# NAG FL Interfaceg02aaf (corrmat_​nearest)

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## 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 <nag.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)
The routine may be called by the names g02aaf or nagf_correg_corrmat_nearest.

## 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) array Input/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: $\mathbf{ldg}$Integer Input
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: $\mathbf{n}$Integer Input
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 ${\mathbf{errtol}}\le 0.0$, is used.
5: $\mathbf{maxits}$Integer Input
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: $\mathbf{maxit}$Integer Input
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) array Output
On exit: contains the nearest correlation matrix.
8: $\mathbf{ldx}$Integer Input
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: $\mathbf{iter}$Integer Output
On exit: the number of Newton steps taken.
10: $\mathbf{feval}$Integer Output
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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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.
${\mathbf{ifail}}=4$
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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)