NAG Library Function Document

nag_rand_matrix_multi_normal (g05rzc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:    $\mathbf{order}$ – Nag_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 $\mathrm{Nag\_ColMajor}$.

2:    $\mathbf{mode}$ – Nag_ModeRNGInput

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 nag_rand_matrix_multi_normal (g05rzc).

$\mathbf{mode}=\mathrm{Nag\_InitializeAndGenerate}$

Set up reference vector and generate variates.

Constraint: $\mathbf{mode}=\mathrm{Nag\_InitializeReference}$, $\mathrm{Nag\_GenerateFromReference}$ or $\mathrm{Nag\_InitializeAndGenerate}$.

3:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of random variates required.

Constraint: $\mathbf{n}\ge 0$.

4:    $\mathbf{m}$ – IntegerInput

On entry: $m$, the number of dimensions of the distribution.

Constraint: $\mathbf{m}>0$.

5:    $\mathbf{xmu}\left[\mathbf{m}\right]$ – const doubleInput

On entry: $a$, the vector of means of the distribution.

6:    $\mathbf{c}\left[\mathit{dim}\right]$ – const doubleInput

Note: the dimension, dim, of the array c must be at least $\mathbf{pdc}\times \mathbf{m}$.

The $\left(i,j\right)$th element of the matrix $C$ is stored in

• $\mathbf{c}\left[\left(j-1\right)\times \mathbf{pdc}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{c}\left[\left(i-1\right)\times \mathbf{pdc}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On entry: the covariance matrix of the distribution. Only the upper triangle need be set.

Constraint: $C$ must be positive semidefinite to machine precision.

7:    $\mathbf{pdc}$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array c.

Constraint: $\mathbf{pdc}\ge \mathbf{m}$.

8:    $\mathbf{r}\left[\mathbf{lr}\right]$ – doubleInput/Output

On entry: if $\mathbf{mode}=\mathrm{Nag\_GenerateFromReference}$, the reference vector as set up by nag_rand_matrix_multi_normal (g05rzc) in a previous call with $\mathbf{mode}=\mathrm{Nag\_InitializeReference}$ or $\mathrm{Nag\_InitializeAndGenerate}$.

On exit: if $\mathbf{mode}=\mathrm{Nag\_InitializeReference}$ or $\mathrm{Nag\_InitializeAndGenerate}$, the reference vector that can be used in subsequent calls to nag_rand_matrix_multi_normal (g05rzc) with $\mathbf{mode}=\mathrm{Nag\_GenerateFromReference}$.

9:    $\mathbf{lr}$ – IntegerInput

On entry: the dimension of the array r. If $\mathbf{mode}=\mathrm{Nag\_GenerateFromReference}$, it must be the same as the value of lr specified in the prior call to nag_rand_matrix_multi_normal (g05rzc) with $\mathbf{mode}=\mathrm{Nag\_InitializeReference}$ or $\mathrm{Nag\_InitializeAndGenerate}$.

Constraint: $\mathbf{lr}\ge \mathbf{m}\times \left(\mathbf{m}+1\right)+1$.

10:  $\mathbf{state}\left[\mathit{dim}\right]$ – IntegerCommunication 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.

11:  $\mathbf{x}\left[\mathit{dim}\right]$ – doubleOutput

Note: the dimension, dim, of the array x must be at least

• $\max \left(1,\mathbf{pdx}\times \mathbf{m}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\mathbf{n}\times \mathbf{pdx}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

Where $\mathbf{X}\left(i,j\right)$ appears in this document, it refers to the array element

• $\mathbf{x}\left[\left(j-1\right)\times \mathbf{pdx}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{x}\left[\left(i-1\right)\times \mathbf{pdx}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On exit: the array of pseudorandom multivariate Normal vectors generated by the function, with $\mathbf{X}\left(i,j\right)$ holding the $j$th dimension for the $i$th variate.

12:  $\mathbf{pdx}$ – IntegerInput

On entry: the stride used in the array x.

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pdx}\ge \mathbf{n}$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdx}\ge \mathbf{m}$.

13:  $\mathbf{fail}$ – NagError *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.
See Section 3.2.1.2 in the Essential Introduction for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_INT

On entry, lr is not large enough, $\mathbf{lr}=〈\mathit{\text{value}}〉$: minimum length required $=〈\mathit{\text{value}}〉$.

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}>0$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

NE_INT_2

On entry, $\mathbf{pdc}=〈\mathit{\text{value}}〉$ and $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdc}\ge \mathbf{m}$.

On entry, $\mathbf{pdx}=〈\mathit{\text{value}}〉$ and $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdx}\ge \mathbf{m}$.

On entry, $\mathbf{pdx}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdx}\ge \mathbf{n}$.

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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction 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 3.6.5 in the Essential Introduction for further information.

NE_POS_DEF

On entry, the covariance matrix $C$ is not positive semidefinite to machine precision.

NE_PREV_CALL

m is not the same as when r was set up in a previous call.
Previous value of $\mathbf{m}=〈\mathit{\text{value}}〉$ and $\mathbf{m}=〈\mathit{\text{value}}〉$.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (g05rzce.c)

10.2  Program Data

10.3  Program Results

Program Results (g05rzce.r)