g02asc 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

#include <nag.h>

void	g02asc (double g[], Integer pdg, Integer n, double alpha, const Integer h[], Integer pdh, double errtol, Integer maxit, Integer m, double x[], Integer pdx, Integer its, double fnorm, NagError *fail)

The function may be called by the names: g02asc or nag_correg_corrmat_fixed.

3 Description

g02asc 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: $g [pdg \times n]$ – double Input/Output: Note: the $(i, j)$ th element of the matrix $G$ is stored in $g [(j - 1) \times pdg + i - 1]$ .

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: $pdg$ – Integer Input: On entry: the stride separating matrix row elements in the array g.

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

Constraint: $n > 0$ .
4: $alpha$ – double Input: On entry: the value of $α$ .
If $alpha < 0.0$ , a value of $0.0$ is used.

Constraint: $alpha < 1.0$ .
5: $h [pdh \times n]$ – const Integer Input: Note: the $(i, j)$ th element of the matrix $H$ is stored in $h [(j - 1) \times pdh + i - 1]$ .

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: $pdh$ – Integer Input: On entry: the stride separating matrix row elements in the array h.

Constraint: $pdh \geq n$ .
7: $errtol$ – double 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: $x [pdx \times n]$ – double Output: Note: the $(i, j)$ th element of the matrix $X$ is stored in $x [(j - 1) \times pdx + i - 1]$ .

On exit: contains the matrix $X$ .
11: $pdx$ – Integer Input: On entry: the stride separating matrix row elements in the array x.

Constraint: $pdx \geq n$ .
12: $its$ – Integer * Output: On exit: the number of iterations taken.
13: $fnorm$ – double * Output: On exit: the value of ${‖ G - X ‖}_{F}$ after the final iteration.
14: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALG_FAIL: Failure during Anderson acceleration.
Consider setting $m = 0$ and recomputing.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONVERGENCE: Function failed to converge in $⟨ value ⟩$ iterations.
A solution may not exist, however, try increasing maxit.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .
NE_INT_2: On entry, $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $m \leq n \times n$ .

On entry, $pdg = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdg \geq n$ .

On entry, $pdh = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdh \geq n$ .

On entry, $pdx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdx \geq n$ .

The fixed element $G_{i j}$ , lies outside the interval $[−1, 1]$ , for $i = ⟨ value ⟩$ and $j = ⟨ value ⟩$ .
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_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, $alpha = ⟨ value ⟩$ .
Constraint: $alpha < 1.0$ .

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 g02asc. 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 c05mdc 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.

g02asc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g02asc 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 g02asc. The total size of these arrays does not exceed

12 \times n^{2}

real elements. All allocated memory is freed before return of g02asc.

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 CL Interfaceg02asc (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 CL Interface
g02asc (corrmat_fixed)