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## 1Purpose

e02bb is the AD Library version of the primal routine e02bbf. Based (in the C++ interface) on overload resolution, e02bb can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

## 2Specification

Fortran Interface
 Subroutine e02bb_AD_f ( ncap7, lamda, c, x, s, ifail)
 Integer, Intent (In) :: ncap7 Integer, Intent (Inout) :: ifail ADTYPE, Intent (In) :: lamda(ncap7), c(ncap7), x ADTYPE, Intent (Out) :: s 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:
C++ Interface
#include <dco.hpp>
namespace nag {
}
}
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

## 3Description

e02bb is the AD Library version of the primal routine e02bbf.
e02bbf evaluates a cubic spline from its B-spline representation. For further information see Section 3 in the documentation for e02bbf.

## 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
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

## 5Arguments

In addition to the arguments present in the interface of the primal routine, e02bb 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.
2: ncap7 – Integer Input
3: lamda(ncap7) – ADTYPE array Input
4: c(ncap7) – ADTYPE array Input
5: Input
6: Output
7: ifail – Integer Input/Output

## 6Error Indicators and Warnings

e02bb preserves all error codes from e02bbf and in addition can return:
${\mathbf{ifail}}=-89$
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.

Not applicable.

## 8Parallelism and Performance

e02bb is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for e02bbf, modified to demonstrate calling the NAG AD Library.
Description of the primal example.
Evaluate at nine equally-spaced points in the interval $1.0\le x\le 9.0$ 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 $\overline{n}$ intervals, in its normalized cubic B-spline form, to be evaluated at $m$ equally-spaced points in the interval ${\mathbf{lamda}}\left(4\right)\le x\le {\mathbf{lamda}}\left(\overline{n}+4\right)$. The program is self-starting in that any number of datasets may be supplied.

Language Source File Data Results
Fortran e02bb_a1w_fe.f90 None e02bb_a1w_fe.r
C++ e02bb_a1w_hcppe.cpp None e02bb_a1w_hcppe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran e02bb_t1w_fe.f90 None e02bb_t1w_fe.r
C++ e02bb_t1w_hcppe.cpp None e02bb_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran e02bb_p0w_fe.f90 None e02bb_p0w_fe.r
C++ e02bb_p0w_hcppe.cpp None e02bb_p0w_hcppe.r