g05ykc generates a quasi-random sequence from a log-normal distribution. It must be preceded by a call to one of the initialization functions g05ylc or g05ync.

2 Specification

#include <nag.h>

void	g05ykc (Nag_OrderType order, const double xmean[], const double std[], Integer n, double quas[], Integer pdquas, Integer iref[], NagError *fail)

The function may be called by the names: g05ykc, nag_rand_quasi_lognormal or nag_quasi_rand_lognormal.

3 Description

g05ykc generates a quasi-random sequence from a log-normal distribution by first generating a uniform quasi-random sequence which is then transformed into a log-normal sequence using the exponential of the inverse of the Normal CDF. The type of uniform sequence used depends on the initialization function called and can include the low-discrepancy sequences proposed by Sobol, Faure or Niederreiter. If the initialization function g05ync was used then the underlying uniform sequence is first scrambled prior to being transformed (see Section 3 in g05ync for details).

4 References

Bratley P and Fox B L (1988) Algorithm 659: implementing Sobol's quasirandom sequence generator ACM Trans. Math. Software 14(1) 88–100

Fox B L (1986) Algorithm 647: implementation and relative efficiency of quasirandom sequence generators ACM Trans. Math. Software 12(4) 362–376

Wichura (1988) Algorithm AS 241: the percentage points of the Normal distribution Appl. Statist. 37 477–484

5 Arguments

Note: the following variables are used in the parameter descriptions:

$idim = idim$ , the number of dimensions required, see g05ylc or g05ync;
$liref = liref$ , the length of iref as supplied to the initialization functions g05ylc or g05ync.

1: $order$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $xmean [\dim]$ – const double Input

On entry: specifies, for each dimension, the mean of the underlying Normal distribution.

Constraint:

| xmean [i - 1] | \leq | - \log (nag_real_safe_small_number) - 10.0 \times std [i - 1] |

, for

i = 1, 2, \dots, idim

3: $std [\dim]$ – const double Input

On entry: specifies, for each dimension, the standard deviation of the underlying Normal distribution.

Constraint:

std [i - 1] \geq 0.0

, for

i = 1, 2, \dots, idim

4: $n$ – Integer Input

On entry: the number of quasi-random numbers required.

Constraint:

n \geq 0

and

n + previous number of generated values \leq 2^{31} - 1

5: $quas [\dim]$ – double Output

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

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

where

QUAS (i, j)

appears in this document, it refers to the array element

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

On exit:

QUAS (i, j)

holds the

i

th value for the

j

th dimension.

6: $pdquas$ – Integer Input

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

Constraints:

if $order = Nag_ColMajor$ , $pdquas \geq n$ ;
if $order = Nag_RowMajor$ , $pdquas \geq idim$ .

7: $iref [\dim]$ – Integer Communication Array

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

liref

On entry: contains information on the current state of the sequence.

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

8: $fail$ – 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_INITIALIZATION: On entry, iref has either not been initialized or has been corrupted.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdquas = ⟨ value ⟩$ and $idim = ⟨ value ⟩$ .
Constraint: $pdquas \geq idim$ .

On entry, $pdquas = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdquas \geq n$ .
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_ARRAY: On entry, $i = ⟨ value ⟩$ and $std [i - 1] = ⟨ value ⟩$ .
Constraint: $std [i - 1] \geq 0$ .

On entry, $i = ⟨ value ⟩$ and $xmean [i - 1] = ⟨ value ⟩$ .
Constraint: $| xmean [i - 1] | \leq ⟨ value ⟩$ .
NE_TOO_MANY_CALLS: There have been too many calls to the generator.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g05ykc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g05ykc 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.

The Sobol, Sobol (A659) and Niederreiter quasi-random number generators in g05ykc have been parallelized, but require quite large problem sizes to see any significant performance gain. Parallelism is only enabled when

order = Nag_ColMajor

. The Faure generator is serial.

9 Further Comments

None.

10 Example

This example calls g05ylc to initialize the generator and then g05ykc to produce a sequence of five four-dimensional quasi-random numbers variates.