NAG Toolbox: nag_fit_1dspline_auto (e02be)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{start}$ – string (length ≥ 1)

Must be set to 'C' or 'W'.

$\mathbf{start}=\text{'C'}$

The function will build up the knot set starting with no interior knots. No values need be assigned to the arguments n, lamda, wrk or iwrk.

$\mathbf{start}=\text{'W'}$

The function will restart the knot-placing strategy using the knots found in a previous call of the function. In this case, the arguments n, lamda, wrk, and iwrk must be unchanged from that previous call. This warm start can save much time in searching for a satisfactory value of $\mathbf{s}$.

Constraint: $\mathbf{start}=\text{'C'}$ or $\text{'W'}$.

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

The values $x_{\mathit{r}}$ of the independent variable (abscissa) $x$, for $\mathit{r}=1,2,\dots ,m$.

Constraint: $x_1<x_2<\cdots <x_m$.

3:     $\mathrm{y}\left(\mathbf{m}\right)$ – double array

The values $y_{\mathit{r}}$ of the dependent variable (ordinate) $y$, for $\mathit{r}=1,2,\dots ,m$.

4:     $\mathrm{w}\left(\mathbf{m}\right)$ – double array

The values $w_{\mathit{r}}$ of the weights, for $\mathit{r}=1,2,\dots ,m$. For advice on the choice of weights, see Weighting of data points in the E02 Chapter Introduction.

Constraint: $\mathbf{w}\left(\mathit{r}\right)>0.0$, for $\mathit{r}=1,2,\dots ,m$.

5:     $\mathrm{s}$ – double scalar

The smoothing factor, $S$.

If $S=0.0$, the function returns an interpolating spline.

If $S$ is smaller than machine precision, it is assumed equal to zero.

For advice on the choice of $S$, see Description and Choice of .

Constraint: $\mathbf{s}\ge 0.0$.

6:     $\mathrm{n}$ – int64int32nag_int scalar

If the warm start option is used, the value of n must be left unchanged from the previous call.

7:     $\mathrm{lamda}\left(\mathbf{nest}\right)$ – double array

If the warm start option is used, the values $\mathbf{lamda}\left(1\right),\mathbf{lamda}\left(2\right),\dots ,\mathbf{lamda}\left(\mathbf{n}\right)$ must be left unchanged from the previous call.

8:     $\mathrm{wrk}\left(\mathit{lwrk}\right)$ – double array

lwrk, the dimension of the array, must satisfy the constraint $\mathit{lwrk}\ge 4\times \mathbf{m}+16\times \mathbf{nest}+41$.

If the warm start option is used on entry, the values $\mathbf{wrk}\left(1\right),\dots ,\mathbf{wrk}\left(n\right)$ must be left unchanged from the previous call.

9:     $\mathrm{iwrk}\left(\mathbf{nest}\right)$ – int64int32nag_int array

If the warm start option is used, on entry, the values $\mathbf{iwrk}\left(1\right),\dots ,\mathbf{iwrk}\left(n\right)$ must be left unchanged from the previous call.

This array is used as workspace.

Optional Input Parameters

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

Default: the dimension of the arrays x, y, w. (An error is raised if these dimensions are not equal.)

$m$, the number of data points.

Constraint: $\mathbf{m}\ge 4$.

2:     $\mathrm{nest}$ – int64int32nag_int scalar

Default: the dimension of the arrays lamda, iwrk. (An error is raised if these dimensions are not equal.)

An overestimate for the number, $n$, of knots required.

Constraint: $\mathbf{nest}\ge 8$. In most practical situations, $\mathbf{nest}=\mathbf{m}/2$ is sufficient. nest never needs to be larger than $\mathbf{m}+4$, the number of knots needed for interpolation $\left(\mathbf{s}=0.0\right)$.

Output Parameters

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

The total number, $n$, of knots of the computed spline.

2:     $\mathrm{lamda}\left(\mathbf{nest}\right)$ – double array

The knots of the spline, i.e., the positions of the interior knots $\mathbf{lamda}\left(5\right),\mathbf{lamda}\left(6\right),\dots ,\mathbf{lamda}\left(\mathbf{n}-4\right)$ as well as the positions of the additional knots

$$\mathbf{lamda}\left(1\right)=\mathbf{lamda}\left(2\right)=\mathbf{lamda}\left(3\right)=\mathbf{lamda}\left(4\right)=x_1$$

and

$$\mathbf{lamda}\left(\mathbf{n}-3\right)=\mathbf{lamda}\left(\mathbf{n}-2\right)=\mathbf{lamda}\left(\mathbf{n}-1\right)=\mathbf{lamda}\left(\mathbf{n}\right)=x_m$$

needed for the B-spline representation.

3:     $\mathrm{c}\left(\mathbf{nest}\right)$ – double array

The coefficient $c_{\mathit{i}}$ of the B-spline $N_{\mathit{i}}\left(x\right)$ in the spline approximation $s\left(x\right)$, for $\mathit{i}=1,2,\dots ,n-4$.

4:     $\mathrm{fp}$ – double scalar

The sum of the squared weighted residuals, $\theta$, of the computed spline approximation. If $\mathbf{fp}=0.0$, this is an interpolating spline. fp should equal $\mathbf{s}$ within a relative tolerance of $0.001$ unless $n=8$ when the spline has no interior knots and so is simply a cubic polynomial. For knots to be inserted, $\mathbf{s}$ must be set to a value below the value of fp produced in this case.

5:     $\mathrm{wrk}\left(\mathit{lwrk}\right)$ – double array

6:     $\mathrm{iwrk}\left(\mathbf{nest}\right)$ – int64int32nag_int array

This array is used as workspace.

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

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

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry,	$\mathbf{start}\ne \text{'C'}$ or $\text{'W'}$,
or	$\mathbf{m}<4$,
or	$\mathbf{s}<0.0$,
or	$\mathbf{s}=0.0$ and $\mathbf{nest}<\mathbf{m}+4$,
or	$\mathbf{nest}<8$,
or	$\mathit{lwrk}<4\times \mathbf{m}+16\times \mathbf{nest}+41$.

   $\mathbf{ifail}=2$

The weights are not all strictly positive.

   $\mathbf{ifail}=3$

The values of $\mathbf{x}\left(\mathit{r}\right)$, for $\mathit{r}=1,2,\dots ,\mathbf{m}$, are not in strictly increasing order.

   $\mathbf{ifail}=4$

The number of knots required is greater than nest. Try increasing nest and, if necessary, supplying larger arrays for the arguments lamda, c, wrk and iwrk. However, if nest is already large, say $\mathbf{nest}>\mathbf{m}/2$, then this error exit may indicate that s is too small.

   $\mathbf{ifail}=5$

The iterative process used to compute the coefficients of the approximating spline has failed to converge. This error exit may occur if s has been set very small. If the error persists with increased s, contact NAG.

   $\mathbf{ifail}=-99$

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 S

Outline of Method Used

Evaluation of Computed Spline

[s, ifail] = e02bb(lamda, c, x);

[s, ifail] = e02bc(lamda, c, x, left);

[dint, ifail] = e02bd(lamda, c);

Example

Open in the MATLAB editor: e02be_example

function e02be_example


fprintf('e02be example results\n\n');

start = 'C';
data = [0.0  -1.100  1.0;
        0.5  -0.372  2.0;
        1.0   0.431  1.5;
        1.5   1.690  1.0;
        2.0   2.110  3.0;
        2.5   3.100  1.0;
        3.0   4.230  0.5;
        4.0   4.350  1.0;
        4.5   4.810  2.0;
        5.0   4.610  2.5;
        5.5   4.790  1.0;
        6.0   5.230  3.0;
        7.0   6.350  1.0;
        7.5   7.190  2.0;
        8.0   7.970  1.0];
x = data(:,1);
y = data(:,2);
w = data(:,3);
m = size(data,1);

s = [1 0.5 0.1];

nest  = m + 4;
n     = int64(nest);
lamda = zeros(nest,1);
wrk   = zeros(4*m+16*nest+41, 1);
iwrk  = zeros(nest, 1, 'int64');
start = 'Cold';

xs(1:2:2*m-1) = x;
xs(2:2:2*m-2) = (x(1:m-1)+x(2:m))/2;

for is = 1:3
  % Get spline
  [n, lamda, c, fp, wrk, iwrk, ifail] = ...
  e02be( ...
         start, x, y, w, s(is), n, lamda, wrk, iwrk);

  % Print details of spline
  fprintf('\nCalling with smoothing factor S = %5.2f\n\n', s(is));
  fprintf('         Knots    Coeffs\n');
  fprintf('%4d%20.4f\n', 1, c(1));
  for j = 2:n-5
    fprintf('%4d%10.4f%10.4f\n', j, lamda(j+2), c(j));
  end
  fprintf('%4d%20.4f\n\n', n-4, c(n-4));
  fprintf('Weighted sum of squared residuals = %7.4f\n', fp);
  if fp==0
    fprintf('(The spline is an interpolating spline)\n');
  elseif n==8
    fprintf('(The spline is the weighted least-squares cubic polynomial)\n');
  end
  fprintf('\n');

  % Evaluate at x and mid-points (xs)
  for i = 1:2*m-1
    [fit(i,is), ifail] = e02bb( ...
                                lamda, c, xs(i),'ncap7',n);
  end

  start = 'Warm';
end

fig1 = figure;
hold on
plot(x,y,'*','Color','Green')
plot(xs,fit(:,1));
xlabel('x');
title('Evaluation of B-spline representation, S = 1.0');
legend('Evaluation points','B-spline','Location','NorthWest');
hold off;

fig2 = figure;
hold on
plot(x,y,'*','Color','Green')
plot(xs,fit(:,2));
xlabel('x');
title('Evaluation of B-spline representation, S = 0.5');
legend('Evaluation points','B-spline','Location','NorthWest');
hold off;

fig3 = figure;
hold on
plot(x,y,'*','Color','Green')
plot(xs,fit(:,3));
xlabel('x');
title('Evaluation of B-spline representation, S = 0.1');
legend('Evaluation points','B-spline','Location','NorthWest');
hold off;

e02be example results


Calling with smoothing factor S =  1.00

         Knots    Coeffs
   1             -1.3201
   2    0.0000    1.3542
   3    4.0000    5.5510
   4    8.0000    4.7031
   5              8.2277

Weighted sum of squared residuals =  1.0003


Calling with smoothing factor S =  0.50

         Knots    Coeffs
   1             -1.1072
   2    0.0000   -0.6571
   3    1.0000    0.4350
   4    2.0000    2.8061
   5    4.0000    4.6824
   6    5.0000    4.6416
   7    6.0000    5.1976
   8    8.0000    6.9008
   9              7.9979

Weighted sum of squared residuals =  0.5001


Calling with smoothing factor S =  0.10

         Knots    Coeffs
   1             -1.0901
   2    0.0000   -0.6401
   3    1.0000    0.0334
   4    1.5000    1.6390
   5    2.0000    2.1243
   6    3.0000    4.5591
   7    4.0000    4.2174
   8    4.5000    4.9105
   9    5.0000    4.5475
  10    5.5000    4.6960
  11    6.0000    5.7370
  12    8.0000    6.8179
  13              7.9953

Weighted sum of squared residuals =  0.1000