# 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.

## 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 (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
 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)

## 3Description

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 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.

#### 3.1.1Mathematical 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
 (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.

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

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 column-wise) if $\mathbf{m}\ge \mathbf{n}$; this argument is overwritten with the first $m$ rows of ${V}^{T}$ (the right singular vectors, stored row-wise) otherwise.
• lda – the first dimension of the array a.
• s – on exit: the singular values of $A$, sorted so that $\mathbf{s}\left(i\right)\ge \mathbf{s}\left(i+\mathrm{1}\right)$.
• 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 $\mathrm{min}\left(m,n\right)$ columns of $U$ (the left singular vectors, stored column-wise). 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).

## 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 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.

Not applicable.

## 8Parallelism and Performance

f08kd_a1w_f is not threaded in any implementation.