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.

2 Specification

#include <nag.h>

void	g02ajc (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)

The function may be called by the names: g02ajc, nag_correg_corrmat_h_weight or nag_nearest_correlation_h_weight.

3 Description

g02ajc finds the nearest correlation matrix,

X

, to an approximate correlation matrix,

G

, using element-wise weighting, this minimizes

{‖ H \circ (G - X) ‖}_{F}

, where

C = A \circ B

denotes the matrix

C

with elements

C_{i j} = A_{i j} \times B_{i j}

You can optionally specify a lower bound on the eigenvalues,

α

, of the computed correlation matrix, forcing the matrix to be strictly positive definite, if

0 < α < 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.

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

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

5 Arguments

1: $g [\dim]$ – double Input/Output: Note: the dimension, dim, of the array g must be at least $pdg \times n$ .

On entry: $G$ , the initial matrix.

On exit: $G$ is overwritten.
2: $pdg$ – Integer Input: On entry: the stride separating row elements of the matrix $G$ 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 $alpha < 0.0$ , $0.0$ is used.

Constraint: $alpha < 1.0$ .
5: $h [\dim]$ – double Input/Output: Note: the dimension, dim, of the array h must be at least $pdh \times n$ .

The $(i, j)$ th element of the matrix $H$ is stored in $h [(j - 1) \times pdh + i - 1]$ .

On entry: the matrix of weights $H$ .

On exit: a symmetric matrix $\frac{1}{2} (H + H^{T})$ with its diagonal elements set to zero and the remaining elements scaled so that the maximum element is $1.0$ .

Constraint: $H [(j - 1) \times pdh + i - 1] \geq 0.0$ , for all $i$ and $j = 1, 2, \dots, n$ , $i \neq j$ .
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 $errtol \leq 0.0$ , $n \times \sqrt{machine precision}$ is used. See Section 7 for further details.
8: $maxit$ – Integer Input: On entry: specifies the maximum number of iterations to be used.
If $maxit \leq 0$ , $200$ is used.
9: $x [\dim]$ – double Output: Note: the dimension, dim, of the array x must be at least $pdx \times n$ .

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

Constraint: $pdx \geq n$ .
11: $iter$ – Integer * Output: On exit: the number of iterations taken.
12: $norm$ – double * Output: On exit: the value of ${‖ H \circ (G - X) ‖}_{F}$ after the final iteration.
13: $fail$ – 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_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.
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, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .
NE_INT_2: 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: $pdx \geq n$ .
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: $alpha < 1.0$ .
NE_WEIGHTS_NOT_POSITIVE: On entry, one or more of the off-diagonal elements of $H$ were negative.

7 Accuracy

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

e_{i} = {‖ H \circ (G - X) ‖}_{F},

where

H

has been scaled as described above, then the algorithm terminates when:

\frac{| e_{i} - e_{i - 1} |}{1 + \max (e_{i}, e_{i - 1})} \leq errtol .

8 Parallelism and Performance

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

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.

9 Further Comments

Arrays are internally allocated by g02ajc. The total size of these arrays is

15 \times n + 5 \times n \times n + \max (2 \times n \times n + 6 \times n + 1, 120 + 9 \times n)

double elements and

5 \times n + 3

Integer elements. All allocated memory is freed before return of g02ajc.

10 Example

This example finds the nearest correlation matrix to:

G = (\begin{array}{r} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 2 \end{array})

weighted by:

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

with minimum eigenvalue

0.04

g02aj: FL CL CPP AD

NAG CL Interfaceg02ajc (corrmat_​h_​weight)

▸▿ 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
g02ajc (corrmat_h_weight)