g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_nearest_correlation (g02aac)

## 1  Purpose

nag_nearest_correlation (g02aac) computes the nearest correlation matrix, in the Frobenius norm, to a given square, input matrix.

## 2  Specification

 #include #include
 void nag_nearest_correlation (Nag_OrderType order, double g[], Integer pdg, Integer n, double errtol, Integer maxits, Integer maxit, double x[], Integer pdx, Integer *iter, Integer *feval, double *nrmgrd, NagError *fail)

## 3  Description

A correlation matrix may be characterised as a real square matrix that is symmetric, has a unit diagonal and is positive semidefinite.
nag_nearest_correlation (g02aac) applies an inexact Newton method to a dual formulation of the problem, as described by Qi and Sun (2006). It applies the improvements suggested by Borsdorf and Higham (2010).
Borsdorf R and Higham N J (2010) A preconditioned (Newton) algorithm for the nearest correlation matrix IMA Journal of Numerical Analysis 30(1) 94–107
Qi H and Sun D (2006) A quadratically convergent Newton method for computing the nearest correlation matrix SIAM J. Matrix AnalAppl 29(2) 360–385

## 5  Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{g}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array g must be at least ${\mathbf{pdg}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $G$ is stored in
• ${\mathbf{g}}\left[\left(j-1\right)×{\mathbf{pdg}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{g}}\left[\left(i-1\right)×{\mathbf{pdg}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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 $I$.
3:    $\mathbf{pdg}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array g.
Constraint: ${\mathbf{pdg}}\ge {\mathbf{n}}$.
4:    $\mathbf{n}$IntegerInput
On entry: the size of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
5:    $\mathbf{errtol}$doubleInput
On entry: the termination tolerance for the Newton iteration. If ${\mathbf{errtol}}\le 0.0$ then  is used.
6:    $\mathbf{maxits}$IntegerInput
On entry: maxits specifies the maximum number of iterations used for the iterative scheme used to solve the linear algebraic equations at each Newton step.
If ${\mathbf{maxits}}\le 0$, $2×{\mathbf{n}}$ is used.
7:    $\mathbf{maxit}$IntegerInput
On entry: specifies the maximum number of Newton iterations.
If ${\mathbf{maxit}}\le 0$, $200$ is used.
8:    $\mathbf{x}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array x must be at least ${\mathbf{pdx}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $X$ is stored in
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: contains the nearest correlation matrix.
9:    $\mathbf{pdx}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
10:  $\mathbf{iter}$Integer *Output
On exit: the number of Newton steps taken.
11:  $\mathbf{feval}$Integer *Output
On exit: the number of function evaluations of the dual problem.
12:  $\mathbf{nrmgrd}$double *Output
On exit: the norm of the gradient of the last Newton step.
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_CONVERGENCE
Machine precision is limiting convergence.
The array returned in x may still be of interest.
Newton iteration fails to converge in $〈\mathit{\text{value}}〉$ iterations.
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, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ 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_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 returned accuracy is controlled by errtol and limited by machine precision.

## 8  Parallelism and Performance

nag_nearest_correlation (g02aac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nearest_correlation (g02aac) 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 (g02aac). The total size of these arrays is $11×{\mathbf{n}}+3×{\mathbf{n}}×{\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2×{\mathbf{n}}×{\mathbf{n}}+6×{\mathbf{n}}+1,120+9×{\mathbf{n}}\right)$ real elements and $5×{\mathbf{n}}+3$ integer elements.

## 10  Example

This example finds the nearest correlation matrix to:
 $G = 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 2$

### 10.1  Program Text

Program Text (g02aace.c)

None.

### 10.3  Program Results

Program Results (g02aace.r)