# NAG Library Routine Document

## 1Purpose

g02abf computes the nearest correlation matrix, in the Frobenius norm or weighted Frobenius norm, and optionally with bounds on the eigenvalues, to a given square, input matrix.

## 2Specification

Fortran Interface
 Subroutine g02abf ( g, ldg, n, opt, w, 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) :: alpha, errtol Real (Kind=nag_wp), Intent (Inout) :: g(ldg,n), w(n), x(ldx,n) Real (Kind=nag_wp), Intent (Out) :: nrmgrd Character (1), Intent (In) :: opt
#include nagmk26.h
 void g02abf_ (double g[], const Integer *ldg, const Integer *n, const char *opt, const double *alpha, double w[], const double *errtol, const Integer *maxits, const Integer *maxit, double x[], const Integer *ldx, Integer *iter, Integer *feval, double *nrmgrd, Integer *ifail, const Charlen length_opt)

## 3Description

Finds the nearest correlation matrix $X$ by minimizing $\frac{1}{2}{‖G-X‖}^{2}$ where $G$ is an approximate correlation matrix.
The norm can either be the Frobenius norm or the weighted Frobenius norm $\frac{1}{2}{‖{W}^{\frac{1}{2}}\left(G-X\right){W}^{\frac{1}{2}}‖}_{F}^{2}$.
You can optionally specify a lower bound on the eigenvalues, $\alpha$, of the computed correlation matrix, forcing the matrix to be positive definite, $0<\alpha <1$.
Note that if the weights vary by several orders of magnitude from one another the algorithm may fail to converge.

## 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: $G$ is overwritten.
2:     $\mathbf{ldg}$ – IntegerInput
On entry: the first dimension of the array g as declared in the (sub)program from which g02abf is called.
Constraint: ${\mathbf{ldg}}\ge {\mathbf{n}}$.
3:     $\mathbf{n}$ – IntegerInput
On entry: the order of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
4:     $\mathbf{opt}$ – Character(1)Input
On entry: indicates the problem to be solved.
${\mathbf{opt}}=\text{'A'}$
The lower bound problem is solved.
${\mathbf{opt}}=\text{'W'}$
The weighted norm problem is solved.
${\mathbf{opt}}=\text{'B'}$
Both problems are solved.
Constraint: ${\mathbf{opt}}=\text{'A'}$, $\text{'W'}$ or $\text{'B'}$.
5:     $\mathbf{alpha}$ – Real (Kind=nag_wp)Input
On entry: the value of $\alpha$.
If ${\mathbf{opt}}=\text{'W'}$, alpha need not be set.
Constraint: $0.0<{\mathbf{alpha}}<1.0$.
6:     $\mathbf{w}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the square roots of the diagonal elements of $W$, that is the diagonal of ${W}^{\frac{1}{2}}$.
If ${\mathbf{opt}}=\text{'A'}$, w is not referenced and need not be set.
On exit: if ${\mathbf{opt}}=\text{'W'}$ or $\text{'B'}$, the array is scaled so $0<{\mathbf{w}}\left(\mathit{i}\right)\le 1$, for $\mathit{i}=1,2,\dots ,n$.
Constraint: ${\mathbf{w}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,n$.
7:     $\mathbf{errtol}$ – Real (Kind=nag_wp)Input
On entry: the termination tolerance for the Newton iteration. If ${\mathbf{errtol}}\le 0.0$ then  is used.
8:     $\mathbf{maxits}$ – IntegerInput
On entry: specifies the maximum number of iterations to be used by the iterative scheme to solve the linear algebraic equations at each Newton step.
If ${\mathbf{maxits}}\le 0$, $2×{\mathbf{n}}$ is used.
9:     $\mathbf{maxit}$ – IntegerInput
On entry: specifies the maximum number of Newton iterations.
If ${\mathbf{maxit}}\le 0$, $200$ is used.
10:   $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: contains the nearest correlation matrix.
11:   $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02abf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
12:   $\mathbf{iter}$ – IntegerOutput
On exit: the number of Newton steps taken.
13:   $\mathbf{feval}$ – IntegerOutput
On exit: the number of function evaluations of the dual problem.
14:   $\mathbf{nrmgrd}$ – Real (Kind=nag_wp)Output
On exit: the norm of the gradient of the last Newton step.
15:   $\mathbf{ifail}$ – 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, all elements of w were not positive.
On entry, ${\mathbf{alpha}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<{\mathbf{alpha}}<1.0$.
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: ${\mathbf{n}}>0$.
On entry, ${\mathbf{opt}}\ne \text{'A'}$, $\text{'W'}$ or $\text{'B'}$.
${\mathbf{ifail}}=2$
Newton iteration fails to converge in $〈\mathit{\text{value}}〉$ iterations. Increase maxit or check the call to the routine.
${\mathbf{ifail}}=3$
The machine precision is limiting convergence. In this instance the returned value of x may be useful.
${\mathbf{ifail}}=4$
${\mathbf{ifail}}=-99$
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

g02abf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02abf 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 g02abf. The total size of these arrays is $12×{\mathbf{n}}+3×{\mathbf{n}}×{\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2×{\mathbf{n}}×{\mathbf{n}}+6×{\mathbf{n}}+1,120+9×{\mathbf{n}}\right)$ real elements and $5×{\mathbf{n}}+3$ integer elements. All allocated memory is freed before return of g02abf.

## 10Example

This example finds the nearest correlation matrix to:
 $G = 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 2$
weighted by ${W}^{\frac{1}{2}}=\mathrm{diag}\left(100,20,20,20\right)$ with minimum eigenvalue $0.02$.

### 10.1Program Text

Program Text (g02abfe.f90)

### 10.2Program Data

Program Data (g02abfe.d)

### 10.3Program Results

Program Results (g02abfe.r)

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