g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_nearest_correlation_bounded (g02abc)

## 1  Purpose

nag_nearest_correlation_bounded (g02abc) computes the nearest correlation matrix, in the Frobenius norm or weighted Frobenius norm, and optionally with bounds on the eigenvalues, to a given square, input matrix.

## 2  Specification

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

## 3  Description

Finds the nearest correlation matrix $X$ by minimizing $\frac{1}{2}{‖G-X‖}^{2}$ where $G$ is an approximate correlation matrix.
The norm can either be the Frobenius norm or the weighted Frobenius norm $\frac{1}{2}{‖{W}^{\frac{1}{2}}\left(G-X\right){W}^{\frac{1}{2}}‖}_{F}^{2}$.
You can optionally specify a lower bound on the eigenvalues, $\alpha$, of the computed correlation matrix, forcing the matrix to be positive definite, $0<\alpha <1$.
Note that if the weights vary by several orders of magnitude from one another the algorithm may fail to converge.

## 4  References

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:    $\mathbf{g}\left[{\mathbf{pdg}}×{\mathbf{n}}\right]$doubleInput/Output
Note: the $\left(i,j\right)$th element of the matrix $G$ is stored in
• ${\mathbf{g}}\left[\left(j-1\right)×{\mathbf{pdg}}+i-1\right]$ 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: $G$ is overwritten.
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 order of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
5:    $\mathbf{opt}$Nag_NearCorr_ProbTypeInput
On entry: indicates the problem to be solved.
${\mathbf{opt}}=\mathrm{Nag_LowerBound}$
The lower bound problem is solved.
${\mathbf{opt}}=\mathrm{Nag_WeightedNorm}$
The weighted norm problem is solved.
${\mathbf{opt}}=\mathrm{Nag_Both}$
Both problems are solved.
Constraint: ${\mathbf{opt}}=\mathrm{Nag_LowerBound}$, $\mathrm{Nag_WeightedNorm}$ or $\mathrm{Nag_Both}$.
6:    $\mathbf{alpha}$doubleInput
On entry: the value of $\alpha$.
If ${\mathbf{opt}}=\mathrm{Nag_WeightedNorm}$, alpha need not be set.
Constraint: $0.0<{\mathbf{alpha}}<1.0$.
7:    $\mathbf{w}\left[{\mathbf{n}}\right]$doubleInput/Output
On entry: the square roots of the diagonal elements of $W$, that is the diagonal of ${W}^{\frac{1}{2}}$.
If ${\mathbf{opt}}=\mathrm{Nag_LowerBound}$, w is not referenced and may be NULL.
On exit: if ${\mathbf{opt}}=\mathrm{Nag_WeightedNorm}$ or $\mathrm{Nag_Both}$, the array is scaled so $0<{\mathbf{w}}\left[\mathit{i}-1\right]\le 1$, for $\mathit{i}=1,2,\dots ,n$.
Constraint: ${\mathbf{w}}\left[\mathit{i}-1\right]>0.0$, for $\mathit{i}=1,2,\dots ,n$.
8:    $\mathbf{errtol}$doubleInput
On entry: the termination tolerance for the Newton iteration. If ${\mathbf{errtol}}\le 0.0$ then  is used.
9:    $\mathbf{maxits}$IntegerInput
On entry: specifies the maximum number of iterations to be used by the iterative scheme to solve the linear algebraic equations at each Newton step.
If ${\mathbf{maxits}}\le 0$, $2×{\mathbf{n}}$ is used.
10:  $\mathbf{maxit}$IntegerInput
On entry: specifies the maximum number of Newton iterations.
If ${\mathbf{maxit}}\le 0$, $200$ is used.
11:  $\mathbf{x}\left[{\mathbf{pdx}}×{\mathbf{n}}\right]$doubleOutput
Note: the $\left(i,j\right)$th element of the matrix $X$ is stored in
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ 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.
12:  $\mathbf{pdx}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
13:  $\mathbf{iter}$Integer *Output
On exit: the number of Newton steps taken.
14:  $\mathbf{feval}$Integer *Output
On exit: the number of function evaluations of the dual problem.
15:  $\mathbf{nrmgrd}$double *Output
On exit: the norm of the gradient of the last Newton step.
16:  $\mathbf{fail}$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
Newton iteration fails to converge in $〈\mathit{\text{value}}〉$ iterations. Increase maxit or check the call to the function.
The machine precision is limiting convergence. In this instance the returned value of x may be useful.
NE_EIGENPROBLEM
An intermediate eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.
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.
An unexpected error has been triggered by this function. Please contact NAG.
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.
NE_REAL
On entry, ${\mathbf{alpha}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<{\mathbf{alpha}}<1.0$.
NE_WEIGHTS_NOT_POSITIVE
On entry, all elements of w were not positive.

## 7  Accuracy

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

## 8  Parallelism and Performance

nag_nearest_correlation_bounded (g02abc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nearest_correlation_bounded (g02abc) 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.

## 9  Further Comments

Arrays are internally allocated by nag_nearest_correlation_bounded (g02abc). The total size of these arrays is $12×{\mathbf{n}}+3×{\mathbf{n}}×{\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2×{\mathbf{n}}×{\mathbf{n}}+6×{\mathbf{n}}+1,120+9×{\mathbf{n}}\right)$ double elements and $5×{\mathbf{n}}+3$ Integer elements. All allocated memory is freed before return of nag_nearest_correlation_bounded (g02abc).

## 10  Example

This example finds the nearest correlation matrix to:
 $G = 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 2$
weighted by ${W}^{\frac{1}{2}}=\mathrm{diag}\left(100,20,20,20\right)$ with minimum eigenvalue $0.02$.

### 10.1  Program Text

Program Text (g02abce.c)

### 10.2  Program Data

Program Data (g02abce.d)

### 10.3  Program Results

Program Results (g02abce.r)