g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_nearest_correlation_shrinking (g02anc)

## 1  Purpose

nag_nearest_correlation_shrinking (g02anc) computes a correlation matrix, subject to preserving a leading principle submatrix and applying the smallest uniform perturbation to the remainder of the approximate input matrix.

## 2  Specification

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

## 3  Description

nag_nearest_correlation_shrinking (g02anc) finds a correlation matrix, $X$, starting from an approximate correlation matrix, $G$, with positive definite leading principle 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 principle submatrix of the input matrix $G$ and positive definite, and $\alpha \in \left[0,1\right]$.
nag_nearest_correlation_shrinking (g02anc) utilizes a shrinking method to find the minimum value of $\alpha$ such that $X$ is positive definite with unit diagonal.

## 4  References

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

## 5  Arguments

1:    $\mathbf{g}\left[{\mathbf{pdg}}×{\mathbf{n}}\right]$doubleInput/Output
Note: the $\left(i,j\right)$th element of the matrix $G$ is stored in ${\mathbf{g}}\left[\left(j-1\right)×{\mathbf{pdg}}+i-1\right]$.
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:    $\mathbf{pdg}$IntegerInput
On entry: the stride separating matrix row elements 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:    $\mathbf{k}$IntegerInput
On entry: $k$, the order of the leading principle submatrix $A$.
Constraint: ${\mathbf{n}}\ge {\mathbf{k}}>0$.
5:    $\mathbf{errtol}$doubleInput
On entry: the termination tolerance for the iteration.
If ${\mathbf{errtol}}\le 0$,  is used. See Section 7 for further details.
6:    $\mathbf{eigtol}$doubleInput
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:    $\mathbf{x}\left[{\mathbf{pdx}}×{\mathbf{n}}\right]$doubleOutput
Note: the $\left(i,j\right)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$.
On exit: contains the matrix $X$.
8:    $\mathbf{pdx}$IntegerInput
On entry: the stride separating matrix row elements in the array x.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
9:    $\mathbf{alpha}$double *Output
On exit: $\alpha$.
10:  $\mathbf{iter}$Integer *Output
On exit: the number of iterations taken.
11:  $\mathbf{eigmin}$double *Output
On exit: the smallest eigenvalue of the leading principle submatrix $A$.
12:  $\mathbf{norm}$double *Output
On exit: the value of ${‖G-X‖}_{F}$ after the final iteration.
13:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction 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, ${\mathbf{k}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge {\mathbf{k}}>0$.
On entry, ${\mathbf{pdg}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdg}}\ge {\mathbf{n}}$.
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 3.6.6 in the Essential Introduction for further information.
NE_MAT_NOT_POS_DEF
The $k$-by-$k$ principle leading submatrix of the initial matrix $G$ is not positive definite.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

## 7  Accuracy

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 sucessfully factorized with a call to nag_dpotrf (f07fdc).
The number of interations taken for the bisection will be:
 $log21errtol .$

## 8  Parallelism and Performance

nag_nearest_correlation_shrinking (g02anc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nearest_correlation_shrinking (g02anc) 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_shrinking (g02anc). 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_shrinking (g02anc).

## 10  Example

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

### 10.1  Program Text

Program Text (g02ance.c)

### 10.2  Program Data

Program Data (g02ance.d)

### 10.3  Program Results

Program Results (g02ance.r)