NAG Library Routine Document
E01BAF
1 Purpose
E01BAF determines a cubic spline interpolant to a given set of data.
2 Specification
| SUBROUTINE E01BAF ( | M, X, Y, LAMDA, C, LCK, WRK, LWRK, IFAIL) |
| INTEGER | M, LCK, LWRK, IFAIL |
| double precision | X(M), Y(M), LAMDA(LCK), C(LCK), WRK(LWRK) |
3 Description
E01BAF determines a cubic spline
sx, defined in the range
x1≤x≤xm, which interpolates (passes exactly through) the set of data points
xi,yi,
for
i=1,2,…,m, where
m≥4 and
x1<x2<⋯<xm. Unlike some other spline interpolation algorithms, derivative end conditions are not imposed. The spline interpolant chosen has
m-4 interior knots
λ5,λ6,…,λm, which are set to the values of
x3,x4,…,xm-2 respectively. This spline is represented in its B-spline form (see
Cox (1975)):
where
Nix denotes the normalized B-spline of degree
3,
defined upon the knots
λi,λi+1,…,λi+4, and
ci denotes its coefficient, whose value is to be determined by the routine.
The use of B-splines requires eight additional knots λ1, λ2,
λ3, λ4,
λm+1, λm+2,
λm+3 and λm+4 to be specified; E01BAF sets the first four of these to x1 and the last four to xm.
The algorithm for determining the coefficients is as described in
Cox (1975) except that
QR factorization is used instead of
LU decomposition. The implementation of the algorithm involves setting up appropriate information for the related
routine
E02BAF followed by a call of that routine. (See
E02BAF for further details.)
Values of the spline interpolant, or of its derivatives or definite integral, can subsequently be computed as detailed in
Section 8.
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 Parameters
- 1: M – INTEGERInput
On entry:
m, the number of data points.
Constraint:
M≥4.
- 2: X(M) – double precision arrayInput
On entry: Xi must be set to xi, the ith data value of the independent variable x, for i=1,2,…,m.
Constraint:
Xi<Xi+1, for i=1,2,…,M-1.
- 3: Y(M) – double precision arrayInput
On entry: Yi must be set to yi, the ith data value of the dependent variable y, for i=1,2,…,m.
- 4: LAMDA(LCK) – double precision arrayOutput
On exit: the value of λi, the ith knot, for i=1,2,…,m+4.
- 5: C(LCK) – double precision arrayOutput
On exit: the coefficient ci of the B-spline Nix, for i=1,2,…,m. The remaining elements of the array are not used.
- 6: LCK – INTEGERInput
On entry: the dimension of the arrays
LAMDA and
C as declared in the (sub)program from which E01BAF is called.
Constraint:
LCK≥M+4.
- 7: WRK(LWRK) – double precision arrayWorkspace
- 8: LWRK – INTEGERInput
On entry: the dimension of the array
WRK as declared in the (sub)program from which E01BAF is called.
Constraint:
LWRK≥6×M+16.
- 9: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
6 Error Indicators and Warnings
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
- IFAIL=1
-
| On entry, | M<4, |
| or | LCK<M+4, |
| or | LWRK<6×M+16. |
- IFAIL=2
-
The
X-values fail to satisfy the condition
X1<X2<X3<⋯<XM.
7 Accuracy
The rounding errors incurred are such that the computed spline is an exact interpolant for a slightly perturbed set of ordinates yi+δyi. The ratio of the root-mean-square value of the δyi to that of the yi is no greater than a small multiple of the relative machine precision.
8 Further Comments
The time taken by E01BAF is approximately proportional to m.
All the
xi are used as knot positions except
x2 and
xm-1. This choice of knots
(see
Cox (1977)) means that
sx is composed of
m-3 cubic arcs as follows. If
m=4, there is just a single arc space spanning the whole interval
x1 to
x4. If
m≥5, the first and last arcs span the intervals
x1 to
x3 and
xm-2 to
xm respectively. Additionally if
m≥6, the
ith arc,
for
i=2,3,…,m-4, spans the interval
xi+1 to
xi+2.
After the call
CALL E01BAF (M, X, Y, LAMDA, C, LCK, WRK, LWRK, IFAIL)
the following operations may be carried out on the interpolant
sx.
The value of
sx at
x=X can be provided in the
double precision variable
S by the call
CALL E02BBF (M+4, LAMDA, C, X, S, IFAIL)
(see
E02BBF).
The values of
sx and its first three derivatives at
x=X can be provided in the
double precision array
S
of dimension
4, by the call
CALL E02BCF (M+4, LAMDA, C, X, LEFT, S, IFAIL)
(see
E02BCF).
Here
LEFT
must specify whether the left- or right-hand value of the third derivative is required (see
E02BCF for details).
The value of the integral of
sx over the range
x1 to
xm can be provided in the
double precision variable
DINT
by
CALL E02BDF (M+4, LAMDA, C, DINT, IFAIL)
(see
E02BDF).
9 Example
This example sets up data from 7 values of the exponential function in the interval 0 to 1. E01BAF is then called to compute a spline interpolant to these data.
The spline is evaluated by
E02BBF, at the data points and at points halfway between each adjacent pair of data points, and the spline values and the values of
ex are printed out.
9.1 Program Text
Program Text (e01bafe.f)
9.2 Program Data
None.
9.3 Program Results
Program Results (e01bafe.r)