# NAG CL Interfaceg02aac (corrmat_​nearest)

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## 1Purpose

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

## 2Specification

 #include
 void g02aac (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)
The function may be called by the names: g02aac, nag_correg_corrmat_nearest or nag_nearest_correlation.

## 3Description

A correlation matrix may be characterised as a real square matrix that is symmetric, has a unit diagonal and is positive semidefinite.
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).

## 4References

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

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{g}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array g must be at least ${\mathbf{pdg}}×{\mathbf{n}}$.
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: $\mathbf{pdg}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $G$ in the array g.
Constraint: ${\mathbf{pdg}}\ge {\mathbf{n}}$.
4: $\mathbf{n}$Integer Input
On entry: the size of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
5: $\mathbf{errtol}$double Input
On entry: the termination tolerance for the Newton iteration. If ${\mathbf{errtol}}\le 0.0$, is used.
6: $\mathbf{maxits}$Integer Input
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 ${\mathbf{maxits}}\le 0$, $100$ is used.
7: $\mathbf{maxit}$Integer Input
On entry: specifies the maximum number of Newton iterations.
If ${\mathbf{maxit}}\le 0$, $200$ is used.
8: $\mathbf{x}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array x must be at least ${\mathbf{pdx}}×{\mathbf{n}}$.
On exit: contains the nearest correlation matrix.
9: $\mathbf{pdx}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $X$ 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 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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: ${\mathbf{n}}>0$.
On entry, ${\mathbf{pdg}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdg}}>0$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}>0$.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

The returned accuracy is controlled by errtol and limited by machine precision.

## 8Parallelism and Performance

g02aac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 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.
It is important to note that the nearest correlation matrix, to the symmetrized matrix supplied, may not be positive definite. That is, at least one of the eigenvalues of the returned correlation matrix may be zero relative to the accuracy requested. Numerically, in such cases, computing the eigenvalues of the returned correlation matrix may yield small and negative eigenvalues; reducing the tolerance used will reduce the size (closer to zero) of such eigenvalues, but some computed eigenvalues may remain negative.
To ensure a correlation matrix that is positive definite with small positive eigenvalues, you can call f07fdc on a copy of the returned correlation matrix ($C={\mathbf{x}}$, say). If ${\mathbf{info}}>0$ then the leading matrix of order $r={\mathbf{info}}-1$ of x is positive definite. x and $r$ can then be passed as arguments g and k to g02anc to obtain a positive definite correlation matrix (with some very small positive eigenvalues) that is very close to the matrix returned by g02aac.
Alternatively, a correlation matrix with zero eigenvalues (to the accuracy required) suggests that some (${\mathbf{n}}-r$) of the correlated variables are redundant and should be considered for removal from the data model.

## 10Example

This example finds the nearest correlation matrix to the matrix $G$, where
 $G = ( 2 −1 0 0 −1 2 −1 0 0 −1 2 −1 0 0 −1 2 ) .$
The example also demonstrates how a modified Cholesky factorization (computed using f01mdc and f01mec) can be used to obtain a bound on the distance to the nearest correlation matrix prior to computing the NCM itself.

### 10.1Program Text

Program Text (g02aace.c)

### 10.2Program Data

Program Data (g02aace.d)

### 10.3Program Results

Program Results (g02aace.r)