# NAG Library Function Document

## 1Purpose

nag_nearest_correlation_target (g02apc) 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

 #include #include
 void nag_nearest_correlation_target (double g[], Integer pdg, Integer n, double theta, double h[], Integer pdh, double errtol, double eigtol, double x[], Integer pdx, double *alpha, Integer *iter, double *eigmin, double *norm, NagError *fail)

## 3Description

Starting from an approximate correlation matrix, $G$, nag_nearest_correlation_target (g02apc) 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 $X$.
nag_nearest_correlation_target (g02apc) 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.

## 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:    $\mathbf{g}\left[{\mathbf{pdg}}×{\mathbf{n}}\right]$doubleInput/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:    $\mathbf{pdg}$IntegerInput
On entry: the stride separating column elements of the matrix $G$ in the array g.
Constraint: ${\mathbf{pdg}}\ge {\mathbf{n}}$.
3:    $\mathbf{n}$IntegerInput
On entry: the order of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
4:    $\mathbf{theta}$doubleInput
On entry: the value of $\theta$. If ${\mathbf{theta}}<0.0$, $0.0$ is used.
Constraint: ${\mathbf{theta}}<1.0$.
5:    $\mathbf{h}\left[{\mathbf{pdh}}×{\mathbf{n}}\right]$doubleInput/Output
Note: the $\left(i,j\right)$th element of the matrix $H$ is stored in ${\mathbf{h}}\left[\left(j-1\right)×{\mathbf{pdh}}+i-1\right]$.
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:    $\mathbf{pdh}$IntegerInput
On entry: the stride separating matrix row elements in the array h.
Constraint: ${\mathbf{pdh}}\ge {\mathbf{n}}$.
7:    $\mathbf{errtol}$doubleInput
On entry: the termination tolerance for the iteration.
If ${\mathbf{errtol}}\le 0$,  is used. See Section 7 for further details.
8:    $\mathbf{eigtol}$doubleInput
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:    $\mathbf{x}\left[{\mathbf{pdx}}×{\mathbf{n}}\right]$doubleOutput
On exit: contains the matrix $X$.
10:  $\mathbf{pdx}$IntegerInput
On entry: the stride separating column elements of the matrix $X$ in the array x.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
11:  $\mathbf{alpha}$double *Output
On exit: the constant $\alpha$ used in the formation of $X$.
12:  $\mathbf{iter}$Integer *Output
On exit: the number of iterations taken.
13:  $\mathbf{eigmin}$double *Output
On exit: the smallest eigenvalue of the target matrix $T$.
14:  $\mathbf{norm}$double *Output
On exit: the value of ${‖G-X‖}_{F}$ after the final iteration.
15:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_EIGENPROBLEM
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
NE_INT_2
On entry, ${\mathbf{pdg}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdg}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdh}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdh}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_MAT_NOT_POS_DEF
The target matrix is not positive definite.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{theta}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{theta}}<1.0$.

## 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 nag_dpotrf (f07fdc).
The number of iterations taken for the bisection will be:
 $log21errtol .$

## 8Parallelism and Performance

nag_nearest_correlation_target (g02apc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nearest_correlation_target (g02apc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Arrays are internally allocated by nag_nearest_correlation_target (g02apc). The total size of these arrays does not exceed $2×{n}^{2}+3×n$ real elements. All allocated memory is freed before return of nag_nearest_correlation_target (g02apc).

## 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 (g02apce.c)

### 10.2Program Data

Program Data (g02apce.d)

### 10.3Program Results

Program Results (g02apce.r)

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