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## 1Purpose

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. The parameter ad_handle can be used to choose whether adjoints are computed using a symbolic adjoint or straightforward algorithmic differentiation.

## 2Specification

Fortran Interface
 Subroutine g02aa_AD_f ( 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:
C++ Interface
#include <dco.hpp>
namespace nag {
 void g02aa ( handle_t &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,

## 3Description

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.

#### 3.1.1Symbolic Strategy

Symbolic strategy may be selected by calling ad_handle.set_strategy(nag::ad::symbolic) prior to calling g02aa. No further changes are needed compared to using the algorithmic strategy.

#### 3.1.2Mathematical Background

The symbolic adjoint is computed by the application of matrix calculus techniques to the Newton iteration in the primal algorithm.
Note that each step of the iteration involves a projection into the set of positive semidefinite matrices. This is done by repeated application of the function $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(0,\lambda \right)$, where $\lambda$ is a matrix eigenvalue. Since this operation is nonsmooth at the origin, the computation of the nearest correlation matrix is technically nondifferentiable. In practice, when this occurs an element of the subdifferential of $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(0,\lambda \right)$ is returned and the resulting adjoint is computed from the generalized Jacobian. Numerical experiments have shown that the adjoint computation is not sensitive to this issue; the resulting adjoints are still useful and are in close agreement with those that would be obtained, for example, via finite differences.
Note that discrepancies between the second order symbolic and algorithmic adjoints are known to occur if certain elements of the input matrix are identically zero. This is a result of the underlying primal code being nondifferentiable in certain places. It is recommended that the symbolic adjoint should be used in this case.
See Hüser (2015) for further details.

## 4References

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
Hüser J (2015) Adjoint Derivatives of a Nearest Correlation Matrix Algorithm MSc Dissertation, RWTH Aachen University
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

## 5Arguments

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.
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: g(ldg, n) – ADTYPE array Input/Output
3: ldg – Integer Input
4: n – Integer Input
5: 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: Output
13: ifail – Integer Input/Output

## 6Error Indicators and Warnings

g02aa preserves all error codes from g02aaf and in addition can return:
${\mathbf{ifail}}=-89$
See Error Handling in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-199$
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.
${\mathbf{ifail}}=-444$
A C++ exception was thrown.
The error message will show the details of the C++ exception text.
${\mathbf{ifail}}=-899$
Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.

Not applicable.

## 8Parallelism and Performance

g02aa is not threaded in any implementation.

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.

## 10Example

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 the matrix $G$, where
 $G = ( 2 −1 0 0 −1 2 −1 0 0 −1 2 −1 0 0 −1 2 ) .$
The example also demonstrates how a modified Cholesky factorization (computed using f01md and f01me) can be used to obtain a bound on the distance to the nearest correlation matrix prior to computing the NCM itself.

Language Source File Data Results
Fortran g02aa_a1t1w_fe.f90 g02aa_a1t1w_fe.d g02aa_a1t1w_fe.r
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_a1_sym_dcoe.cpp None g02aa_a1_sym_dcoe.r
C++ g02aa_a1t1_algo_dcoe.cpp None g02aa_a1t1_algo_dcoe.r
C++ g02aa_a1t1_sym_dcoe.cpp None g02aa_a1t1_sym_dcoe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran g02aa_t1w_fe.f90 g02aa_t1w_fe.d g02aa_t1w_fe.r
Fortran g02aa_t2w_fe.f90 g02aa_t2w_fe.d g02aa_t2w_fe.r
C++ g02aa_t1_dcoe.cpp None g02aa_t1_dcoe.r
C++ g02aa_t2_dcoe.cpp None g02aa_t2_dcoe.r

### 10.3Passive mode

Language Source File Data Results
Fortran g02aa_p0w_fe.f90 g02aa_p0w_fe.d g02aa_p0w_fe.r
C++ g02aa_passive_dcoe.cpp None g02aa_passive_dcoe.r