NAG Library Routine Document

E02BBF

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) – double precision 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) – double precision 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 used.
4: X – double precisionInput: On entry: the argument x at which the cubic spline is to be evaluated.
Constraint: LAMDA4≤X≤LAMDANCAP7-3.
5: S – double precisionOutput: 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.

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: 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.