Note: a1w denotes that first order adjoints are computed in working precision; this has the corresponding argument type nagad_a1w_w_rtype. Also available is the t1w (first order tangent linear) mode, the interface of which is implied by replacing a1w by t1w throughout this document. Additionally, the p0w (passive interface, as alternative to the FL interface) mode is available and can be inferred by replacing of active types by the corresponding passive types. The method of codifying AD implementations in the routine name and corresponding argument types is described in the NAG AD Library Introduction.

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## 1Purpose

e01da_a1w_f is the adjoint version of the primal routine e01daf.

## 2Specification

Fortran Interface
 Subroutine e01da_a1w_f ( ad_handle, 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 Type (nagad_a1w_w_rtype), Intent (In) :: x(mx), y(my), f(mx*my) Type (nagad_a1w_w_rtype), Intent (Out) :: lamda(mx+4), mu(my+4), c(mx*my), wrk((mx+6)*(my+6)) Type (c_ptr), Intent (Inout) :: ad_handle
The routine may be called by the names e01da_a1w_f or nagf_interp_dim2_spline_grid_a1w. The corresponding t1w and p0w variants of this routine are also available.

## 3Description

e01da_a1w_f is the adjoint 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.
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_a1w_f 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 – Type (c_ptr) Input/Output
On entry: a handle to the AD configuration data object, as created by x10aa_a1w_f.
2: mx – Integer Input
3: my – Integer Input
4: x(mx) – Type (nagad_a1w_w_rtype) array Input
5: y(my) – Type (nagad_a1w_w_rtype) array Input
6: f(${\mathbf{mx}}×{\mathbf{my}}$) – Type (nagad_a1w_w_rtype) array Input
7: px – Integer Output
8: py – Integer Output
9: lamda(${\mathbf{mx}}+4$) – Type (nagad_a1w_w_rtype) array Output
10: mu(${\mathbf{my}}+4$) – Type (nagad_a1w_w_rtype) array Output
11: c(${\mathbf{mx}}×{\mathbf{my}}$) – Type (nagad_a1w_w_rtype) array Output
12: wrk($\left({\mathbf{mx}}+6\right)×\left({\mathbf{my}}+6\right)$) – Type (nagad_a1w_w_rtype) array Workspace
13: ifail – Integer Input/Output

## 6Error Indicators and Warnings

e01da_a1w_f preserves all error codes from e01daf and in addition can return:
${\mathbf{ifail}}=-89$
See Section 4.8.2 in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-899$
Dynamic memory allocation failed for AD.
See Section 4.8.1 in the NAG AD Library Introduction for further information.

Not applicable.

## 8Parallelism and Performance

e01da_a1w_f 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.