain unchanged between calls.
3:     iq2×m+1 int64int32nag_int array
must be the same array returned as argument iq in the preceding call to nag_interp_4d_scat_shep (e01tk). It must remain unchanged between calls.
4:     rq15×m+9 – double array
must be the same array returned as argument rq in the preceding call to nag_interp_4d_scat_shep (e01tk). It must remain unchanged between calls.
5:     xe4n – double array
xe1:4i must be set to the evaluation point xi , for i=1,2,,n.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the dimension of the array f and the first dimension of the array x. (An error is raised if these dimensions are not equal.)
must be the same value supplied for argument m in the preceding call to nag_interp_4d_scat_shep (e01tk).
Constraint: m16.
2:     n int64int32nag_int scalar
Default: the second dimension of the array xe.
n, the number of evaluation points.
Constraint: n1.

Output Parameters

1:     qn – double array
qi contains the value of the interpolant, at xi, for i=1,2,,n. If any of these evaluation points lie outside the region of definition of the interpolant the corresponding entries in q are set to the largest machine representable number (see nag_machine_real_largest (x02al)), and nag_interp_4d_scat_shep_eval (e01tl) returns with ifail=3.
2:     qx4n – double array
qxji contains the value of the partial derivatives with respect to xj of the interpolant Q x at xi, for i=1,2,,n, and for each of the four partial derivatives j=1,2,3,4. If any of these evaluation points lie outside the region of definition of the interpolant, the corresponding entries in qx are set to the largest machine representable number (see nag_machine_real_largest (x02al)), and nag_interp_4d_scat_shep_eval (e01tl) returns with ifail=3.
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:
   ifail=1
Constraint: m16.
Constraint: n1.
   ifail=2
On entry, values in iq appear to be invalid. Check that iq has not been corrupted between calls to nag_interp_4d_scat_shep (e01tk) and nag_interp_4d_scat_shep_eval (e01tl).
On entry, values in rq appear to be invalid. Check that rq has not been corrupted between calls to nag_interp_4d_scat_shep (e01tk) and nag_interp_4d_scat_shep_eval (e01tl).
   ifail=3
On entry, at least one evaluation point lies outside the region of definition of the interpolant.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

Computational errors should be negligible in most practical situations.

Further Comments

The time taken for a call to nag_interp_4d_scat_shep_eval (e01tl) will depend in general on the distribution of the data points. If the data points are approximately uniformly distributed, then the time taken should be only On. At worst Omn time will be required.

Example

This program evaluates the function
f x = 1.25 + cos5.4x4 cos6x1 cos6x2 6 + 6 3 x3 - 1 2  
at a set of 30 randomly generated data points and calls nag_interp_4d_scat_shep (e01tk) to construct an interpolating function Q x . It then calls nag_interp_4d_scat_shep_eval (e01tl) to evaluate the interpolant at a set of random points.
To reduce the time taken by this example, the number of data points is limited to 30. Increasing this value improves the interpolation accuracy at the expense of more time.
See also Example in nag_interp_4d_scat_shep (e01tk).
function e01tl_example


fprintf('e01tl example results\n\n');

genid = int64(1);
subid = int64(1);
seed  = [int64(1762543)];
seed2 = [int64(43331)];


% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);

% Generate the data points x
[state, x, ifail] = g05sa(int64(120), state);

% Put x into the right shape for the e01 routines
x = reshape(x, 4, 30);

% Create function values
c1 = cos(6*x(1,:));
c2 = cos(6*x(2,:));
c3 = 6 + 6*(3*x(3,:)-1).^2;
c4 = 1.25 + cos(5.4*x(4,:));
f  = c4.*c1.*c2./c3;


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

% Generate repeatable evaluation points
[state, ifail] = g05kf(genid, subid, seed2);
[state, xe, ifail] = g05sa(int64(32), state);
xe = reshape(xe, 4, 8);

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

% Evaluate function at xe
c1  = cos(6*xe(1,:));
c2  = cos(6*xe(2,:));
c3  = 6 + 6*(3*xe(3,:)-1).^2;
c4  = 1.25 + cos(5.4*xe(4,:));
fun = c4.*c1.*c2./c3;

fprintf('\n i |  f(i)       q(i)   |f(i)-q(i)|\n');
fprintf('---|--------------------+---------+\n');
for i=1:8
  fprintf(' %d |%8.4f  %8.4f  %8.4f \n', i, fun(i), q(i), abs(fun(i)-q(i)));
end


e01tl example results


 i |  f(i)       q(i)   |f(i)-q(i)|
---|--------------------+---------+
 1 | -0.0189   -0.0394    0.0205 
 2 | -0.0186    0.0967    0.1153 
 3 |  0.1147    0.0606    0.0541 
 4 |  0.0096   -0.1313    0.1409 
 5 | -0.1354   -0.1878    0.0524 
 6 |  0.0022   -0.1595    0.1617 
 7 | -0.0095   -0.1179    0.1084 
 8 |  0.0113   -0.3950    0.4063 

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