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.

2 Specification

Fortran Interface

Subroutine f08kd_AD_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 (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:

when ADTYPE is	Real(kind=nag_wp)	then AD is p0w
when ADTYPE is	Type(nagad_a1w_w_rtype)	then AD is a1w
when ADTYPE is	Type(nagad_t1w_w_rtype)	then AD is t1w

C++ Interface

#include <dco.hpp>
#include <nagad.h>
namespace nag {
namespace ad {

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

Note: this function can be used with AD tools other than dco/c++. For details, please contact NAG.

3 Description

f08kd is the AD Library 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).

3.1 Symbolic Adjoint

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:

(i)for argument s,
(ii)the first $\min (m, n)$ columns of u and
(iii)the first $\min (m, n)$ rows of vt.

The symbolic adjoint assumes that the primal routine has successfully converged. Moreover for considering the adjoints of s the first

\min (m, n)

columns of u and the first

\min (m, n)

rows of vt are required. To consider the adjoints of the first

\min (m, n)

columns of u and/or the first

\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

jobz ='N'

the SVD decomposition is performed by calling f08kd 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 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.

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) $σ_{i} \neq σ_{j}$ for all $i \neq j$ , $1 \leq i, j \leq \min (m, n)$ ;
(ii)if $m \neq n$ then $σ_{i} \neq 0$ for all $1 \leq i \leq \min (m, n)$ ,

where

σ_{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

jobz ='O'

, s, u if

jobz \neq'O'

and

m \geq n

, and vt if

jobz \neq'O'

and

m < n

. The adjoints of all other output arguments are ignored.

f08kd increments the adjoints of input argument a according to the first order adjoint model.

4 References

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

5 Arguments

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.

1: ad_handle – nag::ad::handle_t Input/Output: 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: Please consult Overwriting of Inputs in the NAG AD Library Introduction.
6: lda – Integer Input
7: s( $\min (m, n)$ ) – 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( $\max (1, lwork)$ ) – ADTYPE array Workspace
13: lwork – Integer Input
14: iwork( $8 \times \min (m, n)$ ) – Integer array Workspace
15: ifail – Integer Input/Output: On entry: must be set to $0$ , $- 1 or 1$ .
On exit: any errors are indicated as described in Section 6.

6 Error 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:

$ifail = - 89$: An unexpected AD error has been triggered by this routine. Please contact NAG.
See Error Handling in the NAG AD Library Introduction for further information.

$ifail = - 199$: 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.

$ifail = - 444$: A C++ exception was thrown.
The error message will show the details of the C++ exception text.

$ifail = - 899$: 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:

$ifail = 10$: Singular values are not distinct.

$ifail = 11$: At least one singular value is numerically zero.

7 Accuracy

Not applicable.

8 Parallelism and Performance

f08kd is not threaded in any implementation.

9 Further Comments

None.

10 Example

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 \times 6

matrix

A = (\begin{array}{r} 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 \end{array}),

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 \geq n

10.1 Adjoint modes

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

10.2 Tangent 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

10.3 Passive 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

NAG Library Manual, Mark 30.1

Interfaces: FL CL CPP AD

NAG AD Library Introduction

F08 (Lapackeig) Chapter Contents

F08 (Lapackeig) Chapter Introduction (FL)

f08kd: FL CL CPP AD

NAG AD Libraryf08kd (dgesdd)

▸▿ Contents

1 Purpose