nag_fit_1dspline_integ (e02bd) can be used to determine the definite integrals of cubic spline fits and interpolants produced by nag_fit_1dspline_knots (e02ba).

References

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

Parameters

Compulsory Input Parameters

1: $lamda (ncap7)$ – double array: $lamda (j)$ must be set to the value of the $j$ th member of the complete set of knots, $λ_{j}$ , for $j = 1, 2, \dots, \bar{n} + 7$ .

Constraint: the $lamda (j)$ must be in nondecreasing order with $lamda (ncap7 - 3) > lamda (4)$ and satisfy $lamda (1) = lamda (2) = lamda (3) = lamda (4)$ and $lamda (ncap7 - 3) = lamda (ncap7 - 2) = lamda (ncap7 - 1) = lamda (ncap7)$ .
2: $c (ncap7)$ – double array: The coefficient $c_{i}$ of the B-spline $N_{i} (x)$ , for $i = 1, 2, \dots, \bar{n} + 3$ . The remaining elements of the array are not referenced.

Optional Input Parameters

1: $ncap7$ – int64int32nag_int scalar: Default: the dimension of the arrays lamda, c. (An error is raised if these dimensions are not equal.)
$\bar{n} + 7$ , where $\bar{n}$ is the number of intervals of the spline (which is one greater than the number of interior knots, i.e., the knots strictly within the range $a$ to $b$ ) over which the spline is defined.

Constraint: $ncap7 \geq 8$ .

Output Parameters

1: $dint$ – double scalar: The value of the definite integral of $s (x)$ between the limits $x = a$ and $x = b$ , where $a = λ_{4}$ and $b = λ_{\bar{n} + 4}$ .
2: $ifail$ – 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:

$ifail = 1$: $ncap7 < 8$ , i.e., the number of intervals is not positive.

$ifail = 2$

At least one of the following restrictions on the knots is violated:

$lamda (ncap7 - 3) > lamda (4)$ ,
$lamda (j) \geq lamda (j - 1)$ ,

for

j = 2, 3, \dots, ncap7

, with equality in the cases

j = 2, 3, 4, ncap7 - 2, ncap7 - 1

, and ncap7

$ifail = - 99$: 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 are such that the computed value of the integral is exact for a slightly perturbed set of B-spline coefficients

c_{i}

differing in a relative sense from those supplied by no more than

2.2 \times (\bar{n} + 3) \times machine precision

Further Comments

The time taken is approximately proportional to

\bar{n} + 7

Example

This example determines the definite integral over the interval

0 \leq x \leq 6

of a cubic spline having

6

interior knots at the positions

λ = 1

3

3

3

4

4

, the

8

additional knots

0

0

0

0

6

6

6

6

, and the

10

B-spline coefficients

10

12

13

15

22

26

24

18

14

12

The input data items (using the notation of Arguments) comprise the following values in the order indicated:

$\bar{n}$
$lamda (j)$ ,	for $j = 1, 2, \dots, ncap7$
$c (j)$ ,	for $j = 1, 2, \dots, ncap7 - 3$

Open in the MATLAB editor: e02bd_example

function e02bd_example


fprintf('e02bd example results\n\n');

knots = [ 1  3  3  3  4  4];
ncap = size(knots,2) + 1;
ncap7 = ncap + 7;

lamda = zeros(ncap7,1);
lamda(5:ncap+3) = knots;
lamda(ncap+4:ncap7) = 6;

% B-spline coefficients
c = zeros(ncap7,1);
c(1:ncap+3) = [10  12  13  15  22  26  24  18  14  12];

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

fprintf('Definite integral = %10.3e\n',dint);

e02bd example results

Definite integral =  1.000e+02

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

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