NAG Library Function Document

nag_1d_spline_deriv (e02bcc)

+− 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_deriv (e02bcc) evaluates a cubic spline and its first three derivatives from its B-spline representation.

2 Specification

#include <nag.h>

#include <nage02.h>

void	nag_1d_spline_deriv (Nag_DerivType derivs, double x, double s[], Nag_Spline spline, NagError fail)

3 Description

nag_1d_spline_deriv (e02bcc) 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 nag_1d_spline_fit_knots (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 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 4th-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.

nag_1d_spline_deriv (e02bcc) can be used to compute the values and derivatives of cubic spline fits and interpolants produced by nag_1d_spline_fit_knots (e02bac), nag_1d_spline_fit (e02bec) or nag_1d_spline_interpolant (e01bac).

If only values and not derivatives are required, nag_1d_spline_evaluate (e02bbc) may be used instead of nag_1d_spline_deriv (e02bcc), which 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

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

5 Arguments

1: derivs – Nag_DerivTypeInput

On entry: derivs, of type Nag_DerivType, 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 derivs is equal to

Nag_LeftDerivs

Nag_RightDerivs

respectively. If

x

does not coincide with a knot, the value of derivs is immaterial. If

x = spline \to lamda [3]

, right-hand values are computed, and if

x = spline \to lamda [spline \to n - 4]

), left-hand values are formed, regardless of the value of derivs.

Constraint:

derivs = Nag_LeftDerivs

Nag_RightDerivs

2: x – doubleInput

On entry: the argument

x

at which the cubic spline and its derivatives are to be evaluated.

Constraint:

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

3: s[ $4$ ] – doubleOutput

On exit:

s [j]

contains the value of the

j

th derivative of the spline at the argument

x

, for

j = 0, 1, 2, 3

. Note that

s [0]

contains the value of the spline.

4: 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 $λ_{4}$ to $λ_{\bar{n} + 4}$ over which the spline is defined).
Constraint: $spline \to n \geq 8$ .

lamda – doubleInput: 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 – doubleInput: 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_deriv (e02bcc) 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_deriv (e02bcc).

5: 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 $spline \to lamda [3] \leq x \leq spline \to lamda [spline \to n - 4]$ :
$spline \to lamda [3] = ⟨value⟩$ , $x = ⟨value⟩$ , $spline \to lamda [⟨value⟩] = ⟨value⟩$ .
NE_BAD_PARAM: On entry, argument derivs had an illegal value.
NE_INT_ARG_LT: On entry, $spline \to n$ must not be less than 8: $spline \to n = ⟨value⟩$ .
NE_SPLINE_RANGE_INVALID: On entry, the cubic spline range is invalid:
$spline \to lamda [3] = ⟨value⟩$ while $spline \to lamda [spline \to n - 4] = ⟨value⟩$ .
These must satisfy $spline \to lamda [3] < spline \to lamda [spline \to n - 4]$ .

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 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by this function is approximately linear in

\log (\bar{n} + 7)

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

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:

$\bar{n}$	$m$
$spline \to lamda [j]$	for $j = 0, 1, \dots, \bar{n} + 6$
$spline \to c [j]$ ,	for $j = 0, 1, \dots, \bar{n} + 2$
x	m values of x

The 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.

NAG Library Function Documentnag_1d_spline_deriv (e02bcc)

+− 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_deriv (e02bcc)