g02asf computes the nearest correlation matrix, in the Frobenius norm, while fixing elements and optionally with bounds on the eigenvalues, to a given square input matrix.

2 Specification

Fortran Interface

Subroutine g02asf (

g, ldg, n, alpha, h, ldh, errtol, maxit, m, x, ldx, its, fnorm, ifail)

Integer, Intent (In)	::	ldg, n, h(ldh,n), ldh, maxit, m, ldx
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	its
Real (Kind=nag_wp), Intent (In)	::	alpha, errtol
Real (Kind=nag_wp), Intent (Inout)	::	g(ldg,n), x(ldx,n)
Real (Kind=nag_wp), Intent (Out)	::	fnorm

C Header Interface

#include <nag.h>

void	g02asf_ (double g[], const Integer ldg, const Integer n, const double alpha, const Integer h[], const Integer ldh, const double errtol, const Integer maxit, const Integer m, double x[], const Integer ldx, Integer its, double fnorm, Integer *ifail)

The routine may be called by the names g02asf or nagf_correg_corrmat_fixed.

3 Description

g02asf finds the nearest correlation matrix,

X

, to a matrix,

G

, in the Frobenius norm. It uses an alternating projections algorithm with Anderson acceleration. Elements in the input matrix can be fixed by supplying the value

1

in the corresponding element of the matrix

H

. However, note that the algorithm may fail to converge if the fixed elements do not form part of a valid correlation matrix. You can optionally specify a lower bound,

α

, on the eigenvalues of the computed correlation matrix, forcing the matrix to be positive definite with

0 \leq α < 1

4 References

Anderson D G (1965) Iterative Procedures for Nonlinear Integral Equations J. Assoc. Comput. Mach. 12 547–560

Higham N J and Strabić N (2016) Anderson acceleration of the alternating projections method for computing the nearest correlation matrix Numer. Algor. 72 1021–1042

5 Arguments

1: $g (ldg, n)$ – Real (Kind=nag_wp) array Input/Output: On entry: $\tilde{G}$ , the initial matrix.

On exit: the symmetric matrix $G = \frac{1}{2} (\tilde{G} + {\tilde{G}}^{T})$ with the diagonal elements set to $1.0$ .
2: $ldg$ – Integer Input: On entry: the first dimension of the array g as declared in the (sub)program from which g02asf is called.

Constraint: $ldg \geq n$ .
3: $n$ – Integer Input: On entry: the order of the matrix $G$ .

Constraint: $n > 0$ .
4: $alpha$ – Real (Kind=nag_wp) Input: On entry: the value of $α$ .
If $alpha < 0.0$ , a value of $0.0$ is used.

Constraint: $alpha < 1.0$ .
5: $h (ldh, n)$ – Integer array Input: On entry: the symmetric matrix $H$ . If an element of $H$ is $1$ then the corresponding element in $G$ is fixed in the output $X$ . Only the strictly lower triangular part of $H$ need be set.
6: $ldh$ – Integer Input: On entry: the first dimension of the array h as declared in the (sub)program from which g02asf is called.

Constraint: $ldh \geq n$ .
7: $errtol$ – Real (Kind=nag_wp) Input: On entry: the termination tolerance for the iteration.
If $errtol \leq 0.0$ , $\sqrt{machine precision}$ is used. See Section 7 for further details.
8: $maxit$ – Integer Input: On entry: specifies the maximum number of iterations.
If $maxit \leq 0$ , a value of $200$ is used.
9: $m$ – Integer Input: On entry: the number of previous iterates to use in the Anderson acceleration. If $m = 0$ , Anderson acceleration is not used. See Section 7 for further details.
If $m < 0$ , a value of $4$ is used.

Constraint: $m \leq n \times n$ .
10: $x (ldx, n)$ – Real (Kind=nag_wp) array Output: On exit: contains the matrix $X$ .
11: $ldx$ – Integer Input: On entry: the first dimension of the array x as declared in the (sub)program from which g02asf is called.

Constraint: $ldx \geq n$ .
12: $its$ – Integer Output: On exit: the number of iterations taken.
13: $fnorm$ – Real (Kind=nag_wp) Output: On exit: the value of ${‖ G - X ‖}_{F}$ after the final iteration.
14: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .

$ifail = 2$: On entry, $ldg = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldg \geq n$ .

$ifail = 3$: On entry, $ldh = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldh \geq n$ .

$ifail = 4$: On entry, $alpha = ⟨ value ⟩$ .
Constraint: $alpha < 1.0$ .

$ifail = 5$: On entry, $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $m \leq n \times n$ .

$ifail = 6$: On entry, $ldx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldx \geq n$ .

$ifail = 7$: Routine failed to converge in $⟨ value ⟩$ iterations.
A solution may not exist, however, try increasing maxit.

$ifail = 8$: Failure during Anderson acceleration.
Consider setting $m = 0$ and recomputing.

$ifail = 9$: The fixed element $G_{i j}$ , lies outside the interval $[−1, 1]$ , for $i = ⟨ value ⟩$ and $j = ⟨ value ⟩$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Alternating projections is an iterative process where at each iteration the new iterate,

X_{k}

, can be improved by using Anderson acceleration to reduce the overall number of iterations. The alternating projections algorithm terminates at the

k

th iteration when

\frac{{‖ X_{k} - Y_{k} ‖}_{F}}{{‖ X_{k} ‖}_{F}} \leq errtol

where

Y_{k}

is the result of the first of two projections computed at each step.

Without Anderson acceleration this algorithm is guaranteed to converge. There is no theoretical guarantee of convergence of Anderson acceleration and, therefore, when it is used, no guarantee of convergence of g02asf. However, in practice it can be seen to significantly reduce the number of alternating projection iterations. Anderson acceleration is not used when m is set to zero. See c05mdf and Higham and Strabić (2016) and Anderson (1965) for further information.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g02asf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g02asf 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 implementation-specific information.

9 Further Comments

Arrays are internally allocated by g02asf. The total size of these arrays does not exceed

12 \times n^{2}

real elements. All allocated memory is freed before return of g02asf.

10 Example

This example finds the nearest correlation matrix,

X

, to the input,

G

, whilst fixing two diagonal blocks as given by

H

. The minimum eigenvalue of

X

is stipulated to be

0.04

G = (\begin{array}{r} 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 \end{array})

and

H = (\begin{matrix} 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \end{matrix}) .

Only the strictly lower half of

H

is supplied in the example.

g02as: FL CL CPP AD PY MB

NAG FL Interfaceg02asf (corrmat_​fixed)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g02asf (corrmat_fixed)