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## 1Purpose

f08kd is the AD Library version of the primal routine f08kdf (dgesdd). Based (in the C++ interface) on overload resolution, f08kd can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order. The parameter ad_handle can be used to choose whether adjoints are computed using a symbolic adjoint or straightforward algorithmic differentiation.

## 2Specification

Fortran Interface
 Subroutine f08kd_AD_f ( 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)) ADTYPE, Intent (Inout) :: a(lda,*), u(ldu,*), vt(ldvt,*) ADTYPE, Intent (Out) :: s(min(m,n)), work(max(1,lwork)) Character (1), Intent (In) :: jobz 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 {
 void f08kd ( handle_t &ad_handle, const char *jobz, const Integer &m, const Integer &n, ADTYPE a[], const Integer &lda, ADTYPE s[], ADTYPE u[], const Integer &ldu, ADTYPE vt[], const Integer &ldvt, ADTYPE work[], const Integer &lwork, Integer iwork[], Integer &ifail)
}
}
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

f08kd is the AD Library version of the primal routine f08kdf (dgesdd).
f08kdf (dgesdd) computes the singular value decomposition (SVD) of a real $m×n$ 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 can provide symbolic adjoints by setting the symbolic strategy as described in Section 3.3.3 in the Introduction to the NAG AD Library.
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 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 increments the adjoints of input argument a according to the first order adjoint model.

## 4References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Giles M (2017) Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

In addition to the arguments present in the interface of the primal routine, f08kd 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 and AD Strategies in the NAG AD Library Introduction.
2: jobz – character Input
3: m – Integer Input
4: n – Integer Input
5: a(lda, $*$) – ADTYPE array Input/Output
6: lda – Integer Input
7: s($\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$) – ADTYPE array Output
8: u(ldu, $*$) – ADTYPE array Output
9: ldu – Integer Input
10: vt(ldvt, $*$) – ADTYPE array Output
11: ldvt – Integer Input
12: work($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)$) – ADTYPE 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 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 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.
If the symbolic strategy is used, 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 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.
Description of the primal example.
This example finds the singular values and left and right singular vectors of the $4×6$ matrix
 $A = ( 2.27 0.28 -0.48 1.07 -2.35 0.62 -1.54 -1.67 -3.09 1.22 2.93 -7.39 1.15 0.94 0.99 0.79 -1.45 1.03 -1.94 -0.78 -0.21 0.63 2.30 -2.57 ) ,$
together with approximate error bounds for the computed singular values and vectors.
The example program for f08kb illustrates finding a singular value decomposition for the case $m\ge n$.

Language Source File Data Results
Fortran 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

### 10.2Tangent modes

Language Source File Data Results
Fortran f08kd_t1w_fe.f90 f08kd_t1w_fe.d f08kd_t1w_fe.r
C++ f08kd_t1w_hcppe.cpp f08kd_t1w_hcppe.d f08kd_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran f08kd_p0w_fe.f90 f08kd_p0w_fe.d f08kd_p0w_fe.r
C++ f08kd_p0w_hcppe.cpp f08kd_p0w_hcppe.d f08kd_p0w_hcppe.r