g02aa is the AD Library version of the primal routine g02aaf. Based (in the C++ interface) on overload resolution, g02aa can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first and second order.

2 Specification

Fortran Interface

Subroutine g02aa_AD_f (

ad_handle, g, ldg, n, errtol, maxits, maxit, x, ldx, iter, feval, nrmgrd, ifail)

Integer, Intent (In)	::	ldg, n, maxits, maxit, ldx
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	iter, feval
ADTYPE, Intent (In)	::	errtol
ADTYPE, Intent (Inout)	::	g(ldg,n), x(ldx,n)
ADTYPE, Intent (Out)	::	nrmgrd
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
when ADTYPE is	Type(nagad_a1t1w_w_rtype)	then AD is a1t1w
when ADTYPE is	Type(nagad_t2w_w_rtype)	then AD is t2w

C++ Header Interface

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

void g02aa (

void *&ad_handle, ADTYPE g[], const Integer &ldg, const Integer &n, const ADTYPE &errtol, const Integer &maxits, const Integer &maxit, ADTYPE x[], const Integer &ldx, Integer &iter, Integer &feval, ADTYPE &nrmgrd, 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,
dco::gt1s<dco::gt1s<double>::type>::type,
dco::ga1s<dco::gt1s<double>::type>::type,

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

3 Description

g02aa is the AD Library version of the primal routine g02aaf.

g02aaf computes the nearest correlation matrix, in the Frobenius norm, to a given square, input matrix. For further information see Section 3 in the documentation for g02aaf.

4 References

Borsdorf R and Higham N J (2010) A preconditioned (Newton) algorithm for the nearest correlation matrix IMA Journal of Numerical Analysis 30(1) 94–107

Qi H and Sun D (2006) A quadratically convergent Newton method for computing the nearest correlation matrix SIAM J. Matrix AnalAppl 29(2) 360–385

5 Arguments

In addition to the arguments present in the interface of the primal routine, g02aa 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 – Pointer to AD Data Input/Output: On entry: a handle to the AD configuration data object, as created by x10aa.
2: g(ldg, n) – ADTYPE array Input/Output
3: ldg – Integer Input
4: n – Integer Input
5: errtol – ADTYPE Input
6: maxits – Integer Input
7: maxit – Integer Input
8: x(ldx, n) – ADTYPE array Output
9: ldx – Integer Input
10: iter – Integer Output
11: feval – Integer Output
12: nrmgrd – ADTYPE Output
13: ifail – Integer Input/Output

6 Error Indicators and Warnings

g02aa preserves all error codes from g02aaf and in addition can return:

$ifail = - 89$: 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.

$ifail = - 199$: The routine was called using a mode that has not yet been implemented.

$ifail = - 443$: On entry: ad_handle is nullptr.
This check is only made if the overloaded C++ interface is used with arguments not of type double.

$ifail = - 444$: 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 Section 4.8.1 in the NAG AD Library Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g02aa is not threaded in any implementation.

9 Further Comments

Since g is not a pure output and there is overwriting of variables, accessing adjoints later may result in wrong values, so a copy of the active input/output is used to obtain correct derivative values. See the example g02aa_a1_algo_dcoe.cpp for details.

10 Example

The following examples are variants of the example for g02aaf, modified to demonstrate calling the NAG AD Library.

Description of the primal example.

This example finds the nearest correlation matrix to:

G = (\begin{matrix} 2 & −1 & 0 & 0 \\ −1 & 2 & −1 & 0 \\ 0 & −1 & 2 & −1 \\ 0 & 0 & −1 & 2 \end{matrix})

10.1 Adjoint modes

Language	Source File	Data	Results
Fortran	g02aa_a1w_fe.f90	g02aa_a1w_fe.d	g02aa_a1w_fe.r
C++	g02aa_a1_algo_dcoe.cpp	None	g02aa_a1_algo_dcoe.r
C++	g02aa_a1t1_algo_dcoe.cpp	None	g02aa_a1t1_algo_dcoe.r