NAG FL Interface
e02bcf (dim1_​spline_​deriv)

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1 Purpose

e02bcf evaluates a cubic spline and its first three derivatives from its B-spline representation.

2 Specification

Fortran Interface
Subroutine e02bcf ( ncap7, lamda, c, x, left, s, ifail)
Integer, Intent (In) :: ncap7, left
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: lamda(ncap7), c(ncap7), x
Real (Kind=nag_wp), Intent (Out) :: s(4)
C Header Interface
#include <nag.h>
void  e02bcf_ (const Integer *ncap7, const double lamda[], const double c[], const double *x, const Integer *left, double s[], Integer *ifail)
The routine may be called by the names e02bcf or nagf_fit_dim1_spline_deriv.

3 Description

e02bcf evaluates the cubic spline s(x) and its first three derivatives at a prescribed argument x. It is assumed that s(x) is represented in terms of its B-spline coefficients ci, for i=1,2,,n¯+3 and (augmented) ordered knot set λi, for i=1,2,,n¯+7, (see e02baf), i.e.,
s(x) = i=1q ci Ni(x) .  
Here q=n¯+3, n¯ is the number of intervals of the spline and Ni(x) denotes the normalized B-spline of degree 3 (order 4) defined upon the knots λi,λi+1,,λi+4. The prescribed argument x must satisfy
λ4 x λ n¯+4 .  
At a simple knot λi (i.e., one satisfying λi-1<λi<λi+1), the third derivative of the spline is in general discontinuous. At a multiple knot (i.e., two or more knots with the same value), lower derivatives, and even the spline itself, may be discontinuous. Specifically, at a point x=u where (exactly) r knots coincide (such a point is termed a knot of multiplicity r), the values of the derivatives of order 4-j, for j=1,2,,r, are in general discontinuous. (Here 1r4; r>4 is not meaningful.) You must specify whether the value at such a point is required to be the left- or right-hand derivative.
The method employed is based upon:
  1. (i)carrying out a binary search for the knot interval containing the argument x (see Cox (1978)),
  2. (ii)evaluating the nonzero B-splines of orders 1, 2, 3 and 4 by recurrence (see Cox (1972) and Cox (1978)),
  3. (iii)computing all derivatives of the B-splines of order 4 by applying a second recurrence to these computed B-spline values (see de Boor (1972)),
  4. (iv)multiplying the fourth-order B-spline values and their derivative by the appropriate B-spline coefficients, and summing, to yield the values of s(x) and its derivatives.
e02bcf can be used to compute the values and derivatives of cubic spline fits and interpolants produced by e02baf.
If only values and not derivatives are required, e02bbf may be used instead of e02bcf, which 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
de Boor C (1972) On calculating with B-splines J. Approx. Theory 6 50–62

5 Arguments

1: ncap7 Integer Input
On entry: n¯+7, where n¯ is the number of intervals of the spline (which is 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: ncap78.
2: lamda(ncap7) Real (Kind=nag_wp) array Input
On entry: lamda(j) 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 lamda(j) must be in nondecreasing order with lamda(ncap7-3)>lamda(4).
3: c(ncap7) Real (Kind=nag_wp) array Input
On entry: the coefficient ci of the B-spline Ni(x), 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 and its derivatives are to be evaluated.
Constraint: lamda(4)xlamda(ncap7-3).
5: left Integer Input
On entry: specifies whether left- or right-hand values of the spline and its derivatives are to be computed (see Section 3). Left- or right-hand values are formed according to whether left is equal or not equal to 1.
If x does not coincide with a knot, the value of left is immaterial.
If x=lamda(4), right-hand values are computed.
If x=lamda(ncap7-3), left-hand values are formed, regardless of the value of left.
6: s(4) Real (Kind=nag_wp) array Output
On exit: s(j) contains the value of the (j-1)th derivative of the spline at the argument x, for j=1,2,3,4. Note that s(1) contains the value of the spline.
7: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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:
On entry, ncap7=value.
Constraint: ncap78.
On entry, lamda(4)=value, ncap7=value and lamda(ncap7-3)=value.
Constraint: lamda(4)<lamda(ncap7-3).
On entry, x=value, ncap7=value and lamda(ncap7-3)=value.
Constraint: xlamda(ncap7-3).
On entry, x=value and lamda(4)=value.
Constraint: xlamda(4).
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The computed value of s(x) 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+3xλj+4. If cj,cj+1,cj+2 and cj+3 are all of the same sign, then the computed value of s(x) has relative error bounded by 20×machine precision. For full details see Cox (1978).
No complete error analysis is available for the computation of the derivatives of s(x). However, for most practical purposes the absolute errors in the computed derivatives should be small.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
e02bcf is not threaded in any implementation.

9 Further Comments

The time taken is approximately linear in log(n¯+7).
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 log(n¯+7). All the conditions are tested in e02baf, however.

10 Example

Compute, at the 7 arguments x=0, 1, 2, 3, 4, 5, 6, the left- and right-hand values and first 3 derivatives of the cubic spline defined over the interval 0x6 having the 6 interior knots x=1, 3, 3, 3, 4, 4, the 8 additional knots 0, 0, 0, 0, 6, 6, 6, 6, and the 10 B-spline coefficients 10, 12, 13, 15, 22, 26, 24, 18, 14, 12.
The input data items (using the notation of Section 5) comprise the following values in the order indicated:
n¯ m
lamda(j), for j=1,2,,ncap7
c(j), for j=1,2,,ncap7-4
x(i), for i=1,2,,m
This example program is written in a general form that will enable the values and derivatives of a cubic spline having an arbitrary number of knots to be evaluated at a set of arbitrary points. Any number of datasets may be supplied. The only changes required to the program relate to the dimensions of the arrays lamda and c.

10.1 Program Text

Program Text (e02bcfe.f90)

10.2 Program Data

Program Data (e02bcfe.d)

10.3 Program Results

Program Results (e02bcfe.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 −40 −30 −20 −10 0 10 20 30 40 0 1 2 3 4 5 6 x Spline Spline 1st Derivative (left) 1st Derivative (left) 2nd Derivative (left) 2nd Derivative (left) 3rd Derivative (left) 3rd Derivative (left) Example Program Cubic spline and its first three derivatives