# NAG Library Routine Document

## 1Purpose

g05tdf generates a vector of pseudorandom integers from a discrete distribution with a given PDF (probability density function) or CDF (cumulative distribution function) $p$.

## 2Specification

Fortran Interface
 Subroutine g05tdf ( mode, n, p, np, ip1, r, lr, x,
 Integer, Intent (In) :: mode, n, np, ip1, itype, lr Integer, Intent (Inout) :: state(*), ifail Integer, Intent (Out) :: x(n) Real (Kind=nag_wp), Intent (In) :: p(np) Real (Kind=nag_wp), Intent (Inout) :: r(lr)
#include nagmk26.h
 void g05tdf_ (const Integer *mode, const Integer *n, const double p[], const Integer *np, const Integer *ip1, const Integer *itype, double r[], const Integer *lr, Integer state[], Integer x[], Integer *ifail)

## 3Description

g05tdf generates a sequence of $n$ integers ${x}_{i}$, from a discrete distribution defined by information supplied in p. This may either be the PDF or CDF of the distribution. A reference vector is first set up to contain the CDF of the distribution in its higher elements, followed by an index.
Setting up the reference vector and subsequent generation of variates can each be performed by separate calls to g05tdf or may be combined in a single call.
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 g05tdf.
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

## 5Arguments

1:     $\mathbf{mode}$ – IntegerInput
On entry: a code for selecting the operation to be performed by the routine.
${\mathbf{mode}}=0$
Set up reference vector only.
${\mathbf{mode}}=1$
Generate variates using reference vector set up in a prior call to g05tdf.
${\mathbf{mode}}=2$
Set up reference vector and generate variates.
${\mathbf{mode}}=3$
Generate variates without using the reference vector.
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{p}\left({\mathbf{np}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the PDF or CDF of the distribution.
Constraints:
• $0.0\le {\mathbf{p}}\left(\mathit{i}\right)\le 1.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{np}}$;
• if ${\mathbf{itype}}=1$, $\sum _{\mathit{i}=1}^{{\mathbf{np}}}{\mathbf{p}}\left(\mathit{i}\right)=1.0$;
• if ${\mathbf{itype}}=2$, ${\mathbf{p}}\left(\mathit{i}\right)<{\mathbf{p}}\left(j\right)\text{, ​}\mathit{i}.
4:     $\mathbf{np}$ – IntegerInput
On entry: the number of values supplied in p defining the PDF or CDF of the discrete distribution.
Constraint: ${\mathbf{np}}>0$.
5:     $\mathbf{ip1}$ – IntegerInput
On entry: the value of the variate, a whole number, to which the probability in ${\mathbf{p}}\left(1\right)$ corresponds.
6:     $\mathbf{itype}$ – IntegerInput
On entry: indicates the type of information contained in p.
${\mathbf{itype}}=1$
p contains a probability distribution function (PDF).
${\mathbf{itype}}=2$
p contains a cumulative distribution function (CDF).
Constraint: ${\mathbf{itype}}=1$ or $2$.
7:     $\mathbf{r}\left({\mathbf{lr}}\right)$ – Real (Kind=nag_wp) arrayCommunication Array
On entry: if ${\mathbf{mode}}=1$, the reference vector from the previous call to g05tdf.
On exit: the reference vector.
8:     $\mathbf{lr}$ – IntegerInput
On entry: the dimension of the array r as declared in the (sub)program from which g05tdf is called.
Suggested values:
• if ${\mathbf{mode}}\ne 3$, ${\mathbf{lr}}=10+1.4×{\mathbf{np}}$ approximately (for optimum efficiency in generating variates);
• otherwise ${\mathbf{lr}}=1$.
Constraints:
• if ${\mathbf{mode}}=0$ or $2$, ${\mathbf{lr}}\ge {\mathbf{np}}+8$;
• if ${\mathbf{mode}}=1$, lr should remain unchanged from the previous call to g05tdf.
9:     $\mathbf{state}\left(*\right)$ – 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.
10:   $\mathbf{x}\left({\mathbf{n}}\right)$ – Integer arrayOutput
On exit: contains $n$ pseudorandom numbers from the specified discrete distribution.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{mode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mode}}=0$, $1$ or $2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, at least one element of the vector p is less than $0.0$ or greater than $1.0$.
On entry, ${\mathbf{itype}}=1$ and the sum of the elements of p do not equal one.
On entry, ${\mathbf{itype}}=2$ and the values of p are not all in stricly ascending order.
On entry, ${\mathbf{p}}\left({\mathbf{np}}\right)=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{itype}}=2$, ${\mathbf{p}}\left({\mathbf{np}}\right)=1.0$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{np}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{np}}>0$.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{itype}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{itype}}=1$ or $2$.
${\mathbf{ifail}}=7$
On entry, some of the elements of the array r have been corrupted or have not been initialized.
The value of np or ip1 is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{np}}=〈\mathit{\text{value}}〉$ and ${\mathbf{np}}=〈\mathit{\text{value}}〉$.
Previous value of ${\mathbf{ip1}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ip1}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=8$
On entry, lr is too small when ${\mathbf{mode}}=0$ or $2$: ${\mathbf{lr}}=〈\mathit{\text{value}}〉$, minimum length required $\text{}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=9$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

## 8Parallelism and Performance

g05tdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.

None.

## 10Example

This example prints $20$ pseudorandom variates from a discrete distribution whose PDF, $p$, is defined as follows:
 $n p -5 0.01 -4 0.02 -3 0.04 -2 0.08 -1 0.20 -0 0.30 -1 0.20 -2 0.08 -3 0.04 -4 0.02 -5 0.01$
The reference vector is set up and and the variates are generated by a single call to g05tdf, after initialization by g05kff.

### 10.1Program Text

Program Text (g05tdfe.f90)

### 10.2Program Data

Program Data (g05tdfe.d)

### 10.3Program Results

Program Results (g05tdfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017