NAG AD Library
f08kd_a1w_f (dgesdd_a1w)
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.
1
Purpose
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.
2
Specification
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 

C++ Header Interface
#include <nagad.h>
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.
3
Description
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 divideandconquer method.
For further information see
Section 3 in the documentation for
f08kdf (dgesdd).
3.1
Symbolic Adjoint
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:

(i)for argument s,

(ii)the first $\mathrm{min}\left(m,n\right)$ columns of u and

(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.
3.1.1
Mathematical 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

(i)${\sigma}_{i}\ne {\sigma}_{j}$ for all $i\ne j$, $1\le i,j\le \mathrm{min}\left(m,n\right)$;

(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.
3.1.2
Usable adjoints
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 $m<n$. 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.
4
References
Giles M (2017) Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation
5
Arguments
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($8\times \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathbf{m},\mathbf{n}\right)$) – Integer array
Workspace


15:
ifail – Integer
Input/Output

On entry: must be set to $0$, $1\text{\hspace{0.25em}or\hspace{0.25em}}1$.
On exit: any errors are indicated as described in Section 6.
6
Error 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:
 $\mathbf{ifail}=89$
An unexpected AD error has been triggered by this routine. Please
contact
NAG.
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:
7
Accuracy
Not applicable.
8
Parallelism and Performance
f08kd_a1w_f
is not threaded in any implementation.
None.
10
Example
The following examples are variants of the example for
f08kdf (dgesdd),
modified to demonstrate calling the NAG AD Library.