e02bc
is the AD Library version of the primal routine
e02bcf.
Based (in the C++ interface) on overload resolution,
e02bc can be used for primal, tangent and adjoint
evaluation. It supports tangents and adjoints of first order.
Corresponding to the overloaded C++ function, the Fortran interface provides five routines with names reflecting the type used for active real arguments. The actual subroutine and type names are formed by replacing AD and ADTYPE in the above as follows:
The function is overloaded on ADTYPE which represents the type of active arguments. ADTYPE may be any of the following types: double, dco::ga1s<double>::type, dco::gt1s<double>::type
Note: this function can be used with AD tools other than dco/c++. For details, please contact NAG.
3Description
e02bc
is the AD Library version of the primal routine
e02bcf.
e02bcf evaluates a cubic spline and its first three derivatives from its B-spline representation.
For further information see Section 3 in the documentation for e02bcf.
4References
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. Theory6 50–62
5Arguments
In addition to the arguments present in the interface of the primal routine,
e02bc includes some arguments specific to AD.
A brief summary of the AD specific arguments is given below. For the remainder, links are provided to the corresponding argument from the primal routine.
A tooltip popup for all arguments can be found by hovering over the argument name in Section 2 and in this section.
On entry: a configuration object that holds information on the differentiation strategy. Details on setting the AD strategy are described in AD handle object in the NAG AD Library Introduction.
e02bc preserves all error codes from e02bcf and in addition can return:
${\mathbf{ifail}}=-89$
An unexpected AD error has been triggered by this routine. Please
contact NAG.
See Error Handling in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-199$
The routine was called using a strategy that has not yet been implemented.
See AD Strategies in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-444$
A C++ exception was thrown.
The error message will show the details of the C++ exception text.
${\mathbf{ifail}}=-899$
Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.
7Accuracy
Not applicable.
8Parallelism and Performance
e02bc
is not threaded in any implementation.
9Further Comments
None.
10Example
The following examples are variants of the example for
e02bcf,
modified to demonstrate calling the NAG AD Library.
Description of the primal 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 $0\le x\le 6$ 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:
$\overline{\mathit{n}}$
$\mathit{m}$
${\mathbf{lamda}}\left(j\right)$,
for $j=1,2,\dots ,{\mathbf{ncap7}}$
${\mathbf{c}}\left(j\right)$,
for $j=1,2,\dots ,{\mathbf{ncap7}}-4$
${\mathbf{x}}\left(i\right)$,
for $i=1,2,\dots ,m$
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.