g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_quasi_rand_lognormal (g05ykc)

## 1  Purpose

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

## 2  Specification

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

## 3  Description

nag_quasi_rand_lognormal (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 nag_quasi_init_scrambled (g05ync) was used then the underlying uniform sequence is first scrambled prior to being transformed (see Section 3 in nag_quasi_init_scrambled (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:
• $\mathit{idim}={\mathbf{idim}}$, the number of dimensions required, see nag_quasi_init (g05ylc) or nag_quasi_init_scrambled (g05ync);
• $\mathit{liref}={\mathbf{liref}}$, the length of iref as supplied to the initialization functions nag_quasi_init (g05ylc) or nag_quasi_init_scrambled (g05ync).
• $\mathit{tdquas}={\mathbf{n}}$ if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$; otherwise $\mathit{tdquas}=\mathit{idim}$
1:     orderNag_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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     xmean[$\mathit{idim}$]const doubleInput
On entry: specifies, for each dimension, the mean of the underlying Normal distribution.
Constraint: $\left|{\mathbf{xmean}}\left[\mathit{i}-1\right]\right|\le \left|-\mathrm{log}\left({\mathbf{nag_real_safe_small_number}}\right)-10.0×{\mathbf{std}}\left[\mathit{i}-1\right]\right|$, for $\mathit{i}=1,2,\dots ,\mathit{idim}$.
3:     std[$\mathit{idim}$]const doubleInput
On entry: specifies, for each dimension, the standard deviation of the underlying Normal distribution.
Constraint: ${\mathbf{std}}\left[\mathit{i}-1\right]\ge 0.0$, for $\mathit{i}=1,2,\dots ,\mathit{idim}$.
4:     nIntegerInput
On entry: the number of quasi-random numbers required.
Constraint: ${\mathbf{n}}\ge 0$ and ${\mathbf{n}}+\text{previous number of generated values}\le {2}^{31}-1$.
5:     quas[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array quas must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdquas}}×\mathit{idim}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdquas}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Where ${\mathbf{QUAS}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{quas}}\left[\left(j-1\right)×{\mathbf{pdquas}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{quas}}\left[\left(i-1\right)×{\mathbf{pdquas}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: ${\mathbf{QUAS}}\left(i,j\right)$ holds the $i$th value for the $j$th dimension.
6:     pdquasIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array quas.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdquas}}\ge {\mathbf{n}}$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdquas}}\ge \mathit{idim}$.
7:     iref[$\mathit{liref}$]IntegerCommunication Array
On entry: contains information on the current state of the sequence.
On exit: contains updated information on the state of the sequence.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INITIALIZATION
On entry, iref has either not been initialized or has been corrupted.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pdquas}}=〈\mathit{\text{value}}〉$ and $\mathit{idim}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdquas}}\ge \mathit{idim}$.
On entry, ${\mathbf{pdquas}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdquas}}\ge {\mathbf{n}}$.
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.
NE_REAL_ARRAY
On entry, at least one element of xmean is too large, ${\mathbf{xmean}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{xmean}}\left[i\right]\right|\le 〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{std}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{std}}\left[i\right]\ge 0$.
NE_TOO_MANY_CALLS
There have been too many calls to the generator.

Not applicable.

None.

## 9  Example

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

### 9.1  Program Text

Program Text (g05ykce.c)

None.

### 9.3  Program Results

Program Results (g05ykce.r)