NAG Library Function Document

nag_1d_spline_evaluate (e02bbc)

+− 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

nag_1d_spline_evaluate (e02bbc) evaluates a cubic spline from its B-spline representation.

2 Specification

#include <nag.h>

#include <nage02.h>

void	nag_1d_spline_evaluate (double x, double s, Nag_Spline spline, NagError *fail)

3 Description

nag_1d_spline_evaluate (e02bbc) evaluates the cubic spline

s (x)

at a prescribed argument

x

from its augmented knot set

λ_{i}

, for

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

, (see nag_1d_spline_fit_knots (e02bac)) 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}

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 (1978).

It is expected that a common use of nag_1d_spline_evaluate (e02bbc) will be the evaluation of the cubic spline approximations produced by nag_1d_spline_fit_knots (e02bac). A generalization of nag_1d_spline_evaluate (e02bbc) which also forms the derivative of

s (x)

is nag_1d_spline_deriv (e02bcc). nag_1d_spline_deriv (e02bcc) takes about 50% longer than nag_1d_spline_evaluate (e02bbc).

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 Arguments

1: x – doubleInput

On entry: the argument

x

at which the cubic spline is to be evaluated.

Constraint:

spline \to lamda [3] \leq x \leq spline \to lamda [spline \to n - 4]

2: s – double *Output

On exit: the value of the spline,

s (x)

3: 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 (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: $spline \to n \geq 8$ .

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 $spline \to lamda [spline \to n - 4] > spline \to lamda [3]$ .

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

Under normal usage, the call to nag_1d_spline_evaluate (e02bbc) will follow a call to nag_1d_spline_fit_knots (e02bac), nag_1d_spline_interpolant (e01bac) or nag_1d_spline_fit (e02bec). In that case, the structure spline will have been set up correctly for input to nag_1d_spline_evaluate (e02bbc).

4: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ABSCI_OUTSIDE_KNOT_INTVL: On entry, x must satisfy $spline \to lamda [3] \leq x \leq spline \to lamda [spline \to n - 4]$ :
$spline \to lamda [3] = ⟨value⟩$ , $x = ⟨value⟩$ , $spline \to lamda [⟨value⟩] = ⟨value⟩$ .
In this case s is set arbitrarily to zero.
NE_INT_ARG_LT: On entry, $spline \to n$ must not be less than 8: $spline \to n = ⟨value⟩$ .

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 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_1d_spline_evaluate (e02bbc) is approximately C

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

seconds, where C is a machine-dependent constant.

Note: the function does not test all the conditions on the knots given in the description of

spline \to 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 nag_1d_spline_fit_knots (e02bac), however, and the knots returned by nag_1d_spline_interpolant (e01bac) or nag_1d_spline_fit (e02bec) will satisfy the conditions.

10 Example

Evaluate at 9 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

spline \to lamda [3] \leq x \leq spline \to lamda [\bar{n} + 3]

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

NAG Library Function Documentnag_1d_spline_evaluate (e02bbc)

+− 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 Library Function Document

nag_1d_spline_evaluate (e02bbc)