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 nagmk26.h

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

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: $ncap7$ – IntegerInput: 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) arrayInput: 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) arrayInput: 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: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . 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:

$ifail = 1$: $ncap7 < 8$ , 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.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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 $j = 1, 2, \dots, ncap7 - 3$

NAG Library Routine Document

e02bdf (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

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