g05tdf generates a vector of pseudorandom integers from a discrete distribution with a given PDF (probability density function) or CDF (cumulative distribution function) $p$.
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.
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
Arguments
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 ${\mathbf{itype}}=2$, ${\mathbf{p}}\left(\mathit{i}\right)<{\mathbf{p}}\left(j\right)\text{,}\mathit{i}<j\text{ and}{\mathbf{p}}\left({\mathbf{np}}\right)=1.0$.
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 ${\mathbf{mode}}\ne 3$, ${\mathbf{lr}}=10+1.4\times {\mathbf{np}}$ 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.
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).
6
Error 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, ${\mathbf{mode}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{mode}}=0$, $1$ or $2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
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, ${\mathbf{p}}\left({\mathbf{np}}\right)=\u2329\mathit{\text{value}}\u232a$.
Constraint: if ${\mathbf{itype}}=2$, ${\mathbf{p}}\left({\mathbf{np}}\right)=1.0$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{np}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{np}}>0$.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{itype}}=\u2329\mathit{\text{value}}\u232a$.
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 ${\mathbf{np}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{np}}=\u2329\mathit{\text{value}}\u232a$.
Previous value of ${\mathbf{ip1}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{ip1}}=\u2329\mathit{\text{value}}\u232a$.
${\mathbf{ifail}}=8$
On entry, lr is too small when ${\mathbf{mode}}=0$ or $2$: ${\mathbf{lr}}=\u2329\mathit{\text{value}}\u232a$, minimum length required $\text{}=\u2329\mathit{\text{value}}\u232a$.
${\mathbf{ifail}}=9$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
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.
7
Accuracy
Not applicable.
8
Parallelism 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.
9
Further Comments
None.
10
Example
This example prints $20$ pseudorandom variates from a discrete distribution whose PDF, $p$, is defined as follows: