# NAG Library Routine Document

## 1Purpose

g02aaf computes the nearest correlation matrix, in the Frobenius norm, to a given square, input matrix.

## 2Specification

Fortran Interface
 Subroutine g02aaf ( g, ldg, n, x, ldx, iter,
 Integer, Intent (In) :: ldg, n, maxits, maxit, ldx Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iter, feval Real (Kind=nag_wp), Intent (In) :: errtol Real (Kind=nag_wp), Intent (Inout) :: g(ldg,n), x(ldx,n) Real (Kind=nag_wp), Intent (Out) :: nrmgrd
#include nagmk26.h
 void g02aaf_ (double g[], const Integer *ldg, const Integer *n, const double *errtol, const Integer *maxits, const Integer *maxit, double x[], const Integer *ldx, Integer *iter, Integer *feval, double *nrmgrd, Integer *ifail)

## 3Description

A correlation matrix may be characterised as a real square matrix that is symmetric, has a unit diagonal and is positive semidefinite.
g02aaf applies an inexact Newton method to a dual formulation of the problem, as described by Qi and Sun (2006). It applies the improvements suggested by Borsdorf and Higham (2010).

## 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
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

1:     $\mathbf{g}\left({\mathbf{ldg}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: $G$, the initial matrix.
On exit: a symmetric matrix $\frac{1}{2}\left(G+{G}^{\mathrm{T}}\right)$ with the diagonal set to $I$.
2:     $\mathbf{ldg}$ – IntegerInput
On entry: the first dimension of the array g as declared in the (sub)program from which g02aaf is called.
Constraint: ${\mathbf{ldg}}\ge {\mathbf{n}}$.
3:     $\mathbf{n}$ – IntegerInput
On entry: the size of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
4:     $\mathbf{errtol}$ – Real (Kind=nag_wp)Input
On entry: the termination tolerance for the Newton iteration. If ${\mathbf{errtol}}\le 0.0$ then  is used.
5:     $\mathbf{maxits}$ – IntegerInput
On entry: maxits specifies the maximum number of iterations used for the iterative scheme used to solve the linear algebraic equations at each Newton step.
If ${\mathbf{maxits}}\le 0$, $100$ is used.
6:     $\mathbf{maxit}$ – IntegerInput
On entry: specifies the maximum number of Newton iterations.
If ${\mathbf{maxit}}\le 0$, $200$ is used.
7:     $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: contains the nearest correlation matrix.
8:     $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02aaf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
9:     $\mathbf{iter}$ – IntegerOutput
On exit: the number of Newton steps taken.
10:   $\mathbf{feval}$ – IntegerOutput
On exit: the number of function evaluations of the dual problem.
11:   $\mathbf{nrmgrd}$ – Real (Kind=nag_wp)Output
On exit: the norm of the gradient of the last Newton step.
12:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ldg}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldg}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{ldx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=2$
Newton iteration fails to converge in $〈\mathit{\text{value}}〉$ iterations.
${\mathbf{ifail}}=3$
Machine precision is limiting convergence.
The array returned in x may still be of interest.
${\mathbf{ifail}}=4$
An intermediate eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The returned accuracy is controlled by errtol and limited by machine precision.

## 8Parallelism and Performance

g02aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02aaf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Arrays are internally allocated by g02aaf. The total size of these arrays is $11×{\mathbf{n}}+3×{\mathbf{n}}×{\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2×{\mathbf{n}}×{\mathbf{n}}+6×{\mathbf{n}}+1,120+9×{\mathbf{n}}\right)$ real elements and $5×{\mathbf{n}}+3$ integer elements.

## 10Example

This example finds the nearest correlation matrix to:
 $G = 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 2$

### 10.1Program Text

Program Text (g02aafe.f90)

### 10.2Program Data

Program Data (g02aafe.d)

### 10.3Program Results

Program Results (g02aafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017