# NAG Library Routine Document

## 1Purpose

g02apf computes a correlation matrix, by using a positive definite target matrix derived from weighting the approximate input matrix, with an optional bound on the minimum eigenvalue.

## 2Specification

Fortran Interface
 Subroutine g02apf ( g, ldg, n, h, ldh, x, ldx, iter, norm,
 Integer, Intent (In) :: ldg, n, ldh, ldx Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iter Real (Kind=nag_wp), Intent (In) :: theta, errtol, eigtol Real (Kind=nag_wp), Intent (Inout) :: g(ldg,n), h(ldh,n), x(ldx,n) Real (Kind=nag_wp), Intent (Out) :: alpha, eigmin, norm
#include nagmk26.h
 void g02apf_ (double g[], const Integer *ldg, const Integer *n, const double *theta, double h[], const Integer *ldh, const double *errtol, const double *eigtol, double x[], const Integer *ldx, double *alpha, Integer *iter, double *eigmin, double *norm, Integer *ifail)

## 3Description

Starting from an approximate correlation matrix, $G$, g02apf finds a correlation matrix, $X$, which has the form
 $X = α T + 1-α G ,$
where $\alpha \in \left[0,1\right]$ and $T=H\circ G$ is a target matrix. $C=A\circ B$ denotes the matrix $C$ with elements ${C}_{ij}={A}_{ij}×{B}_{ij}$. $H$ is a matrix of weights that defines the target matrix. The target matrix must be positive definite and thus have off-diagonal elements less than $1$ in magnitude. A value of $1$ in $H$ essentially fixes an element in $G$ so it is unchanged in $X$.
g02apf utilizes a shrinking method to find the minimum value of $\alpha$ such that $X$ is positive definite with unit diagonal and with a smallest eigenvalue of at least $\theta \in \left[0,1\right)$ times the smallest eigenvalue of the target matrix.
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 elements set to $1.0$.
2:     $\mathbf{ldg}$ – IntegerInput
On entry: the first dimension of the array g as declared in the (sub)program from which g02apf 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:     $\mathbf{theta}$ – Real (Kind=nag_wp)Input
On entry: the value of $\theta$. If ${\mathbf{theta}}<0.0$, $0.0$ is used.
Constraint: ${\mathbf{theta}}<1.0$.
5:     $\mathbf{h}\left({\mathbf{ldh}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the matrix of weights $H$.
On exit: a symmetric matrix $\frac{1}{2}\left(H+{H}^{\mathrm{T}}\right)$ with its diagonal elements set to $\mathrm{1.0}$.
6:     $\mathbf{ldh}$ – IntegerInput
On entry: the first dimension of the array h as declared in the (sub)program from which g02apf is called.
Constraint: ${\mathbf{ldh}}\ge {\mathbf{n}}$.
7:     $\mathbf{errtol}$ – 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.
8:     $\mathbf{eigtol}$ – Real (Kind=nag_wp)Input
On entry: the tolerance used in determining the definiteness of the target matrix $T=H\circ G$.
If ${\lambda }_{\mathrm{min}}\left(T\right)>{\mathbf{n}}×{\lambda }_{\mathrm{max}}\left(T\right)×{\mathbf{eigtol}}$, where ${\lambda }_{\mathrm{min}}\left(T\right)$ and ${\lambda }_{\mathrm{max}}\left(T\right)$ denote the minimum and maximum eigenvalues of $T$ respectively, $T$ is positive definite.
If ${\mathbf{eigtol}}\le 0$, machine precision is used.
9:     $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: contains the matrix $X$.
10:   $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02apf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
11:   $\mathbf{alpha}$ – Real (Kind=nag_wp)Output
On exit: the constant $\alpha$ used in the formation of $X$.
12:   $\mathbf{iter}$ – IntegerOutput
On exit: the number of iterations taken.
13:   $\mathbf{eigmin}$ – Real (Kind=nag_wp)Output
On exit: the smallest eigenvalue of the target matrix $T$.
14:   $\mathbf{norm}$ – Real (Kind=nag_wp)Output
On exit: the value of ${‖G-X‖}_{F}$ after the final iteration.
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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
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, ${\mathbf{theta}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{theta}}<1.0$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{ldh}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldh}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{ldx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=6$
The target matrix is not positive definite.
${\mathbf{ifail}}=7$
${\mathbf{ifail}}=-99$
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.
Note: when $\theta$ is zero $X$ is still positive definite, in 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

g02apf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02apf 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 g02apf. The total size of these arrays does not exceed $2×{n}^{2}+3×n$ real elements. All allocated memory is freed before return of g02apf.

## 10Example

This example finds the smallest $\alpha$ such that $\alpha \left(H\circ G\right)+\left(1-\alpha \right)G$ is a correlation matrix. The $2$ by $2$ leading principal submatrix of the input is preserved, and the last $2$ by $2$ diagonal block is weighted to give some emphasis to the off diagonal elements.
 $G = 1.0000 -0.0991 0.5665 -0.5653 -0.3441 -0.0991 1.0000 -0.4273 0.8474 0.4975 0.5665 -0.4273 1.0000 -0.1837 -0.0585 -0.5653 0.8474 -0.1837 1.0000 -0.2713 -0.3441 0.4975 -0.0585 -0.2713 1.0000$
and
 $H = 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.5000 0.0000 0.0000 0.0000 0.5000 1.0000 .$

### 10.1Program Text

Program Text (g02apfe.f90)

### 10.2Program Data

Program Data (g02apfe.d)

### 10.3Program Results

Program Results (g02apfe.r)

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