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.


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.


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)


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).

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 minm,n columns of u and
(iii) the first minm,n rows of vt.
The symbolic adjoint assumes that the primal routine has successfully converged. Moreover for considering the adjoints of s the first minm,n columns of u and the first minm,n rows of vt are required. To consider the adjoints of the first minm,n columns of u and/or the first minm,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 jobz='N' the SVD decomposition is performed by calling f08kd_a1w_f with jobz='S' (you must ensure that all arrays are allocated as specified for jobz='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 jobz='A' (you must ensure that all arrays are allocated as specified for jobz='A'). The results are stored according to the value jobz you provided.

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) σiσj for all ij, 1i,jminm,n;
(ii) if mn then σi0 for all 1iminm,n,
where σi denotes the ith singular value of matrix A. Please see Giles (2017) for more details.

Usable adjoints

You can set or access the adjoints of the output arguments a if jobz='O', s, u if jobz'O' and mn, and vt if jobz'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.


Giles M (2017) Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation


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:

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:
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.
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:
Singular values are not distinct.
At least one singular value is numerically zero.


Not applicable.

Parallelism and Performance

f08kd_a1w_f is not threaded in any implementation.

Further Comments



The following examples are variants of the example for f08kdf (dgesdd), modified to demonstrate calling the NAG AD Library.
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