NAG Library Routine Document
E02BBF
1 Purpose
E02BBF evaluates a cubic spline from its B-spline representation.
2 Specification
INTEGER |
NCAP7, IFAIL |
REAL (KIND=nag_wp) |
LAMDA(NCAP7), C(NCAP7), X, S |
|
3 Description
E02BBF evaluates the cubic spline
sx at a prescribed argument
x from its augmented knot set
λi, for
i=1,2,…,n+7, (see
E02BAF) and from the coefficients
ci, for
i=1,2,…,qin its B-spline representation
Here
q=n-+3, where
n- is the number of intervals of the spline, and
Nix denotes the normalized B-spline of degree
3 defined upon the knots
λi,λi+1,…,λi+4. The prescribed argument
x must satisfy
λ4≤x≤λn-+4.
It is assumed that λj≥λj-1, for j=2,3,…,n-+7, and λn-+4>λ4.
If
x is a point at which
4 knots coincide,
sx is discontinuous at
x; in this case,
S contains the value defined as
x is approached from the right.
The method employed is that of evaluation by taking convex combinations due to
de Boor (1972). For further details of the algorithm and its use see
Cox (1972) and
Cox and Hayes (1973).
It is expected that a common use of E02BBF will be the evaluation of the cubic spline approximations produced by
E02BAF. A generalization of E02BBF which also forms the derivative of
sx is
E02BCF.
E02BCF takes about
50% longer than E02BBF.
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
Cox M G and Hayes J G (1973) Curve fitting: a guide and suite of algorithms for the non-specialist user
NPL Report NAC26 National Physical Laboratory
de Boor C (1972) On calculating with B-splines
J. Approx. Theory 6 50–62
5 Parameters
- 1: NCAP7 – INTEGERInput
On entry: n-+7, where n- is the number of intervals (one greater than the number of interior knots, i.e., the knots strictly within the range λ4 to λn-+4) over which the spline is defined.
Constraint:
NCAP7≥8.
- 2: LAMDA(NCAP7) – REAL (KIND=nag_wp) arrayInput
On entry: LAMDAj must be set to the value of the jth member of the complete set of knots, λj, for j=1,2,…,n-+7.
Constraint:
the LAMDAj must be in nondecreasing order with LAMDANCAP7-3> LAMDA4.
- 3: C(NCAP7) – REAL (KIND=nag_wp) arrayInput
On entry: the coefficient
ci of the B-spline Nix, for i=1,2,…,n-+3. The remaining elements of the array are not referenced.
- 4: X – REAL (KIND=nag_wp)Input
On entry: the argument x at which the cubic spline is to be evaluated.
Constraint:
LAMDA4≤X≤LAMDANCAP7-3.
- 5: S – REAL (KIND=nag_wp)Output
On exit: the value of the spline, sx.
- 6: 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.
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.
On exit:
IFAIL=0 unless the routine detects an error or a warning has been flagged (see
Section 6).
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
The parameter
X does not satisfy
LAMDA4≤X≤LAMDANCAP7-3.
In this case the value of
S is set arbitrarily to zero.
- IFAIL=2
NCAP7<8, i.e., the number of interior knots is negative.
7 Accuracy
The computed value of
sx has negligible error in most practical situations. Specifically, this value has an
absolute error bounded in modulus by
18×cmax×machine precision, where
cmax is the largest in modulus of
cj,cj+1,cj+2 and
cj+3, and
j is an integer such that
λj+3≤x≤λj+4. If
cj,cj+1,cj+2 and
cj+3 are all of the same sign, then the computed value of
sx has a
relative error not exceeding
20×machine precision in modulus. For further details see
Cox (1978).
8 Further Comments
The time taken is approximately
C×1+0.1×logn-+7 seconds, where
C is a machine-dependent constant.
Note: the routine does not test all the conditions on the knots given in the description of
LAMDA in
Section 5, since to do this would result in a computation time approximately linear in
n-+7 instead of
logn-+7. All the conditions are tested in
E02BAF, however.
9 Example
Evaluate at nine equally-spaced points in the interval 1.0≤x≤9.0 the cubic spline with (augmented) knots 1.0, 1.0, 1.0, 1.0, 3.0, 6.0, 8.0, 9.0, 9.0, 9.0, 9.0 and normalized cubic B-spline coefficients 1.0, 2.0, 4.0, 7.0, 6.0, 4.0, 3.0.
The example program is written in a general form that will enable a cubic spline with n- intervals, in its normalized cubic B-spline form, to be evaluated at m equally-spaced points in the interval LAMDA4≤x≤LAMDAn-+4. The program is self-starting in that any number of datasets may be supplied.
9.1 Program Text
Program Text (e02bbfe.f90)
9.2 Program Data
Program Data (e02bbfe.d)
9.3 Program Results
Program Results (e02bbfe.r)