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## 1Purpose

e01da is the AD Library version of the primal routine e01daf. Based (in the C++ interface) on overload resolution, e01da can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

## 2Specification

Fortran Interface
 Subroutine e01da_AD_f ( mx, my, x, y, f, px, py, lamda, mu, c, wrk, ifail)
 Integer, Intent (In) :: mx, my Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: px, py ADTYPE, Intent (In) :: x(mx), y(my), f(mx*my) ADTYPE, Intent (Out) :: lamda(mx+4), mu(my+4), c(mx*my), wrk((mx+6)*(my+6)) 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

e01da is the AD Library version of the primal routine e01daf.
e01daf computes a bicubic spline interpolating surface through a set of data values, given on a rectangular grid in the $x$-$y$ plane. For further information see Section 3 in the documentation for e01daf.

## 4References

Anthony G T, Cox M G and Hayes J G (1982) DASL – Data Approximation Subroutine Library National Physical Laboratory
Cox M G (1975) An algorithm for spline interpolation J. Inst. Math. Appl. 15 95–108
de Boor C (1972) On calculating with B-splines J. Approx. Theory 6 50–62
Hayes J G and Halliday J (1974) The least squares fitting of cubic spline surfaces to general data sets J. Inst. Math. Appl. 14 89–103

## 5Arguments

In addition to the arguments present in the interface of the primal routine, e01da 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: mx – Integer Input
3: my – Integer Input
4: x(mx) – ADTYPE array Input
5: y(my) – ADTYPE array Input
6: f(${\mathbf{mx}}×{\mathbf{my}}$) – ADTYPE array Input
7: px – Integer Output
8: py – Integer Output
9: lamda(${\mathbf{mx}}+4$) – ADTYPE array Output
10: mu(${\mathbf{my}}+4$) – ADTYPE array Output
11: c(${\mathbf{mx}}×{\mathbf{my}}$) – ADTYPE array Output
12: wrk($\left({\mathbf{mx}}+6\right)×\left({\mathbf{my}}+6\right)$) – ADTYPE array Workspace
13: ifail – Integer Input/Output

## 6Error Indicators and Warnings

e01da preserves all error codes from e01daf 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

e01da is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for e01daf, modified to demonstrate calling the NAG AD Library.
Description of the primal example.
This example reads in values of ${m}_{x}$, ${x}_{\mathit{q}}$, for $\mathit{q}=1,2,\dots ,{m}_{x}$, ${m}_{y}$ and ${y}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,{m}_{y}$, followed by values of the ordinates ${f}_{q,r}$ defined at the grid points $\left({x}_{q},{y}_{r}\right)$.
It then calls e01da to compute a bicubic spline interpolant of the data values, and prints the values of the knots and B-spline coefficients. Finally it evaluates the spline at a small sample of points on a rectangular grid.

Language Source File Data Results
Fortran e01da_a1w_fe.f90 e01da_a1w_fe.d e01da_a1w_fe.r
C++ e01da_a1w_hcppe.cpp e01da_a1w_hcppe.d e01da_a1w_hcppe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran e01da_t1w_fe.f90 e01da_t1w_fe.d e01da_t1w_fe.r
C++ e01da_t1w_hcppe.cpp e01da_t1w_hcppe.d e01da_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran e01da_p0w_fe.f90 e01da_p0w_fe.d e01da_p0w_fe.r
C++ e01da_p0w_hcppe.cpp e01da_p0w_hcppe.d e01da_p0w_hcppe.r