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.

2 Specification

Fortran Interface

Subroutine e02bc_AD_f (

ncap7, lamda, c, x, left, s, ifail)

Integer, Intent (In)	::	ncap7, left
Integer, Intent (Inout)	::	ifail
ADTYPE, Intent (In)	::	lamda(ncap7), c(ncap7), x
ADTYPE, Intent (Out)	::	s(4)
Type (c_ptr), Intent (Inout)	::	ad_handle

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:

when ADTYPE is	Real(kind=nag_wp)	then AD is p0w
when ADTYPE is	Type(nagad_a1w_w_rtype)	then AD is a1w
when ADTYPE is	Type(nagad_t1w_w_rtype)	then AD is t1w

C++ Interface

#include <dco.hpp>
#include <nagad.h>
namespace nag {
namespace ad {

void e02bc (

handle_t &ad_handle, const Integer &ncap7, const ADTYPE lamda[], const ADTYPE c[], const ADTYPE &x, const Integer &left, ADTYPE s[], Integer &ifail)

}
}

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.

3 Description

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.

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

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.

1: ad_handle – nag::ad::handle_t Input/Output: 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.
2: ncap7 – Integer Input
3: lamda(ncap7) – ADTYPE array Input
4: c(ncap7) – ADTYPE array Input
5: x – ADTYPE Input
6: left – Integer Input
7: s( $4$ ) – ADTYPE array Output
8: ifail – Integer Input/Output

6 Error Indicators and Warnings

e02bc preserves all error codes from e02bcf and in addition can return:

$ifail = - 89$: 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.

$ifail = - 199$: 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.

$ifail = - 444$: A C++ exception was thrown.
The error message will show the details of the C++ exception text.

$ifail = - 899$: Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

e02bc is not threaded in any implementation.

9 Further Comments

None.

10 Example

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 \leq x \leq 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:

$\bar{n}$	$m$
$lamda (j)$ ,	for $j = 1, 2, \dots, ncap7$
$c (j)$ ,	for $j = 1, 2, \dots, ncap7 - 4$
$x (i)$ ,	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.