# NAG Library Function Document

## 1Purpose

nag_nearest_correlation_shrinking (g02anc) 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

 #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)

## 3Description

nag_nearest_correlation_shrinking (g02anc) 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 $\alpha \in \left[0,1\right]$.
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.

## 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 set to $I$.
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:    $\mathbf{k}$IntegerInput
On entry: $k$, the order of the leading principal 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
On exit: contains the matrix $X$.
8:    $\mathbf{pdx}$IntegerInput
On entry: the stride separating column elements of the matrix $X$ 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 principal 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 3.7 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, ${\mathbf{k}}=〈\mathit{\text{value}}〉$ 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_MAT_NOT_POS_DEF
The $k$ by $k$ principal 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 2.7.5 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 nag_dpotrf (f07fdc).
The number of iterations taken for the bisection will be:
 $log21errtol .$

## 8Parallelism 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).

## 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 (g02ance.c)

### 10.2Program Data

Program Data (g02ance.d)

### 10.3Program Results

Program Results (g02ance.r)

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