# NAG CL Interfaceg05tdc (int_​general)

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## 1Purpose

g05tdc generates a vector of pseudorandom integers from a discrete distribution with a given PDF (probability density function) or CDF (cumulative distribution function) $p$.

## 2Specification

 #include
 void g05tdc (Nag_ModeRNG mode, Integer n, const double p[], Integer np, Integer ip1, Nag_DiscreteDistrib itype, double r[], Integer lr, Integer state[], Integer x[], NagError *fail)
The function may be called by the names: g05tdc, nag_rand_int_general or nag_rand_gen_discrete.

## 3Description

g05tdc 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 g05tdc or may be combined in a single call.
One of the initialization functions g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to g05tdc.

## 4References

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: $\mathbf{mode}$Nag_ModeRNG Input
On entry: a code for selecting the operation to be performed by the function.
${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$
Set up reference vector only.
${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$
Generate variates using reference vector set up in a prior call to g05tdc.
${\mathbf{mode}}=\mathrm{Nag_InitializeAndGenerate}$
Set up reference vector and generate variates.
${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$
Generate variates without using the reference vector.
Constraint: ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$, $\mathrm{Nag_GenerateFromReference}$, $\mathrm{Nag_InitializeAndGenerate}$ or $\mathrm{Nag_GenerateWithoutReference}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{p}\left[{\mathbf{np}}\right]$const double Input
On entry: the PDF or CDF of the distribution.
Constraints:
• $0.0\le {\mathbf{p}}\left[\mathit{i}-1\right]\le 1.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{np}}$;
• if ${\mathbf{itype}}=\mathrm{Nag_PDF}$, $\sum _{\mathit{i}=1}^{{\mathbf{np}}}{\mathbf{p}}\left[\mathit{i}-1\right]=1.0$;
• if ${\mathbf{itype}}=\mathrm{Nag_CDF}$, ${\mathbf{p}}\left[\mathit{i}-1\right]<{\mathbf{p}}\left[j-1\right]\text{, ​}\mathit{i}.
4: $\mathbf{np}$Integer Input
On entry: the number of values supplied in p defining the PDF or CDF of the discrete distribution.
Constraint: ${\mathbf{np}}>0$.
5: $\mathbf{ip1}$Integer Input
On entry: the value of the variate, a whole number, to which the probability in ${\mathbf{p}}\left[0\right]$ corresponds.
6: $\mathbf{itype}$Nag_DiscreteDistrib Input
On entry: indicates the type of information contained in p.
${\mathbf{itype}}=\mathrm{Nag_PDF}$
p contains a probability distribution function (PDF).
${\mathbf{itype}}=\mathrm{Nag_CDF}$
p contains a cumulative distribution function (CDF).
Constraint: ${\mathbf{itype}}=\mathrm{Nag_PDF}$ or $\mathrm{Nag_CDF}$.
7: $\mathbf{r}\left[{\mathbf{lr}}\right]$double Communication Array
On entry: if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, the reference vector from the previous call to g05tdc.
On exit: the reference vector.
8: $\mathbf{lr}$Integer Input
On entry: the dimension of the array r.
Suggested values:
• if ${\mathbf{mode}}\ne \mathrm{Nag_GenerateWithoutReference}$, ${\mathbf{lr}}=10+1.4×{\mathbf{np}}$ approximately (for optimum efficiency in generating variates);
• otherwise ${\mathbf{lr}}=1$.
Constraints:
• if ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$, ${\mathbf{lr}}\ge {\mathbf{np}}+8$;
• if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, lr should remain unchanged from the previous call to g05tdc.
9: $\mathbf{state}\left[\mathit{dim}\right]$Integer Communication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
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 Output
On exit: contains $n$ pseudorandom numbers from the specified discrete distribution.
11: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, lr is too small when ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$: ${\mathbf{lr}}=⟨\mathit{\text{value}}⟩$, minimum length required $\text{}=⟨\mathit{\text{value}}⟩$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{np}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{np}}>0$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_PREV_CALL
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 ${\mathbf{ip1}}=⟨\mathit{\text{value}}⟩$.
NE_REAL_ARRAY
On entry, at least one element of the vector p is less than $0.0$ or greater than $1.0$.
On entry, ${\mathbf{itype}}=\mathrm{Nag_CDF}$ and the values of p are not all in stricly ascending order.
On entry, ${\mathbf{itype}}=\mathrm{Nag_PDF}$ and the sum of the elements of p do not equal one.
On entry, ${\mathbf{p}}\left[{\mathbf{np}}-1\right]=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{itype}}=\mathrm{Nag_CDF}$, ${\mathbf{p}}\left[{\mathbf{np}}-1\right]=1.0$.
NE_REF_VEC
On entry, some of the elements of the array r have been corrupted or have not been initialized.

Not applicable.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g05tdc 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 function. 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 g05tdc, after initialization by g05kfc.

### 10.1Program Text

Program Text (g05tdce.c)

None.

### 10.3Program Results

Program Results (g05tdce.r)