G02 Chapter Contents
G02 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG02AAF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G02AAF computes the nearest correlation matrix, in the Frobenius norm, to a given square, input matrix.

## 2  Specification

 SUBROUTINE G02AAF ( G, LDG, N, ERRTOL, MAXITS, MAXIT, X, LDX, ITER, FEVAL, NRMGRD, IFAIL)
 INTEGER LDG, N, MAXITS, MAXIT, LDX, ITER, FEVAL, IFAIL REAL (KIND=nag_wp) G(LDG,N), ERRTOL, X(LDX,N), NRMGRD

## 3  Description

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).
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  Parameters

1:     G(LDG,N) – 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:     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:     N – INTEGERInput
On entry: the size of the matrix $G$.
Constraint: ${\mathbf{N}}>0$.
4:     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:     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 ${\mathbf{MAXITS}}\le 0$, $2×{\mathbf{N}}$ is used.
6:     MAXIT – INTEGERInput
On entry: specifies the maximum number of Newton iterations.
If ${\mathbf{MAXIT}}\le 0$, $200$ is used.
7:     X(LDX,N) – REAL (KIND=nag_wp) arrayOutput
On exit: contains the nearest correlation matrix.
8:     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:     ITER – INTEGEROutput
On exit: the number of Newton steps taken.
10:   FEVAL – INTEGEROutput
On exit: the number of function evaluations of the dual problem.
11:   NRMGRD – REAL (KIND=nag_wp)Output
On exit: the norm of the gradient of the last Newton step.
12:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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, ${\mathbf{N}}\le 0$, or ${\mathbf{LDG}}<{\mathbf{N}}$, or ${\mathbf{LDX}}<{\mathbf{N}}$.
${\mathbf{IFAIL}}=2$
The routine fails to converge in MAXIT iterations. Increase MAXIT or check the call to the routine.
${\mathbf{IFAIL}}=3$
Machine precision is limiting convergence. In this instance the returned value of X may be useful.
${\mathbf{IFAIL}}=4$
${\mathbf{IFAIL}}=-999$
Internal memory allocation failed.

## 7  Accuracy

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

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.

## 9  Example

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

### 9.1  Program Text

Program Text (g02aafe.f90)

### 9.2  Program Data

Program Data (g02aafe.d)

### 9.3  Program Results

Program Results (g02aafe.r)