Integer, Intent (In)	::	ncap7
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	lamda(ncap7), c(ncap7)
Real (Kind=nag_wp), Intent (Out)	::	dint

C Header Interface

#include <nag.h>

void	e02bdf_ (const Integer ncap7, const double lamda[], const double c[], double dint, Integer *ifail)

The routine may be called by the names e02bdf or nagf_fit_dim1_spline_integ.

3 Description

e02bdf computes the definite integral of the cubic spline

s (x)

between the limits

x = a

and

x = b

, where

a

and

b

are respectively the lower and upper limits of the range over which

s (x)

is defined. It is assumed that

s (x)

is represented in terms of its B-spline coefficients

c_{i}

, for

i = 1, 2, \dots, \bar{n} + 3

and (augmented) ordered knot set

λ_{i}

, for

i = 1, 2, \dots, \bar{n} + 7

, with

λ_{i} = a

, for

i = 1, 2, 3, 4

and

λ_{i} = b

, for

i = \bar{n} + 4, \dots, \bar{n} + 7

, (see e02baf), i.e.,

s (x) = \sum_{i = 1}^{q} c_{i} N_{i} (x) .

Here

q = \bar{n} + 3

\bar{n}

is the number of intervals of the spline and

N_{i} (x)

denotes the normalized B-spline of degree

3

(order

4

) defined upon the knots

λ_{i}, λ_{i + 1}, \dots, λ_{i + 4}

The method employed uses the formula given in Section 3 of Cox (1975).

e02bdf can be used to determine the definite integrals of cubic spline fits and interpolants produced by e02baf.

4 References

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

5 Arguments

1: $ncap7$ – Integer Input: On entry: $\bar{n} + 7$ , 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: $ncap7 \geq 8$ .
2: $lamda (ncap7)$ – Real (Kind=nag_wp) array Input: On entry: $lamda (j)$ 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 $lamda (ncap7 - 3) = lamda (ncap7 - 2) = lamda (ncap7 - 1) = lamda (ncap7)$ .
3: $c (ncap7)$ – Real (Kind=nag_wp) array Input: On entry: 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.
4: $dint$ – Real (Kind=nag_wp) Output: On exit: the value of the definite integral of $s (x)$ between the limits $x = a$ and $x = b$ , where $a = λ_{4}$ and $b = λ_{\bar{n} + 4}$ .
5: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $ncap7 = ⟨ value ⟩$ .
Constraint: $ncap7 \geq 8$ .

$ifail = 2$: On entry, $J = ⟨ value ⟩$ , $lamda (J) = ⟨ value ⟩$ and $lamda (J - 1) = ⟨ value ⟩$ .
Constraint: $lamda (J) \geq lamda (J - 1)$ .

On entry, $lamda (1) = ⟨ value ⟩$ and $lamda (2) = ⟨ value ⟩$ .
Constraint: $lamda (1) = lamda (2)$ .

On entry, $lamda (2) = ⟨ value ⟩$ and $lamda (3) = ⟨ value ⟩$ .
Constraint: $lamda (2) = lamda (3)$ .

On entry, $lamda (3) = ⟨ value ⟩$ and $lamda (4) = ⟨ value ⟩$ .
Constraint: $lamda (3) = lamda (4)$ .

On entry, $lamda (4) = ⟨ value ⟩$ , $NCAP4 = ⟨ value ⟩$ and $lamda (NCAP4) = ⟨ value ⟩$ .
Constraint: $lamda (4) < lamda (NCAP4)$ .

On entry, $ncap7 = ⟨ value ⟩$ , $lamda (ncap7 - 1) = ⟨ value ⟩$ and $lamda (ncap7) = ⟨ value ⟩$ .
Constraint: $lamda (ncap7 - 1) = lamda (ncap7)$ .

On entry, $ncap7 = ⟨ value ⟩$ , $lamda (ncap7 - 2) = ⟨ value ⟩$ and $lamda (ncap7 - 1) = ⟨ value ⟩$ .
Constraint: $lamda (ncap7 - 2) = lamda (ncap7 - 1)$ .

On entry, $ncap7 = ⟨ value ⟩$ , $lamda (ncap7 - 3) = ⟨ value ⟩$ and $lamda (ncap7 - 3) = ⟨ value ⟩$ .
Constraint: $lamda (ncap7 - 3) = lamda (ncap7 - 2)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 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

8 Parallelism and Performance

e02bdf is not threaded in any implementation.

9 Further Comments

The time taken is approximately proportional to

\bar{n} + 7

10 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 Section 5) comprise the following values in the order indicated:

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

e02bd: FL CL CPP AD

NAG FL Interfacee02bdf (dim1_​spline_​integ)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
e02bdf (dim1_spline_integ)