g05yqf generates a quasi-random sequence from a Normal (Gaussian) distribution. Values are generated for a subset of dimensions. It must be preceded by a call to one of the initialization routines g05ylf or g05ynf.

2 Specification

Fortran Interface

Subroutine g05yqf (

sorder, n, xmean, std, fdim, ldim, quas, ldquas, iref, ifail)

Integer, Intent (In)	::	sorder, n, fdim, ldim, ldquas
Integer, Intent (Inout)	::	iref(*), ifail
Real (Kind=nag_wp), Intent (In)	::	xmean(), std()
Real (Kind=nag_wp), Intent (Inout)	::	quas(ldquas,*)

C Header Interface

#include <nag.h>

void	g05yqf_ (const Integer sorder, const Integer n, const double xmean[], const double std[], const Integer fdim, const Integer ldim, double quas[], const Integer ldquas, Integer iref[], Integer ifail)

The routine may be called by the names g05yqf or nagf_rand_quasi_normal_bydim.

3 Description

g05yqf generates a quasi-random sequence, for a specified subset of dimensions, from a Normal distribution by first generating a uniform quasi-random sequence, for the specified subset of dimensions, which is then transformed into a Normal sequence using the inverse of the Normal CDF. The type of uniform sequence used depends on the initialization routine called and can include the low-discrepancy sequences proposed by Sobol, Faure or Niederreiter. If the initialization routine g05ynf was used then the underlying uniform sequence is first scrambled prior to being transformed (see Section 3 in g05ynf 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 g05ylf or g05ynf
$liref = liref$ , the length of iref as supplied to the initialization routine g05ylf or g05ynf

1: $sorder$ – Integer Input

On entry: the order in which the generated values are returned.

Constraint:

sorder = 1

2

2: $n$ – 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

3: $xmean (*)$ – Real (Kind=nag_wp) array Input

Note: the dimension of the array xmean must be at least

idim

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

4: $std (*)$ – Real (Kind=nag_wp) array Input

Note: the dimension of the array std must be at least

idim

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

Constraint:

std (i) \geq 0.0

, for

i = 1, 2, \dots, idim

5: $fdim$ – Integer Input

On entry: the first dimension to return.

Constraint:

1 \leq fdim \leq ldim

6: $ldim$ – Integer Input

On entry: the last dimension to return.

Constraint:

ldim \leq idim

7: $quas (ldquas, *)$ – Real (Kind=nag_wp) array Output

Note: the second dimension of the array quas must be at least

ldim - fdim + 1

sorder = 1

and at least

n

sorder = 2

On exit: contains the n quasi-random numbers for the required dimensions of a sequence with idim dimensions.

For

i = 1, 2, \dots, n

j = fdim, fdim + 1, \dots, ldim

and

k = j - fdim + 1

, the

i

th value for the

j

th dimension is held in

$quas (i, k)$ , if $sorder = 1$ ,
$quas (k, i)$ , if $sorder = 2$ .

8: $ldquas$ – Integer Input

On entry: the first dimension of the array quas as declared in the (sub)program from which g05yqf is called.

Constraints:

if $sorder = 1$ , $ldquas \geq n$ ;
if $sorder = 2$ , $ldquas \geq ldim - fdim + 1$ .

9: $iref (*)$ – Integer array Communication Array

Note: the dimension 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.

10: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 11$: On entry, $sorder = ⟨ value ⟩$ .
Constraint: $sorder = 1$ or $2$ .

$ifail = 21$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = 22$: On entry, value of n would result in too many calls to the generator: $n = ⟨ value ⟩$ , generator has previously been called $⟨ value ⟩$ times.

$ifail = 41$: On entry, $i = ⟨ value ⟩$ and $std (i) = ⟨ value ⟩$ .
Constraint: $std (i) \geq 0.0$ .

$ifail = 61$: On entry, $fdim = ⟨ value ⟩$ and $ldim = ⟨ value ⟩$ .
Constraint: $1 \leq fdim \leq ldim$

$ifail = 71$: On entry, $ldim = ⟨ value ⟩$ and $idim = ⟨ value ⟩$ .
Constraint: $ldim \leq idim$

$ifail = 91$: On entry, $ldquas = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldquas \geq n$ .

$ifail = 92$: On entry, $ldquas = ⟨ value ⟩$ and $ldim - fdim + 1 = ⟨ value ⟩$ .
Constraint: $ldquas \geq ldim - fdim + 1$ .

$ifail = 111$: On entry, iref has either not been initialized or has been corrupted.

$ifail = 113$: On entry, iref is too short to use with g05yqf.

$ifail = 114$: On entry, the specified dimensions are out of sync.
A different number of values have been generated from at least one of the specified dimensions.

$ifail = 115$: g05yqf can not be used with the Faure generator.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g05yqf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g05yqf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example generates a sequence of

10

values from a Normal distribution with mean

3.1

and standard deviation

2.1

, from dimensions

3

7

of an

8

dimension Sobol sequence.

g05yq: FL CL CPP AD PY MB

NAG FL Interfaceg05yqf (quasi_​normal_​bydim)

▸▿ 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 FL Interface
g05yqf (quasi_normal_bydim)