NAG Library Routine Document

G05SMF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G05SMF generates a vector of pseudorandom numbers from a log-normal distribution with parameters

μ

and

σ^{2}

2 Specification

SUBROUTINE G05SMF ( N, XMU, VAR, STATE, X, IFAIL)

INTEGER N, STATE(*), IFAIL

REAL (KIND=nag_wp) XMU, VAR, X(N)

3 Description

The distribution has PDF (probability density function)

\begin{matrix} f (x) = \frac{1}{x σ \sqrt{2 π}} \exp (- \frac{{(\ln x - μ)}^{2}}{2 σ^{2}}) & if ​ x > 0, \\ f (x) = 0 & otherwise, \end{matrix}

i.e.,

\ln x

is normally distributed with mean

μ

and variance

σ^{2}

. G05SMF evaluates

\exp y_{i}

, where the

y_{i}

are generated by G05SKF from a Normal distribution with mean

μ

and variance

σ^{2}

, for

i = 1, 2, \dots, n

One of the initialization routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05SMF.

4 References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

5 Parameters

1: N – INTEGERInput: On entry: $n$ , the number of pseudorandom numbers to be generated.
Constraint: $N \geq 0$ .
2: XMU – REAL (KIND=nag_wp)Input: On entry: $μ$ , the mean of the distribution of $\ln x$ .
3: VAR – REAL (KIND=nag_wp)Input: On entry: $σ^{2}$ , the variance of the distribution of $\ln x$ .
Constraint: $VAR \geq 0.0$ .
4: STATE( $*$ ) – INTEGER arrayCommunication Array: Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.
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: X(N) – REAL (KIND=nag_wp) arrayOutput: On exit: the $n$ pseudorandom numbers from the specified log-normal distribution.
6: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . 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:

$IFAIL = 1$: On entry, $N < 0$ .

$IFAIL = 2$: On entry, unable to calculate $\exp (XMU)$ as XMU is too large.

$IFAIL = 3$: On entry, $VAR < 0.0$ .

$IFAIL = 4$: On entry, STATE vector was not initialized or has been corrupted.

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

This example prints five pseudorandom numbers from a log-normal distribution with mean

1.0

and variance

2.0

, generated by a single call to G05SMF, after initialization by G05KFF.

NAG Library Routine DocumentG05SMF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G05SMF