NAG AD Library Routine Document
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 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 (Out)  ::  iwork(8*min(m,n)), info  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) 

3
Description
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 NAG AD Library Introduction 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 (dgesdd) 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
f08kd_a1w_f provides access to all the arguments available in the primal routine. There are also additional arguments specific to AD. A tooltip popup for each argument can be found by hovering over the argument name in
Section 2 and a summary of the arguments are provided below:
 ad_handle – a handle to the AD configuration data object, as created by x10aa_a1w_f. Symbolic adjoint mode may be selected by calling x10ac_a1w_f with this handle.
 jobz –
specifies options for computing all or part of the matrix $U$.
 m –
$m$, the number of rows of the matrix $A$.
 n –
$n$, the number of columns of the matrix $A$.
 a –
on entry: the $m$ by $n$ matrix $A$.
on exit: if $\mathbf{jobz}=\text{"O"}$, this argument is overwritten with the first $n$ columns of $U$ (the left singular vectors, stored columnwise) if $\mathbf{m}\ge \mathbf{n}$; this argument is overwritten with the first $m$ rows of ${V}^{T}$ (the right singular vectors, stored rowwise) otherwise.
 lda –
the first dimension of the array a.
 s –
on exit: the singular values of $A$, sorted so that $\mathbf{s}\left(i\right)\ge \mathbf{s}(i+\mathrm{1})$.
 u –
on exit:.
If $\mathbf{jobz}=\text{"A"}$ or $\mathbf{jobz}=\text{"O"}$ and $\mathbf{m}<\mathbf{n}$, u contains the $m$ by $m$ orthogonal matrix ${U}^{T}$.
If $\mathbf{jobz}=\text{"S"}$, u contains the first $\mathrm{min}(m,n)$ columns of $U$ (the left singular vectors, stored columnwise).
If $\mathbf{jobz}=\text{"O"}$ and $\mathbf{m}\ge \mathbf{n}$, or $\mathbf{jobz}=\text{"N"}$, u is not referenced.
 ldu –
the first dimension of the array u.
 vt –
on exit: if $\mathbf{jobz}=\text{"A"}$ or $\mathbf{jobz}=\text{"O"}$ and $\mathbf{m}\ge \mathbf{n}$, vt contains the $n$ by $n$ orthogonal matrix ${V}^{T}$.
 ldvt –
the first dimension of the array vt.
 work –
workspace.
 lwork –
the dimension of the array work.
If $\text{this argument}=\mathrm{1}$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to this argument is issued.
 iwork –
workspace.
 ifail –
on exit: $\mathbf{ifail}=\mathrm{0}$ unless the routine detects an error (see 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 5.2 in the NAG AD Library Introduction for further information.
 $\mathbf{ifail}=899$
Dynamic memory allocation failed for AD.
See
Section 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.
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.