void	nag_nearest_correlation_k_factor (Nag_OrderType order, double g[], Integer pdg, Integer n, Integer k, double errtol, Integer maxit, double x[], Integer pdx, Integer iter, Integer feval, double nrmpgd, NagError fail)

3

Description

A correlation matrix

C

with

k

-factor structure may be characterised as a real square matrix that is symmetric, has a unit diagonal, is positive semidefinite and can be written as

C = X X^{T} + diag (I - X X^{T})

, where

I

is the identity matrix and

X

has

n

rows and

k

columns.

X

is often referred to as the factor loading matrix.

nag_nearest_correlation_k_factor (g02aec) applies a spectral projected gradient method to the modified problem

{\min (‖G - X X^{T} + diag (X X^{T} - I)‖)}_{F}

such that

{‖x_{i}^{T}‖}_{2} \leq 1

, for

i = 1, 2, \dots, n

, where

x_{i}

is the

i

th row of the factor loading matrix,

X

, which gives us the solution.

4

References

Birgin E G, Martínez J M and Raydan M (2001) Algorithm 813: SPG–software for convex-constrained optimization ACM Trans. Math. Software 27 340–349

Borsdorf R, Higham N J and Raydan M (2010) Computing a nearest correlation matrix with factor structure. SIAM J. Matrix Anal. Appl. 31(5) 2603–2622

5

Arguments

1: $order$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $g [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array g must be at least

pdg \times n

On entry:

G

, the initial matrix.

On exit: a symmetric matrix

\frac{1}{2} (G + G^{T})

with the diagonal elements set to unity.

3: $pdg$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

G

in the array g.

Constraint:

pdg \geq n

4: $n$ – IntegerInput

On entry:

n

, the order of the matrix

G

Constraint:

n > 0

5: $k$ – IntegerInput

On entry:

k

, the number of factors and columns of

X

Constraint:

0 < k \leq n

6: $errtol$ – doubleInput

On entry: the termination tolerance for the projected gradient norm. See references for further details. If

errtol \leq 0.0

then

0.01

is used. This is often a suitable default value.

7: $maxit$ – IntegerInput

On entry: specifies the maximum number of iterations in the spectral projected gradient method.

maxit \leq 0

40000

is used.

8: $x [\dim]$ – doubleOutput

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times k)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

X

is stored in

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On exit: contains the matrix

X

9: $pdx$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq k$ .

10: $iter$ – Integer *Output

On exit: the number of steps taken in the spectral projected gradient method.

11: $feval$ – Integer *Output

On exit: the number of evaluations of

{‖G - X X^{T} + diag (X X^{T} - I)‖}_{F}

12: $nrmpgd$ – double *Output

On exit: the norm of the projected gradient at the final iteration.

13: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_CONVERGENCE: Spectral gradient method fails to converge in $〈value〉$ iterations.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .
NE_INT_2: On entry, $k = 〈value〉$ and $n = 〈value〉$ .
Constraint: $0 < k \leq n$ .

On entry, $pdg = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdg \geq n$ .

On entry, $pdx = 〈value〉$ and $k = 〈value〉$ .
Constraint: $pdx \geq k$ .

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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 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

nag_nearest_correlation_k_factor (g02aec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_nearest_correlation_k_factor (g02aec) 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 nag_nearest_correlation_k_factor (g02aec). The total size of these arrays is

n \times n + 4 \times n \times k + (nb + 3) \times n + n + 50

double elements and

6 \times n

Integer elements. There is an additional

n \times k

double elements if

order = Nag_RowMajor

. Here

nb

is the block size required for optimal performance by nag_dsytrd (f08fec) and nag_dormtr (f08fgc) which are called internally. All allocated memory is freed before return of nag_nearest_correlation_k_factor (g02aec).

See nag_mv_factor (g03cac) for constructing the factor loading matrix from a known correlation matrix.

10

Example

This example finds the nearest correlation matrix with

k = 2

factor structure to:

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

NAG Library Function Document

nag_nearest_correlation_k_factor (g02aec)

▸▿ 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

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