# NAG CL Interfaceg02apc (corrmat_​target)

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## 1Purpose

g02apc computes a correlation matrix, by using a positive definite target matrix derived from weighting the approximate input matrix, with an optional bound on the minimum eigenvalue.

## 2Specification

 #include
 void g02apc (double g[], Integer pdg, Integer n, double theta, double h[], Integer pdh, double errtol, double eigtol, double x[], Integer pdx, double *alpha, Integer *iter, double *eigmin, double *norm, NagError *fail)
The function may be called by the names: g02apc, nag_correg_corrmat_target or nag_nearest_correlation_target.

## 3Description

Starting from an approximate correlation matrix, $G$, g02apc finds a correlation matrix, $X$, which has the form
 $X = α T + (1-α) G ,$
where $\alpha \in \left[0,1\right]$ and $T=H\circ G$ is a target matrix. $C=A\circ B$ denotes the matrix $C$ with elements ${C}_{ij}={A}_{ij}×{B}_{ij}$. $H$ is a matrix of weights that defines the target matrix. The target matrix must be positive definite and thus have off-diagonal elements less than $1$ in magnitude. A value of $1$ in $H$ essentially fixes an element in $G$ so it is unchanged in $X$.
g02apc utilizes a shrinking method to find the minimum value of $\alpha$ such that $X$ is positive definite with unit diagonal and with a smallest eigenvalue of at least $\theta \in \left[0,1\right)$ times the smallest eigenvalue of the target matrix.

## 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: $\mathbf{g}\left[{\mathbf{pdg}}×{\mathbf{n}}\right]$double Input/Output
On entry: $G$, the initial matrix.
On exit: a symmetric matrix $\frac{1}{2}\left(G+{G}^{\mathrm{T}}\right)$ with the diagonal elements set to $1.0$.
2: $\mathbf{pdg}$Integer Input
On entry: the stride separating row elements of the matrix $G$ in the array g.
Constraint: ${\mathbf{pdg}}\ge {\mathbf{n}}$.
3: $\mathbf{n}$Integer Input
On entry: the order of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
4: $\mathbf{theta}$double Input
On entry: the value of $\theta$. If ${\mathbf{theta}}<0.0$, $0.0$ is used.
Constraint: ${\mathbf{theta}}<1.0$.
5: $\mathbf{h}\left[{\mathbf{pdh}}×{\mathbf{n}}\right]$double Input/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 $1.0$.
6: $\mathbf{pdh}$Integer Input
On entry: the stride separating matrix row elements in the array h.
Constraint: ${\mathbf{pdh}}\ge {\mathbf{n}}$.
7: $\mathbf{errtol}$double Input
On entry: the termination tolerance for the iteration.
If ${\mathbf{errtol}}\le 0.0$, is used. See Section 7 for further details.
8: $\mathbf{eigtol}$double Input
On entry: the tolerance used in determining the definiteness of the target matrix $T=H\circ G$.
If ${\lambda }_{\mathrm{min}}\left(T\right)>{\mathbf{n}}×{\lambda }_{\mathrm{max}}\left(T\right)×{\mathbf{eigtol}}$, where ${\lambda }_{\mathrm{min}}\left(T\right)$ and ${\lambda }_{\mathrm{max}}\left(T\right)$ denote the minimum and maximum eigenvalues of $T$ respectively, $T$ is positive definite.
If ${\mathbf{eigtol}}\le 0$, machine precision is used.
9: $\mathbf{x}\left[{\mathbf{pdx}}×{\mathbf{n}}\right]$double Output
On exit: contains the matrix $X$.
10: $\mathbf{pdx}$Integer Input
On entry: the stride separating row elements of the matrix $X$ in the array x.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
11: $\mathbf{alpha}$double * Output
On exit: the constant $\alpha$ used in the formation of $X$.
12: $\mathbf{iter}$Integer * Output
On exit: the number of iterations taken.
13: $\mathbf{eigmin}$double * Output
On exit: the smallest eigenvalue of the target matrix $T$.
14: $\mathbf{norm}$double * Output
On exit: the value of ${‖G-X‖}_{F}$ after the final iteration.
15: $\mathbf{fail}$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_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, ${\mathbf{pdh}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdh}}\ge {\mathbf{n}}$.
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_MAT_NOT_POS_DEF
The target matrix is not positive definite.
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.
NE_REAL
On entry, ${\mathbf{theta}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{theta}}<1.0$.

## 7Accuracy

The algorithm uses a bisection method. It is terminated when the computed $\alpha$ is within errtol of the minimum value.
Note: when $\theta$ is zero $X$ is still positive definite, in that it can be successfully factorized with a call to f07fdc.
The number of iterations taken for the bisection will be:
 $⌈log2(1errtol)⌉ .$

## 8Parallelism and Performance

g02apc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02apc 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 g02apc. The total size of these arrays does not exceed $2×{n}^{2}+3×n$ real elements. All allocated memory is freed before return of g02apc.

## 10Example

This example finds the smallest $\alpha$ such that $\alpha \left(H\circ G\right)+\left(1-\alpha \right)G$ is a correlation matrix. The $2×2$ leading principal submatrix of the input is preserved, and the last $2×2$ diagonal block is weighted to give some emphasis to the off diagonal elements.
 $G = ( 1.0000 -0.0991 0.5665 -0.5653 -0.3441 -0.0991 1.0000 -0.4273 0.8474 0.4975 0.5665 -0.4273 1.0000 -0.1837 -0.0585 -0.5653 0.8474 -0.1837 1.0000 -0.2713 -0.3441 0.4975 -0.0585 -0.2713 1.0000 )$
and
 $H = ( 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.5000 0.0000 0.0000 0.0000 0.5000 1.0000 ) .$

### 10.1Program Text

Program Text (g02apce.c)

### 10.2Program Data

Program Data (g02apce.d)

### 10.3Program Results

Program Results (g02apce.r)