NAG Library Function Document
nag_1d_spline_interpolant (e01bac) determines a cubic spline interpolant to a given set of data.
||nag_1d_spline_interpolant (Integer m,
const double x,
const double y,
nag_1d_spline_interpolant (e01bac) determines a cubic spline
, defined in the range
, which interpolates (passes exactly through) the set of data points
. Unlike some other spline interpolation algorithms, derivative end conditions are not imposed. The spline interpolant chosen has
, which are set to the values of
respectively. This spline is represented in its B-spline form (see Cox (1975)
denotes the normalized B-spline of degree 3, defined upon the knots
denotes its coefficient, whose value is to be determined by the function.
The use of B-splines requires eight additional knots , , , , , , and to be specified; the function sets the first four of these to and the last four to .
The algorithm for determining the coefficients is as described in Cox (1975)
factorization is used instead of
decomposition. The implementation of the algorithm involves setting up appropriate information for the related function nag_1d_spline_fit_knots (e02bac)
followed by a call of that function. (For further details of nag_1d_spline_fit_knots (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
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
m – IntegerInput
On entry: , the number of data points.
x[m] – const doubleInput
On entry: must be set to , the th data value of the independent variable , for .
, for .
y[m] – const doubleInput
On entry: must be set to , the th data value of the dependent variable , for .
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 . This is set to .
- lamda – double *Output
On exit: the pointer to which storage of size is internally allocated. contains the th knot, for .
- c – double *Output
On exit: the pointer to which storage of size is internally allocated. contains the coefficient of the B-spline , for .
Note that when the information contained in the pointers
is no longer of use, or before a new call to nag_1d_spline_interpolant (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 – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, .
The sequence x
is not strictly increasing:
The rounding errors incurred are such that the computed spline is an exact interpolant for a slightly perturbed set of ordinates . The ratio of the root-mean-square value of the to that of the is no greater than a small multiple of the relative machine precision.
8 Parallelism and Performance
The time taken by nag_1d_spline_interpolant (e01bac) is approximately proportional to .
are used as knot positions except
. This choice of knots (see Cox (1977)
) means that
is composed of
cubic arcs as follows. If
, there is just a single arc space spanning the whole interval
, the first and last arcs span the intervals
respectively. Additionally if
th arc, for
, spans the interval
After the call
e01bac(m, x, y, &spline, &fail)
the following operations may be carried out on the interpolant .
The value of
can be provided in the variable sval
by calling the function
e02bbc(xval, &sval, &spline, &fail)
The values of
and its first three derivatives at
can be provided in the array sdif
of dimension 4, by the call
e02bcc(derivs, xval, sdif, &spline, &fail)
must specify whether the left- or right-hand value of the third derivative is required (see nag_1d_spline_deriv (e02bcc)
for details). The value of the integral of
over the range
can be provided in the variable sint
e02bdc(&spline, &sint, &fail)
The following example program sets up data from 7 values of the exponential function in the interval 0 to 1. nag_1d_spline_interpolant (e01bac) is then called to compute a spline interpolant to these data.
The spline is evaluated by nag_1d_spline_evaluate (e02bbc)
, at the data points and at points halfway between each adjacent pair of data points, and the spline values and the values of
are printed out.
10.1 Program Text
Program Text (e01bace.c)
10.2 Program Data
10.3 Program Results
Program Results (e01bace.r)