NAG Library Routine Document
G02AAF
1 Purpose
G02AAF computes the nearest correlation matrix, in the Frobenius norm, to a given square, input matrix.
2 Specification
SUBROUTINE G02AAF ( 
G, LDG, N, ERRTOL, MAXITS, MAXIT, X, LDX, ITER, FEVAL, NRMGRD, IFAIL) 
INTEGER 
LDG, N, MAXITS, MAXIT, LDX, ITER, FEVAL, IFAIL 
REAL (KIND=nag_wp) 
G(LDG,N), ERRTOL, X(LDX,N), NRMGRD 

3 Description
A correlation matrix may be characterised as a real square matrix that is symmetric, has a unit diagonal and is positive semidefinite.
G02AAF 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).
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: $\mathrm{G}\left({\mathbf{LDG}},{\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

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 $I$.
 2: $\mathrm{LDG}$ – INTEGERInput

On entry: the first dimension of the array
G as declared in the (sub)program from which G02AAF is called.
Constraint:
${\mathbf{LDG}}\ge {\mathbf{N}}$.
 3: $\mathrm{N}$ – INTEGERInput

On entry: the size of the matrix $G$.
Constraint:
${\mathbf{N}}>0$.
 4: $\mathrm{ERRTOL}$ – REAL (KIND=nag_wp)Input

On entry: the termination tolerance for the Newton iteration. If ${\mathbf{ERRTOL}}\le 0.0$ then ${\mathbf{N}}\times \sqrt{\mathit{machineprecision}}$ is used.
 5: $\mathrm{MAXITS}$ – 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 ${\mathbf{MAXITS}}\le 0$, $2\times {\mathbf{N}}$ is used.
 6: $\mathrm{MAXIT}$ – INTEGERInput

On entry: specifies the maximum number of Newton iterations.
If ${\mathbf{MAXIT}}\le 0$, $200$ is used.
 7: $\mathrm{X}\left({\mathbf{LDX}},{\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: contains the nearest correlation matrix.
 8: $\mathrm{LDX}$ – INTEGERInput

On entry: the first dimension of the array
X as declared in the (sub)program from which G02AAF is called.
Constraint:
${\mathbf{LDX}}\ge {\mathbf{N}}$.
 9: $\mathrm{ITER}$ – INTEGEROutput

On exit: the number of Newton steps taken.
 10: $\mathrm{FEVAL}$ – INTEGEROutput

On exit: the number of function evaluations of the dual problem.
 11: $\mathrm{NRMGRD}$ – REAL (KIND=nag_wp)Output

On exit: the norm of the gradient of the last Newton step.
 12: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

On entry, ${\mathbf{LDG}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{LDG}}\ge {\mathbf{N}}$.
On entry, ${\mathbf{LDX}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{LDX}}\ge {\mathbf{N}}$.
On entry, ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{N}}>0$.
 ${\mathbf{IFAIL}}=2$

Newton iteration fails to converge in $\u2329\mathit{\text{value}}\u232a$ iterations.
 ${\mathbf{IFAIL}}=3$

Machine precision is limiting convergence.
The array returned in
X may still be of interest.
 ${\mathbf{IFAIL}}=4$

An intermediate eigenproblem could not be solved. This should not occur. Please contact
NAG with details of your call.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7 Accuracy
The returned accuracy is controlled by
ERRTOL and limited by
machine precision.
8 Parallelism and Performance
G02AAF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
G02AAF 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 routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
Arrays are internally allocated by G02AAF. The total size of these arrays is $11\times {\mathbf{N}}+3\times {\mathbf{N}}\times {\mathbf{N}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2\times {\mathbf{N}}\times {\mathbf{N}}+6\times {\mathbf{N}}+1,120+9\times {\mathbf{N}}\right)$ real elements and $5\times {\mathbf{N}}+3$ integer elements.
10 Example
This example finds the nearest correlation matrix to:
10.1 Program Text
Program Text (g02aafe.f90)
10.2 Program Data
Program Data (g02aafe.d)
10.3 Program Results
Program Results (g02aafe.r)