NAG CL Interface
e01bac (dim1_spline)

decomposition. The implementation of the algorithm involves setting up appropriate information for the related function e02bac followed by a call of that function. (For further details of e02bac, see the function document.)

Values of the spline interpolant, or of its derivatives or definite integral, can subsequently be computed as detailed in Section 9.

4 References

Cox M G (1975) An algorithm for spline interpolation J. Inst. Math. Appl. 15 95–108

Cox M G (1977) A survey of numerical methods for data and function approximation The State of the Art in Numerical Analysis (ed D A H Jacobs) 627–668 Academic Press

5 Arguments

1: $m$ – Integer Input

On entry:

m

, the number of data points.

Constraint:

m \geq 4

2: $x [m]$ – const double Input

On entry:

x [i - 1]

must be set to

x_{i}

, the

i

th data value of the independent variable

x

, for

i = 1, 2, \dots, m

Constraint:

x [i] < x [i + 1]

, for

i = 0, 1, \dots, m - 2

3: $y [m]$ – const double Input

On entry:

y [i - 1]

must be set to

y_{i}

, the

i

th data value of the dependent variable

y

, for

i = 1, 2, \dots, m

4: $spline$ – Nag_Spline *

Pointer to structure of type Nag_Spline with the following members:

n – IntegerOutput: On exit: the size of the storage internally allocated to $lamda$ . This is set to $m + 4$ .

lamda – double *Output: On exit: the pointer to which storage of size $n$ is internally allocated. $lamda [i - 1]$ contains the $i$ th knot, for $i = 1, 2, \dots, m + 4$ .

c – double *Output: On exit: the pointer to which storage of size $n - 4$ is internally allocated. $c [i - 1]$ contains the coefficient $c_{i}$ of the B-spline $N_{i} (x)$ , for $i = 1, 2, \dots, m$ .

Note that when the information contained in the pointers

lamda

and

c

is no longer of use, or before a new call to e01bac with the same spline, you should free this storage using the NAG macro NAG_FREE. This storage will not have been allocated if this function returns with

fail . code \neq

NE_NOERROR.

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_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 4$ .
NE_NOT_STRICTLY_INCREASING: The sequence x is not strictly increasing: $x [⟨ value ⟩] = ⟨ value ⟩$ , $x [⟨ value ⟩] = ⟨ value ⟩$ .

7 Accuracy

The rounding errors incurred are such that the computed spline is an exact interpolant for a slightly perturbed set of ordinates

y_{i} + δ y_{i}

. The ratio of the root-mean-square value of the

δ y_{i}

to that of the

y_{i}

is no greater than a small multiple of the relative machine precision.

8 Parallelism and Performance

e01bac is not threaded in any implementation.

9 Further Comments

The time taken by e01bac is approximately proportional to

m

All the

x_{i}

are used as knot positions except

x_{2}

and

x_{m - 1}

. This choice of knots (see Cox (1977)) means that

s (x)

is composed of

m - 3

cubic arcs as follows. If

m = 4

, there is just a single arc space spanning the whole interval

x_{1}

x_{4}

. If

m \geq 5

, the first and last arcs span the intervals

x_{1}

x_{3}

and

x_{m - 2}

x_{m}

respectively. Additionally if

m \geq 6

, the

i

th arc, for

i = 2, 3, \dots, m - 4

, spans the interval

x_{i + 1}

x_{i + 2}

After the call

e01bac(m, x, y, &spline, &fail)

the following operations may be carried out on the interpolant

s (x)

The value of

s (x)

x = xval

can be provided in the variable sval by calling the function

e02bbc(xval, &sval, &spline, &fail)

The values of

s (x)

and its first three derivatives at

x = xval

can be provided in the array sdif of dimension

4

, by the call

e02bcc(derivs, xval, sdif, &spline, &fail)

Here derivs must specify whether the left- or right-hand value of the third derivative is required (see e02bcc for details). The value of the integral of

s (x)

over the range

x_{1}

x_{m}

can be provided in the variable sint by

e02bdc(&spline, &sint, &fail)

10 Example

The following example program sets up data from 7 values of the exponential function in the interval 0 to

1

. e01bac is then called to compute a spline interpolant to these data.

The spline is evaluated by e02bbc, at the data points and at points halfway between each adjacent pair of data points, and the spline values and the values of