NAG Library Function Document

nag_nearest_correlation_h_weight (g02ajc)

+− 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

1 Purpose

nag_nearest_correlation_h_weight (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>

#include <nagg02.h>

void	nag_nearest_correlation_h_weight (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)

3 Description

nag_nearest_correlation_h_weight (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 (To appear) An inexact accelerated proximal gradient method for large scale linearly constrained convex SDP

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

On exit: $G$ is overwritten.
2: pdg – IntegerInput: On entry: the stride separating matrix row elements in the array g.

Constraint: $pdg \geq n$ .
3: n – IntegerInput: On entry: the order of the matrix $G$ .
Constraint: $n > 0$ .
4: alpha – doubleInput: On entry: the value of $α$ .
If $alpha < 0.0$ , $0.0$ is used.

Constraint: $alpha < 1.0$ .
5: h[ $pdh \times n$ ] – doubleInput/Output: Note: 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 – IntegerInput: On entry: the stride separating matrix row elements in the array h.

Constraint: $pdh \geq n$ .
7: errtol – doubleInput: On entry: the termination tolerance for the iteration. If $errtol \leq 0.0$ then $n \times \sqrt{machine precision}$ is used. See Section 7 for further details.
8: maxit – IntegerInput: On entry: specifies the maximum number of iterations to be used.
If $maxit \leq 0$ , $200$ is used.
9: x[ $pdx \times n$ ] – doubleOutput: Note: the $(i, j)$ th element of the matrix $X$ is stored in $x [(j - 1) \times pdx + i - 1]$ .
On exit: contains the nearest correlation matrix.
10: pdx – IntegerInput: On entry: the stride separating matrix row elements 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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_CONVERGENCE: Function fails 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.
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

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

nag_nearest_correlation_h_weight (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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

Arrays are internally allocated by nag_nearest_correlation_h_weight (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 nag_nearest_correlation_h_weight (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

NAG Library Function Documentnag_nearest_correlation_h_weight (g02ajc)