# NAG Library Routine Document

## 1Purpose

g02anf computes a correlation matrix, subject to preserving a leading principal submatrix and applying the smallest relative perturbation to the remainder of the approximate input matrix.

## 2Specification

Fortran Interface
 Subroutine g02anf ( g, ldg, n, k, x, ldx, iter, norm,
 Integer, Intent (In) :: ldg, n, k, ldx Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iter Real (Kind=nag_wp), Intent (In) :: errtol, eigtol Real (Kind=nag_wp), Intent (Inout) :: g(ldg,n), x(ldx,n) Real (Kind=nag_wp), Intent (Out) :: alpha, eigmin, norm
#include <nagmk26.h>
 void g02anf_ (double g[], const Integer *ldg, const Integer *n, const Integer *k, const double *errtol, const double *eigtol, double x[], const Integer *ldx, double *alpha, Integer *iter, double *eigmin, double *norm, Integer *ifail)

## 3Description

g02anf finds a correlation matrix, $X$, starting from an approximate correlation matrix, $G$, with positive definite leading principal submatrix of order $k$. The returned correlation matrix, $X$, has the following structure:
 $X = α A 0 0 I + 1-α G$
where $A$ is the $k$ by $k$ leading principal submatrix of the input matrix $G$ and positive definite, and $\alpha \in \left[0,1\right]$.
g02anf utilizes a shrinking method to find the minimum value of $\alpha$ such that $X$ is positive definite with unit diagonal.

## 4References

Higham N J, Strabić N and Šego V (2014) Restoring definiteness via shrinking, with an application to correlation matrices with a fixed block MIMS EPrint 2014.54 Manchester Institute for Mathematical Sciences, The University of Manchester, UK

## 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 g02anf 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:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the order of the leading principal submatrix $A$.
Constraint: ${\mathbf{n}}\ge {\mathbf{k}}>0$.
5:     $\mathbf{errtol}$ – Real (Kind=nag_wp)Input
On entry: the termination tolerance for the iteration.
If ${\mathbf{errtol}}\le 0$,  is used. See Section 7 for further details.
6:     $\mathbf{eigtol}$ – Real (Kind=nag_wp)Input
On entry: the tolerance used in determining the definiteness of $A$.
If ${\lambda }_{\mathrm{min}}\left(A\right)>{\mathbf{n}}×{\lambda }_{\mathrm{max}}\left(A\right)×{\mathbf{eigtol}}$, where ${\lambda }_{\mathrm{min}}\left(A\right)$ and ${\lambda }_{\mathrm{max}}\left(A\right)$ denote the minimum and maximum eigenvalues of $A$ respectively, $A$ is positive definite.
If ${\mathbf{eigtol}}\le 0$, machine precision is used.
7:     $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: contains the matrix $X$.
8:     $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02anf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
9:     $\mathbf{alpha}$ – Real (Kind=nag_wp)Output
On exit: $\alpha$.
10:   $\mathbf{iter}$ – IntegerOutput
On exit: the number of iterations taken.
11:   $\mathbf{eigmin}$ – Real (Kind=nag_wp)Output
On exit: the smallest eigenvalue of the leading principal submatrix $A$.
12:   $\mathbf{norm}$ – Real (Kind=nag_wp)Output
On exit: the value of ${‖G-X‖}_{F}$ after the final iteration.
13:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{ldg}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldg}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge {\mathbf{k}}>0$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{ldx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=5$
The $k$ by $k$ principal leading submatrix of the initial matrix $G$ is not positive definite.
${\mathbf{ifail}}=6$
${\mathbf{ifail}}=-99$
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 algorithm uses a bisection method. It is terminated when the computed $\alpha$ is within errtol of the minimum value. The positive definiteness of $X$ is such that it can be successfully factorized with a call to f07fdf (dpotrf).
The number of iterations taken for the bisection will be:
 $log21errtol .$

## 8Parallelism and Performance

g02anf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02anf 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 g02anf. The total size of these arrays does not exceed $2×{n}^{2}+3×n$ real elements. All allocated memory is freed before return of g02anf.

## 10Example

This example finds the smallest uniform perturbation $\alpha$ to $G$, such that the output is a correlation matrix and the $k$ by $k$ leading principal submatrix of the input is preserved,
 $G = 1.0000 -0.0991 0.5665 -0.5653 -0.3441 -0.0991 1.0000 -0.4273 0.8474 0.4975 0.5665 -0.4273 1.0000 -0.1837 -0.0585 -0.5653 0.8474 -0.1837 1.0000 -0.2713 -0.3441 0.4975 -0.0585 -0.2713 1.0000 .$

### 10.1Program Text

Program Text (g02anfe.f90)

### 10.2Program Data

Program Data (g02anfe.d)

### 10.3Program Results

Program Results (g02anfe.r)