Interfaces: FL CL AD

e02bd: FL CL AD

NAG CL Interface
e02bdc (dim1_spline_integ)

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NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG CL Interface Introduction

E02 (Fit) Chapter Contents

E02 (Fit) Chapter Introduction

e02bd: FL CL AD

▸▿ 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

e02bdc computes the definite integral of a cubic spline from its B-spline representation.

2 Specification

#include <nag.h>

void	e02bdc (Nag_Spline spline, double integral, NagError *fail)

The function may be called by the names: e02bdc, nag_fit_dim1_spline_integ or nag_1d_spline_intg.

3 Description

e02bdc 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 e02bac), 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).

e02bdc can be used to determine the definite integrals of cubic spline fits and interpolants produced by e01bac, e02bac and e02bec.

4 References

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

5 Arguments

1: $spline$ – Nag_Spline *

Pointer to structure of type Nag_Spline with the following members:

n – 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: $spline \to n \geq 8$ .

lamda – double *Input: On entry: a pointer to which memory of size $spline \to n$ must be allocated. $spline \to lamda [j - 1]$ 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 $λ_{j}$ must be in nondecreasing order with $spline \to lamda [spline \to n - 4] > spline \to lamda [3]$ and satisfy
$spline \to lamda [0] = spline \to lamda [1] = spline \to lamda [2] = spline \to lamda [3]$
and
$spline \to lamda [spline \to n - 4] = spline \to lamda [spline \to n - 3] =$
$spline \to lamda [spline \to n - 2] = spline \to lamda [spline \to n - 1]$

c – double *Input: On entry: a pointer to which memory of size $spline \to n - 4$ must be allocated. $spline \to c$ holds the coefficient $c_{i}$ of the B-spline $N_{i} (x)$ , for $i = 1, 2, \dots, \bar{n} + 3$ .

2: $integral$ – double * 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}

3: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_INT_ARG_LT: On entry, $spline \to n$ must not be less than 8: $spline \to n = 〈value〉$ .
NE_KNOTS_CONS: On entry, the knots must satisfy the following constraints:
$spline \to lamda [spline \to n - 4] > spline \to lamda [3]$ , $spline \to lamda [j] \geq spline \to lamda [j - 1]$ , for $j = 1, 2, \dots, spline \to n - 1$ , with equality in the cases $j = 1, 2, 3$ , $spline \to n - 3$ , $spline \to n - 2$ and $spline \to n - 1$ .

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

e02bdc is not threaded in any implementation.

9 Further Comments

Under normal usage, the call to e02bdc will follow a call to e01bac, e02bac or e02bec. In that case, the structure spline will have been set up correctly for input to e02bdc.

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:

$\bar{n} + 7$
$spline . lamda [j - 1]$ ,	for $j = 1, 2, \dots, spline . n$
$spline . c [j - 1]$ ,	for $j = 1, 2, \dots, spline . n - 3$

The example program is written in a general form that will enable the definite integral of a cubic spline having an arbitrary number of knots to be computed. Any number of datasets may be supplied. The only changes required to the program relate to the size of

spline . lamda

and the storage allocated to

spline . c

within the structure spline.

Interfaces: FL CL AD

NAG CL Interface Introduction

E02 (Fit) Chapter Contents

E02 (Fit) Chapter Introduction

e02bd: FL CL AD

NAG CL Interfacee02bdc (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 CL Interface
e02bdc (dim1_spline_integ)