f08kd_ad_f : NAG AD Library Routine Document, Mark 26

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)

Description

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).

3.1

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 $\min (m, n)$ 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_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.

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_a1w_f increments the adjoints of input argument a according to the first order adjoint model.

References

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

Arguments

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 $jobz = "O"$ , this argument is overwritten with the first $n$ columns of $U$ (the left singular vectors, stored column-wise) if $m \geq n$ ; 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 $s (i) \geq s (i + 1)$ .
u – on exit:. If $jobz = "A"$ or $jobz = "O"$ and $m < n$ , u contains the $m$ by $m$ orthogonal matrix $U^{T}$ . If $jobz = "S"$ , u contains the first $\min (m, n)$ columns of $U$ (the left singular vectors, stored column-wise). If $jobz = "O"$ and $m \geq n$ , or $jobz = "N"$ , u is not referenced.
ldu – the first dimension of the array u.
vt – on exit: if $jobz = "A"$ or $jobz = "O"$ and $m \geq n$ , 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 $this argument = - 1$ , 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: $ifail = 0$ unless the routine detects an error (see Section 6).

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:

$ifail = - 89$: 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.

$ifail = - 899$: Dynamic memory allocation failed for AD.
See Section 5.1 in the NAG AD Library Introduction for further information.

Language	Source File	Data	Results
Fortan	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

NAG AD Library Routine Document

f08kd_a1w_f (dgesdd_a1w)

▸▿ Contents

1

Purpose

2

Specification

3

Description

3.1

Symbolic Adjoint

3.1.1

Mathematical Background

3.1.2

Usable adjoints

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

NAG AD Library Routine Document

f08kd_a1w_f (dgesdd_a1w)

▸▿ Contents

1 Purpose

2 Specification

3 Description

3.1 Symbolic Adjoint

3.1.1 Mathematical Background

3.1.2 Usable adjoints

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example