# NAG C Library Function Document

## 1Purpose

nag_nearest_correlation_h_weight (g02ajc) computes the nearest correlation matrix, using element-wise weighting in the Frobenius norm and optionally with bounds on the eigenvalues, to a given square, input matrix.

## 2Specification

 #include #include
 void nag_nearest_correlation_h_weight (double g[], Integer pdg, Integer n, double alpha, double h[], Integer pdh, double errtol, Integer maxit, double x[], Integer pdx, Integer *iter, double *norm, NagError *fail)

## 3Description

nag_nearest_correlation_h_weight (g02ajc) finds the nearest correlation matrix, $X$, to an approximate correlation matrix, $G$, using element-wise weighting, this minimizes ${‖H\circ \left(G-X\right)‖}_{F}$, where $C=A\circ B$ denotes the matrix $C$ with elements ${C}_{ij}={A}_{ij}×{B}_{ij}$.
You can optionally specify a lower bound on the eigenvalues, $\alpha$, of the computed correlation matrix, forcing the matrix to be strictly positive definite, if $0<\alpha <1$.
Zero elements in $H$ should be used when you wish to put no emphasis on the corresponding element of $G$. The algorithm scales $H$ so that the maximum element is $1$. It is this scaled matrix that is used in computing the norm above and for the stopping criteria described in Section 7.
Note that if the elements in $H$ vary by several orders of magnitude from one another the algorithm may fail to converge.

## 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
Jiang K, Sun D and Toh K-C (2012) An inexact accelerated proximal gradient method for large scale linearly constrained convex SDP SIAM J. Optim. 22(3) 1042–1064
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:    $\mathbf{g}\left[{\mathbf{pdg}}×{\mathbf{n}}\right]$doubleInput/Output
On entry: $G$, the initial matrix.
On exit: $G$ is overwritten.
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:    $\mathbf{alpha}$doubleInput
On entry: the value of $\alpha$.
If ${\mathbf{alpha}}<0.0$, $0.0$ is used.
Constraint: ${\mathbf{alpha}}<1.0$.
5:    $\mathbf{h}\left[{\mathbf{pdh}}×{\mathbf{n}}\right]$doubleInput/Output
Note: the $\left(i,j\right)$th element of the matrix $H$ is stored in ${\mathbf{h}}\left[\left(j-1\right)×{\mathbf{pdh}}+i-1\right]$.
On entry: the matrix of weights $H$.
On exit: a symmetric matrix $\frac{1}{2}\left(H+{H}^{\mathrm{T}}\right)$ with its diagonal elements set to zero and the remaining elements scaled so that the maximum element is $1.0$.
Constraint: $\mathit{H}\left[\left(\mathit{j}-1\right)×{\mathbf{pdh}}+\mathit{i}-1\right]\ge 0.0$, for all $i$ and $j=1,2,\dots ,n$, $i\ne j$.
6:    $\mathbf{pdh}$IntegerInput
On entry: the stride separating matrix row elements in the array h.
Constraint: ${\mathbf{pdh}}\ge {\mathbf{n}}$.
7:    $\mathbf{errtol}$doubleInput
On entry: the termination tolerance for the iteration. If ${\mathbf{errtol}}\le 0.0$ then  is used. See Section 7 for further details.
8:    $\mathbf{maxit}$IntegerInput
On entry: specifies the maximum number of iterations to be used.
If ${\mathbf{maxit}}\le 0$, $200$ is used.
9:    $\mathbf{x}\left[{\mathbf{pdx}}×{\mathbf{n}}\right]$doubleOutput
On exit: contains the nearest correlation matrix.
10:  $\mathbf{pdx}$IntegerInput
On entry: the stride separating column elements of the matrix $X$ in the array x.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
11:  $\mathbf{iter}$Integer *Output
On exit: the number of iterations taken.
12:  $\mathbf{norm}$double *Output
On exit: the value of ${‖H\circ \left(G-X\right)‖}_{F}$ 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.
NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
Function fails to converge in $〈\mathit{\text{value}}〉$ iterations.
Increase maxit or check the call to the function.
NE_EIGENPROBLEM
Failure to solve intermediate eigenproblem. 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: ${\mathbf{pdg}}\ge$ or ${\mathbf{n}}$.
On entry, ${\mathbf{pdh}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdh}}\ge$ or ${\mathbf{n}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge$ or ${\mathbf{n}}$.
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_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.
NE_REAL
On entry, ${\mathbf{alpha}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{alpha}}<1.0$.
NE_WEIGHTS_NOT_POSITIVE
On entry, one or more of the off-diagonal elements of $H$ were negative.

## 7Accuracy

The returned accuracy is controlled by errtol and limited by machine precision. If ${e}_{i}$ is the value of norm at the $i$th iteration, that is
 $ei = H∘G-XF ,$
where $H$ has been scaled as described above, then the algorithm terminates when:
 $ei-ei-1 1+ maxei,ei-1 ≤ errtol .$

## 8Parallelism and Performance

nag_nearest_correlation_h_weight (g02ajc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nearest_correlation_h_weight (g02ajc) 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.

## 9Further Comments

Arrays are internally allocated by nag_nearest_correlation_h_weight (g02ajc). The total size of these arrays is $15×{\mathbf{n}}+5×{\mathbf{n}}×{\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2×{\mathbf{n}}×{\mathbf{n}}+6×{\mathbf{n}}+1,120+9×{\mathbf{n}}\right)$ double elements and $5×{\mathbf{n}}+3$ Integer elements. All allocated memory is freed before return of nag_nearest_correlation_h_weight (g02ajc).

## 10Example

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:
 $H = 0.0 10.0 0.0 0.0 10.0 0.0 1.5 1.5 0.0 1.5 0.0 0.0 0.0 1.5 0.0 0.0$
with minimum eigenvalue $0.04$.

### 10.1Program Text

Program Text (g02ajce.c)

### 10.2Program Data

Program Data (g02ajce.d)

### 10.3Program Results

Program Results (g02ajce.r)