NAG Library Routine Document

E02BBF

+− 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

E02BBF evaluates a cubic spline from its B-spline representation.

2 Specification

SUBROUTINE E02BBF (

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

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

3 Description

E02BBF evaluates the cubic spline

s (x)

at a prescribed argument

x

from its augmented knot set

λ_{i}

, for

i = 1, 2, \dots, n + 7

, (see E02BAF) and from the coefficients

c_{i}

, for

i = 1, 2, \dots, q

in its B-spline representation

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

Here

q = \bar{n} + 3

, where

\bar{n}

is the number of intervals of the spline, and

N_{i} (x)

denotes the normalized B-spline of degree

3

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}

It is assumed that

λ_{j} \geq λ_{j - 1}

, for

j = 2, 3, \dots, \bar{n} + 7

, and

λ_{\bar{n} + 4} > λ_{4}

x

is a point at which

4

knots coincide,

s (x)

is discontinuous at

x

; in this case, S contains the value defined as

x

is approached from the right.

The method employed is that of evaluation by taking convex combinations due to de Boor (1972). For further details of the algorithm and its use see Cox (1972) and Cox and Hayes (1973).

It is expected that a common use of E02BBF will be the evaluation of the cubic spline approximations produced by E02BAF. A generalization of E02BBF which also forms the derivative of

s (x)

is E02BCF. E02BCF 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

Cox M G and Hayes J G (1973) Curve fitting: a guide and suite of algorithms for the non-specialist user NPL Report NAC26 National Physical Laboratory

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 (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 is to be evaluated.
Constraint: $LAMDA (4) \leq X \leq LAMDA (NCAP7 - 3)$ .
5: S – REAL (KIND=nag_wp)Output: On exit: the value of the spline, $s (x)$ .
6: 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:

$IFAIL = 1$: The parameter X does not satisfy $LAMDA (4) \leq X \leq LAMDA (NCAP7 - 3)$ .

In this case the value of S is set arbitrarily to zero.

$IFAIL = 2$: $NCAP7 < 8$ , i.e., the number of interior knots is negative.

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 a relative error not exceeding

20 \times machine precision

in modulus. For further details see Cox (1978).

8 Further Comments

The time taken is approximately

C \times (1 + 0.1 \times \log (\bar{n} + 7))

seconds, where C is a machine-dependent constant.

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

Evaluate at nine equally-spaced points in the interval

1.0 \leq x \leq 9.0

the cubic spline with (augmented) knots

1.0

1.0

1.0

1.0

3.0

6.0

8.0

9.0

9.0

9.0

9.0

and normalized cubic B-spline coefficients

1.0

2.0

4.0

7.0

6.0

4.0

3.0

The example program is written in a general form that will enable a cubic spline with

\bar{n}

intervals, in its normalized cubic B-spline form, to be evaluated at

m

equally-spaced points in the interval

LAMDA (4) \leq x \leq LAMDA (\bar{n} + 4)

. The program is self-starting in that any number of datasets may be supplied.

NAG Library Routine DocumentE02BBF

+− 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

E02BBF