Note:a1w denotes that first order adjoints are computed in working precision; this has the corresponding argument type nagad_a1w_w_rtype.
Also available is the t1w (first order tangent linear) mode, the interface of which is implied by replacing a1w by t1w throughout this document.
Additionally, the p0w (passive interface, as alternative to the FL interface) mode is available and can be inferred by replacing of active types by the corresponding passive types.
The method of codifying AD implementations in the routine name and corresponding argument types is described in the NAG AD Library Introduction.
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.
The routine may be called by the names f08kd_a1w_f or nagf_lapackeig_dgesdd_a1w. The corresponding t1w and p0w variants of this routine are also available.
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 $m\times n$ 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).
3.1Symbolic 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:
The symbolic adjoint assumes that the primal routine has successfully converged. Moreover for considering the adjoints of s the first $\mathrm{min}(m,n)$ columns of u and the first $\mathrm{min}(m,n)$ rows of vt are required. To consider the adjoints of the first $\mathrm{min}(m,n)$ columns of u and/or the first $\mathrm{min}(m,n)$ 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
(i)${\sigma}_{i}\ne {\sigma}_{j}$ for all $i\ne j$, $1\le i,j\le \mathrm{min}(m,n)$;
(ii)if $m\ne n$ then ${\sigma}_{i}\ne 0$ for all $1\le i\le \mathrm{min}(m,n)$,
where ${\sigma}_{i}$ denotes the $i$th singular value of matrix $A$. Please see Giles (2017) for more details.
3.1.2Usable 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.
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_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.
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.
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:
${\mathbf{ifail}}=-89$
An unexpected AD error has been triggered by this routine. Please
contact NAG.
See Section 4.8.2 in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-899$
Dynamic memory allocation failed for AD.
See Section 4.8.1 in the NAG AD Library Introduction for further information.
In symbolic mode the following may be returned:
7Accuracy
Not applicable.
8Parallelism and Performance
f08kd_a1w_f
is not threaded in any implementation.
9Further Comments
None.
10Example
The following examples are variants of the example for
f08kdf (dgesdd),
modified to demonstrate calling the NAG AD Library.