NAG Toolbox: nag_interp_nd_scat_shep (e01zm)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{x}\left(\mathbf{d},\mathbf{m}\right)$ – double array

$\mathbf{x}\left(1:\mathbf{d},\mathit{r}\right)$ must be set to the Cartesian coordinates of the data point ${\mathbf{x}}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.

Constraint: these coordinates must be distinct, and must not all lie on the same $\left(d-1\right)$-dimensional hypersurface.

2:     $\mathrm{f}\left(\mathbf{m}\right)$ – double array

$\mathbf{f}\left(r\right)$ must be set to the data value $f_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.

Optional Input Parameters

1:     $\mathrm{d}$ – int64int32nag_int scalar

Default: the first dimension of the array x.

$d$, the number of dimensions.

Constraint: $\mathbf{d}\ge 2$.

2:     $\mathrm{m}$ – 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.)

$m$, the number of data points.

Note: on the basis of experimental results reported in Berry and Minser (1999), when $\mathbf{d}\ge 5$ it is recommended to use $\mathbf{m}\ge 4000$.

Constraint: $\mathbf{m}\ge \left(\mathbf{d}+1\right)\times \left(\mathbf{d}+2\right)/2+2$.

3:     $\mathrm{nw}$ – int64int32nag_int scalar

Default: $\mathbf{nw}=-1$

The number $N_w$ of data points that determines each radius of influence $R_w$, appearing in the definition of each of the weights $w_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$ (see Description). Note that $R_w$ is different for each weight. If $\mathbf{nw}\le 0$ the default value $\mathbf{nw}=\min \left(2\times \left(\mathbf{d}+1\right)\times \left(\mathbf{d}+2\right),\mathbf{m}-1\right)$ is used instead.

Constraint: $\mathbf{nw}\le \mathbf{m}-1$.

4:     $\mathrm{nq}$ – int64int32nag_int scalar

Default: $\mathbf{nq}=-1$

The number $N_q$ of data points to be used in the least squares fit for coefficients defining the quadratic functions $q_r\left(\mathbf{x}\right)$ (see Description). If $\mathbf{nq}\le 0$ the default value $\mathbf{nq}=\min \left(\left(\mathbf{d}+1\right)\times \left(\mathbf{d}+2\right)\times 6/5,\mathbf{m}-1\right)$ is used instead.

Constraint: $\mathbf{nq}\le 0$ or $\left(\mathbf{d}+1\right)\times \left(\mathbf{d}+2\right)/2-1\le \mathbf{nq}\le \mathbf{m}-1$.

Output Parameters

1:     $\mathrm{iq}\left(2\times \mathbf{m}+1\right)$ – int64int32nag_int array

Integer data defining the interpolant $Q\left(\mathbf{x}\right)$.

2:     $\mathrm{rq}\left(*\right)$ – double array

The dimension of the array rq will be $\left(\left(\mathbf{d}+1\right)\times \left(\mathbf{d}+2\right)/2\right)\times \mathbf{m}+2\times \mathbf{d}+1$

Real data defining the interpolant $Q\left(\mathbf{x}\right)$.

3:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

Constraint: $\mathbf{d}\ge 2$.

Constraint: $\mathbf{m}\ge \left(\mathbf{d}+1\right)\times \left(\mathbf{d}+2\right)/2+2$.

Constraint: $\mathbf{nq}\le 0$ or $\mathbf{nq}\ge \left(\mathbf{d}+1\right)\times \left(\mathbf{d}+2\right)/2-1$.

Constraint: $\mathbf{nq}\le \mathbf{m}-1$.

Constraint: $\mathbf{nw}\le \mathbf{m}-1$.

On entry, $\left(\left(\mathbf{d}+1\right)\times \left(\mathbf{d}+2\right)/2\right)\times \mathbf{m}+2\times \mathbf{d}+1$ exceeds the largest machine integer.

   $\mathbf{ifail}=2$

There are duplicate nodes in the dataset.

   $\mathbf{ifail}=3$

On entry, all the data points lie on the same hypersurface. No unique solution exists.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Timing

Choice of Nw and Nq

Example

Open in the MATLAB editor: e01zm_example

function e01zm_example


fprintf('e01zm example results\n\n');

% Data: 30 6-dimensional points and function values
nd = 6;
npts = 30;
x = zeros(npts,nd);
x = [0.81, 0.15, 0.44, 0.83, 0.21, 0.64;
     0.91, 0.96, 0.00, 0.09, 0.98, 0.37;
     0.13, 0.88, 0.22, 0.21, 0.73, 1.00;
     0.91, 0.49, 0.39, 0.79, 0.47, 0.71;
     0.63, 0.41, 0.72, 0.68, 0.65, 0.83;
     0.10, 0.13, 0.77, 0.47, 0.22, 0.09;
     0.28, 0.93, 0.24, 0.90, 0.96, 0.21;
     0.55, 0.01, 0.04, 0.41, 0.26, 0.79;
     0.96, 0.19, 0.95, 0.66, 0.99, 0.68;
     0.96, 0.32, 0.53, 0.96, 0.84, 0.47;
     0.16, 0.05, 0.16, 0.30, 0.58, 0.90;
     0.97, 0.14, 0.36, 0.72, 0.78, 0.06;
     0.96, 0.73, 0.28, 0.75, 0.28, 0.68;
     0.49, 0.48, 0.58, 0.19, 0.25, 0.67;
     0.80, 0.34, 0.64, 0.57, 0.08, 0.13;
     0.14, 0.24, 0.12, 0.06, 0.63, 0.89;
     0.42, 0.45, 0.03, 0.68, 0.66, 0.17;
     0.92, 0.19, 0.48, 0.67, 0.28, 0.54;
     0.79, 0.32, 0.15, 0.13, 0.40, 0.03;
     0.96, 0.26, 0.93, 0.89, 0.61, 0.81;
     0.66, 0.83, 0.41, 0.17, 0.09, 0.60;
     0.04, 0.70, 0.40, 0.54, 0.37, 0.41;
     0.85, 0.33, 0.15, 0.03, 0.36, 5.77;
     0.93, 0.58, 0.88, 0.81, 0.40, 0.66;
     0.68, 0.29, 0.88, 0.60, 0.47, 0.96;
     0.76, 0.26, 0.09, 0.41, 0.14, 0.30;
     0.74, 0.26, 0.33, 0.64, 0.36, 0.72;
     0.39, 0.68, 0.69, 0.37, 0.12, 0.75;
     0.66, 0.52, 0.17, 1.00, 0.43, 0.19;
     0.17, 0.08, 0.35, 0.71, 0.17, 0.57];
x=x';
f = [6.39;  2.50; 9.34; 7.52; 6.91; 4.68; 45.40; 5.48;  2.75;  7.43; ...
     6.05;  0.41; 8.68; 2.38; 3.70; 1.34; 15.18; 4.35;  1.50;  3.43; ...
     3.10; 14.33; 0.35; 4.30; 3.77; 4.16;  6.75; 5.22; 16.23; 10.62];

% Interpolation points
nip = 6;
xe = zeros(nd,nip);
for i=1:npts
  xe(1:nd,i) = 0.1*i;
end

% Generate the interpolant
[iq, rq, ifail] = e01zm(x, f);

% Evaluate the interpolant
[q, qx, ifail] = e01zn(x, f, iq, rq, xe);

T = array2table(double([xe' q]));
T.Properties.VariableNames = {'x_1','x_2','x_3','x_4','x_5','x_6','q'};

fprintf('Interpolated Values\n\n');
disp(T(1:nip,1:nd+1));

e01zm example results

Interpolated Values

    x_1    x_2    x_3    x_4    x_5    x_6       q    
    ___    ___    ___    ___    ___    ___    ________

    0.1    0.1    0.1    0.1    0.1    0.1     -6.5071
    0.2    0.2    0.2    0.2    0.2    0.2     -3.5708
    0.3    0.3    0.3    0.3    0.3    0.3    -0.80057
    0.4    0.4    0.4    0.4    0.4    0.4      1.7257
    0.5    0.5    0.5    0.5    0.5    0.5      3.9278
    0.6    0.6    0.6    0.6    0.6    0.6      5.8951