), 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 $4$ th-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.

e02bcc can be used to compute the values and derivatives of cubic spline fits and interpolants produced by e02bac, e02bec or e01bac.

If only values and not derivatives are required, e02bbc may be used instead of e02bcc, which takes about 50% longer than 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_DerivType Input

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: $x$ – double Input

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: $s [4]$ – double Output

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 e02bcc will follow a call to e02bac, e01bac or e02bec. In that case, the structure spline will have been set up correctly for input to e02bcc.

5: $fail$ – 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_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

e02bcc is not threaded in any implementation.

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 e02bac, however, and the knots returned by e01bac or 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.