Note: _a1w_ denotes that first order adjoints are computed in working precision; this has the corresponding argument type nagad_a1w_w_rtype. Further implementations, for example for higher order differentiation or using the tangent linear approach, may become available at later marks of the NAG AD Library. The method of codifying AD implementations in the routine name and corresponding argument types is described in the NAG AD Library Introduction.

## 1Purpose

f08kd_a1w_f is the adjoint version of the primal routine f08kdf (dgesdd). Depending on the value of ad_handle, f08kd_a1w_f uses algorithmic differentiation or symbolic adjoints to compute adjoints of the primal.

## 2Specification

Fortran Interface
 Subroutine f08kd_a1w_f ( ad_handle, jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, iwork, ifail)
 Integer, Intent (In) :: m, n, lda, ldu, ldvt, lwork Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iwork(8*min(m,n)) Type (nagad_a1w_w_rtype), Intent (Inout) :: a(lda,*), u(ldu,*), vt(ldvt,*) Type (nagad_a1w_w_rtype), Intent (Out) :: s(min(m,n)), work(max(1,lwork)) Character (1), Intent (In) :: jobz Type (c_ptr), Intent (In) :: ad_handle
 void f08kd_a1w_f_ ( void *&ad_handle, const char *jobz, const Integer &m, const Integer &n, nagad_a1w_w_rtype a[], const Integer &lda, nagad_a1w_w_rtype s[], nagad_a1w_w_rtype u[], const Integer &ldu, nagad_a1w_w_rtype vt[], const Integer &ldvt, nagad_a1w_w_rtype work[], const Integer &lwork, Integer iwork[], Integer &ifail, const Charlen length_jobz)
The routine may be called by the names f08kd_a1w_f or nagf_lapackeig_dgesdd_a1w.

## 3Description

f08kd_a1w_f is the adjoint version of the primal routine f08kdf (dgesdd).
f08kdf (dgesdd) computes the singular value decomposition (SVD) of a real $m$ by $n$ matrix $A$, optionally computing the left and/or right singular vectors, by using a divide-and-conquer method. For further information see Section 3 in the documentation for f08kdf (dgesdd).

f08kd_a1w_f can provide symbolic adjoints by setting the symbolic mode as described in Section 3.2.2 in the X10 Chapter introduction. Please see Section 4 in the Introduction to the NAG AD Library for API description on how to use symbolic adjoints.
The symbolic adjoint allows you to compute the adjoints of the output arguments:
1. (i)for argument s,
2. (ii)the first $\mathrm{min}\left(m,n\right)$ columns of u and
3. (iii)the first $\mathrm{min}\left(m,n\right)$ rows of vt.
The symbolic adjoint assumes that the primal routine has successfully converged. Moreover for considering the adjoints of s the first $\mathrm{min}\left(m,n\right)$ columns of u and the first $\mathrm{min}\left(m,n\right)$ rows of vt are required. To consider the adjoints of the first $\mathrm{min}\left(m,n\right)$ columns of u and/or the first $\mathrm{min}\left(m,n\right)$ rows of vt the algorithm requires the computation of all entries of the matrices $U$ and $V$.
Hence (to compute the desired adjoint) if the routine is run with $\mathbf{jobz}=\text{'N'}$ the SVD decomposition is performed by calling f08kd_a1w_f with $\mathbf{jobz}=\text{'S'}$ (you must ensure that all arrays are allocated as specified for $\mathbf{jobz}=\text{'S'}$). The results are stored according to the value jobz you provided.
For all other settings of jobz the SVD decomposition is performed by calling the f08kdf with $\mathbf{jobz}=\text{'A'}$ (you must ensure that all arrays are allocated as specified for $\mathbf{jobz}=\text{'A'}$). The results are stored according to the value jobz you provided.

#### 3.1.1Mathematical Background

The symbolic adjoint uses the SVD decomposition computed by the primal routine to obtain the adjoints. To compute the adjoints it is required that
1. (i)${\sigma }_{i}\ne {\sigma }_{j}$ for all $i\ne j$, $1\le i,j\le \mathrm{min}\left(m,n\right)$;
2. (ii)if $m\ne n$ then ${\sigma }_{i}\ne 0$ for all $1\le i\le \mathrm{min}\left(m,n\right)$,
where ${\sigma }_{i}$ denotes the $i$th singular value of matrix $A$. Please see Giles (2017) for more details.

You can set or access the adjoints of the output arguments a if $\mathbf{jobz}=\text{'O'}$, s, u if $\mathbf{jobz}\ne \text{'O'}$ and $m\ge n$, and vt if $\mathbf{jobz}\ne \text{'O'}$ and $m. The adjoints of all other output arguments are ignored.
f08kd_a1w_f increments the adjoints of input argument a according to the first order adjoint model.

## 4References

Giles M (2017) Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation

## 5Arguments

In addition to the arguments present in the interface of the primal routine, f08kd_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
On entry: a handle to the AD configuration data object, as created by x10aa_a1w_f. Symbolic adjoint mode may be selected by calling x10aa_a1w_f with this handle.
2: jobz – character Input
3: m – Integer Input
4: n – Integer Input
5: a(lda, $*$) – Type (nagad_a1w_w_rtype) array Input/Output
6: lda – Integer Input
7: s($\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathbf{m},\mathbf{n}\right)$) – Type (nagad_a1w_w_rtype) array Output
8: u(ldu, $*$) – Type (nagad_a1w_w_rtype) array Output
9: ldu – Integer Input
10: vt(ldvt, $*$) – Type (nagad_a1w_w_rtype) array Output
11: ldvt – Integer Input
12: work($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathbf{lwork}\right)$) – Type (nagad_a1w_w_rtype) array Workspace
13: lwork – Integer Input
14: iwork($8×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathbf{m},\mathbf{n}\right)$) – Integer array Workspace
15: ifail – Integer Input/Output
On entry: must be set to $0$, .
On exit: any errors are indicated as described in Section 6.

## 6Error Indicators and Warnings

f08kd_a1w_f uses the standard NAG ifail mechanism. Any errors indicated via info values returned by f08kdf may be indicated with the same value returned by ifail. In addition, this routine may return:
$\mathbf{ifail}=-89$
See Section 4.5.2 in the NAG AD Library Introduction for further information.
$\mathbf{ifail}=-899$
Dynamic memory allocation failed for AD.
See Section 4.5.1 in the NAG AD Library Introduction for further information.
In symbolic mode the following may be returned:
$\mathbf{ifail}=10$
Singular values are not distinct.
$\mathbf{ifail}=11$
At least one singular value is numerically zero.

Not applicable.

## 8Parallelism and Performance

f08kd_a1w_f is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for f08kdf (dgesdd), modified to demonstrate calling the NAG AD Library.
 Language Source File Data Results Fortan f08kd_a1w_fe.f90 f08kd_a1w_fe.d f08kd_a1w_fe.r C++ f08kd_a1w_hcppe.cpp f08kd_a1w_hcppe.d f08kd_a1w_hcppe.r