NAG Library Routine Document

E02BCF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

E02BCF evaluates a cubic spline and its first three derivatives from its B-spline representation.

2 Specification

SUBROUTINE E02BCF (

NCAP7, LAMDA, C, X, LEFT, S, IFAIL)

INTEGER	NCAP7, LEFT, IFAIL
REAL (KIND=nag_wp)	LAMDA(NCAP7), C(NCAP7), X, S(4)

3 Description

E02BCF evaluates the cubic spline

s (x)

and its first three derivatives at a prescribed argument

x

. 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

, (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 prescribed argument

x

must satisfy

λ_{4} \leq x \leq λ_{\bar{n} + 4} .

At a simple knot

λ_{i}

(i.e., one satisfying

λ_{i - 1} < λ_{i} < λ_{i + 1}

), the third derivative of the spline is in general discontinuous. At a multiple knot (i.e., two or more knots with the same value), lower derivatives, and even the spline itself, may be discontinuous. Specifically, at a point

x = u

where (exactly)

r

knots coincide (such a point is termed a knot of multiplicity

r

), the values of the derivatives of order

4 - j

, for

j = 1, 2, \dots, r

, are in general discontinuous. (Here

1 \leq r \leq 4

;

r > 4

is not meaningful.) You must specify whether the value at such a point is required to be the left- or right-hand derivative.

The method employed is based upon:

(i)	carrying out a binary search for the knot interval containing the argument $x$ (see Cox (1978)),
(ii)	evaluating the nonzero B-splines of orders $1$ , $2$ , $3$ and $4$ by recurrence (see Cox (1972) and Cox (1978)),
(iii)	computing all derivatives of the B-splines of order $4$ by applying a second recurrence to these computed B-spline values (see de Boor (1972)),
(iv)	multiplying the fourth-order B-spline values and their derivative by the appropriate B-spline coefficients, and summing, to yield the values of $s (x)$ and its derivatives.

E02BCF can be used to compute the values and derivatives of cubic spline fits and interpolants produced by E02BAF.

If only values and not derivatives are required, E02BBF may be used instead of E02BCF, which takes about

50 %

longer than E02BBF.

4 References

Cox M G (1972) The numerical evaluation of B-splines J. Inst. Math. Appl. 10 134–149

Cox M G (1978) The numerical evaluation of a spline from its B-spline representation J. Inst. Math. Appl. 21 135–143

de Boor C (1972) On calculating with B-splines J. Approx. Theory 6 50–62

5 Parameters

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 $λ_{4}$ to $λ_{\bar{n} + 4}$ 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 $LAMDA (NCAP7 - 3) > LAMDA (4)$ .
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: X – REAL (KIND=nag_wp)Input: On entry: the argument $x$ at which the cubic spline and its derivatives are to be evaluated.
Constraint: $LAMDA (4) \leq X \leq LAMDA (NCAP7 - 3)$ .
5: LEFT – INTEGERInput: On entry: specifies whether left- or right-hand values of the spline and its derivatives are to be computed (see Section 3). Left- or right-hand values are formed according to whether LEFT is equal or not equal to $1$ .
If $x$ does not coincide with a knot, the value of LEFT is immaterial.

If $x = LAMDA (4)$ , right-hand values are computed.

If $x = LAMDA (NCAP7 - 3)$ , left-hand values are formed, regardless of the value of LEFT.
6: S( $4$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: $S (j)$ contains the value of the $(j - 1)$ th derivative of the spline at the argument $x$ , for $j = 1, 2, 3, 4$ . Note that $S (1)$ contains the value of the spline.
7: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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.

$IFAIL = 2$: Either $LAMDA (4) \geq LAMDA (NCAP7 - 3)$ , i.e., the range over which $s (x)$ is defined is null or negative in length, or X is an invalid argument, i.e., $X < LAMDA (4)$ or $X > LAMDA (NCAP7 - 3)$ .

7 Accuracy

The computed value of

s (x)

has negligible error in most practical situations. Specifically, this value has an absolute error bounded in modulus by

18 \times c_{\max} \times machine precision

, where

c_{\max}

is the largest in modulus of

c_{j}, c_{j + 1}, c_{j + 2}

and

c_{j + 3}

, and

j

is an integer such that

λ_{j + 3} \leq x \leq λ_{j + 4}

. If

c_{j}, c_{j + 1}, c_{j + 2}

and

c_{j + 3}

are all of the same sign, then the computed value of

s (x)

has relative error bounded by

20 \times machine precision

. For full details see Cox (1978).

No complete error analysis is available for the computation of the derivatives of

s (x)

. However, for most practical purposes the absolute errors in the computed derivatives should be small.

8 Further Comments

The time taken is approximately linear in

\log (\bar{n} + 7)

Note: the routine does not test all the conditions on the knots given in the description of LAMDA in Section 5, since to do this would result in a computation time approximately linear in

\bar{n} + 7

instead of

\log (\bar{n} + 7)

. All the conditions are tested in E02BAF, however.

9 Example

Compute, at the

7

arguments

x = 0

1

2

3

4

5

6

, the left- and right-hand values and first

3

derivatives of the cubic spline defined over the interval

0 \leq x \leq 6

having the

6

interior knots

x = 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:

$\bar{n}$	$m$
$LAMDA (j)$ ,	for $j = 1, 2, \dots, NCAP7$
$C (j)$ ,	for $j = 1, 2, \dots, NCAP7 - 4$
$X (i)$ ,	for $i = 1, 2, \dots, m$

This example program is written in a general form that will enable the values and derivatives of a cubic spline having an arbitrary number of knots to be evaluated at a set of arbitrary points. Any number of datasets may be supplied. The only changes required to the program relate to the dimensions of the arrays LAMDA and C.

NAG Library Routine DocumentE02BCF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

E02BCF