are constructed from such pairs, chosen at random, to produce a unit diagonal element corresponding to the first element. This is repeated until all diagonal elements are

1

to within a given tolerance

ε

The randomness of

C

should be interpreted only to the extent that

A

is a random orthogonal matrix and

C

is computed from

A

using the

P_{i}

which are chosen as arbitrarily as possible.

One of the initialization functions g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to g05pyc.

4 References

Lin S P and Bendel R B (1985) Algorithm AS 213: Generation of population correlation on matrices with specified eigenvalues Appl. Statist. 34 193–198

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the dimension of the correlation matrix to be generated.

Constraint:

n \geq 1

2: $d [n]$ – const double Input

On entry: the

n

eigenvalues,

λ_{i}

, for

i = 1, 2, \dots, n

Constraints:

$d [i - 1] \geq 0.0$ , for $i = 1, 2, \dots, n$ ;
$\sum_{i = 1}^{n} d [i - 1] = n$ to within eps.

3: $eps$ – double Input

On entry: the maximum acceptable error in the diagonal elements.

Suggested value:

eps = 0.00001

Constraint:

eps \geq n \times machine precision

(see Chapter X02).

4: $state [\dim]$ – Integer Communication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

5: $c [n \times pdc]$ – double Output

On exit: a random correlation matrix,

C

, of dimension

n

6: $pdc$ – Integer Input

On entry: the stride separating column elements of the matrix

C

in the array c.

Constraint:

pdc \geq n

7: $fail$ – 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_DIAG_ELEMENTS: The diagonals of the returned matrix are not unity, try increasing the value of eps, or rerun the code using a different seed.
NE_EIGVAL_SUM: On entry, the eigenvalues do not sum to n.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

On entry, $pdc = ⟨ value ⟩$ .
Constraint: $pdc > 0$ .
NE_INT_2: On entry, $pdc = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdc \geq n$ .
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_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
NE_NEGATIVE_EIGVAL: On entry, an eigenvalue is negative.
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, $eps = ⟨ value ⟩$ .
Constraint: $eps \geq n \times machine precision$ .

7 Accuracy

The maximum error in a diagonal element is given by eps.

8 Parallelism and Performance

g05pyc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g05pyc 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

The time taken by g05pyc is approximately proportional to

n^{2}

10 Example

Following initialization of the pseudorandom number generator by a call to g05kfc, a

3 \times 3

correlation matrix with eigenvalues of

0.7

0.9

and

1.4

is generated and printed.

g05py: FL CL CPP AD

NAG CL Interfaceg05pyc (matrix_​corr)

▸▿ 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
g05pyc (matrix_corr)