decomposition. The implementation of the algorithm involves setting up appropriate information for the related function nag_fit_1dspline_knots (e02ba) followed by a call of that function. (See nag_fit_1dspline_knots (e02ba) for further details.)

Values of the spline interpolant, or of its derivatives or definite integral, can subsequently be computed as detailed in Further Comments.

References

Cox M G (1975) An algorithm for spline interpolation J. Inst. Math. Appl. 15 95–108

Cox M G (1977) A survey of numerical methods for data and function approximation The State of the Art in Numerical Analysis (ed D A H Jacobs) 627–668 Academic Press

Parameters

Compulsory Input Parameters

1: $x (m)$ – double array: $x (i)$ must be set to $x_{i}$ , the $i$ th data value of the independent variable $x$ , for $i = 1, 2, \dots, m$ .

Constraint: $x (i) < x (i + 1)$ , for $i = 1, 2, \dots, m - 1$ .
2: $y (m)$ – double array: $y (i)$ must be set to $y_{i}$ , the $i$ th data value of the dependent variable $y$ , for $i = 1, 2, \dots, m$ .

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the dimension of the arrays x, y. (An error is raised if these dimensions are not equal.)
$m$ , the number of data points.

Constraint: $m \geq 4$ .

Output Parameters

1: $lamda (lck)$ – double array: $lck = m + 4$ .
The value of $λ_{i}$ , the $i$ th knot, for $i = 1, 2, \dots, m + 4$ .
2: $c (lck)$ – double array: $lck = m + 4$ .
The coefficient $c_{i}$ of the B-spline $N_{i} (x)$ , for $i = 1, 2, \dots, m$ . The remaining elements of the array are not used.
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:

$ifail = 1$

On entry,	$m < 4$ ,
or	$lck < m + 4$ ,
or	$lwrk < 6 \times m + 16$ .

$ifail = 2$: The x-values fail to satisfy the condition

$x (1) < x (2) < x (3) < \dots < x (m)$ .

$ifail = - 99$: 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

The rounding errors incurred are such that the computed spline is an exact interpolant for a slightly perturbed set of ordinates

y_{i} + δ y_{i}

. The ratio of the root-mean-square value of the

δ y_{i}

to that of the

y_{i}

is no greater than a small multiple of the relative machine precision.

Further Comments

The time taken by nag_interp_1d_spline (e01ba) is approximately proportional to

m

All the

x_{i}

are used as knot positions except

x_{2}

and

x_{m - 1}

. This choice of knots (see Cox (1977)) means that

s (x)

is composed of

m - 3

cubic arcs as follows. If

m = 4

, there is just a single arc space spanning the whole interval

x_{1}

x_{4}

. If

m \geq 5

, the first and last arcs span the intervals

x_{1}

x_{3}

and

x_{m - 2}

x_{m}

respectively. Additionally if

m \geq 6

, the

i

th arc, for

i = 2, 3, \dots, m - 4

, spans the interval

x_{i + 1}

x_{i + 2}

After the call

[lamda, c, ifail] = e01ba(x, y, lck);

the following operations may be carried out on the interpolant

s (x)

The value of

s (x)

x = x

can be provided in the double variable s by the call

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

(see nag_fit_1dspline_eval (e02bb)).

The values of

s (x)

and its first three derivatives at

x = x

can be provided in the double array s of dimension

4

, by the call

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

(see nag_fit_1dspline_deriv (e02bc)).

Here left must specify whether the left- or right-hand value of the third derivative is required (see nag_fit_1dspline_deriv (e02bc) for details).

The value of the integral of

s (x)

over the range

x_{1}

x_{m}

can be provided in the double variable dint by

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

(see nag_fit_1dspline_integ (e02bd)).

Example

This example sets up data from

7

values of the exponential function in the interval

0

1

. nag_interp_1d_spline (e01ba) is then called to compute a spline interpolant to these data.

The spline is evaluated by nag_fit_1dspline_eval (e02bb), at the data points and at points halfway between each adjacent pair of data points, and the spline values and the values of

e^{x}

are printed out.

Open in the MATLAB editor: e01ba_example

function e01ba_example


fprintf('e01ba example results\n\n');

x = [0     0.2     0.4     0.6     0.75     0.9     1];
y = exp(x);
[lamda, c, ifail] = e01ba(x, y);

fprintf('\n   j    knot lamda(j+2)   b-spline coeff c(j)\n\n');
j = 1;
fprintf('%4d%35.4f\n', j, c(1));
m = size(x,2);
for j = 2:m - 1;
  fprintf('%4d%15.4f%20.4f\n', j, lamda(j+2), c(j));
end
fprintf('%4d%35.4f\n', m, c(m));
fprintf('\n   R        Abscissa            Ordinate             Spline\n\n');
for r = 1:m;
  [fit, ifail] = e02bb( ...
                      lamda, c, x(r));

  fprintf('%4d%15.4f%20.4f%20.4f\n', r, x(r), y(r), fit);
  if r<m;
    xarg = (x(r)+x(r+1))/2;

    [fit, ifail] = e02bb( ...
                          lamda, c, xarg);
    fprintf('%19.4f%40.4f\n', xarg, fit);
  end
end

e01ba example results


   j    knot lamda(j+2)   b-spline coeff c(j)

   1                             1.0000
   2         0.0000              1.1336
   3         0.4000              1.3726
   4         0.6000              1.7827
   5         0.7500              2.1744
   6         1.0000              2.4918
   7                             2.7183

   R        Abscissa            Ordinate             Spline

   1         0.0000              1.0000              1.0000
             0.1000                                  1.1052
   2         0.2000              1.2214              1.2214
             0.3000                                  1.3498
   3         0.4000              1.4918              1.4918
             0.5000                                  1.6487
   4         0.6000              1.8221              1.8221
             0.6750                                  1.9640
   5         0.7500              2.1170              2.1170
             0.8250                                  2.2819
   6         0.9000              2.4596              2.4596
             0.9500                                  2.5857
   7         1.0000              2.7183              2.7183

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox