void	nag_rand_matrix_multi_students_t (Nag_OrderType order, Nag_ModeRNG mode, Integer n, Integer df, Integer m, const double xmu[], const double c[], Integer pdc, double r[], Integer lr, Integer state[], double x[], Integer pdx, NagError *fail)

3 Description

When the covariance matrix is nonsingular (i.e., strictly positive definite), the distribution has probability density function

f (x) = \frac{Γ (\frac{(ν + m)}{2})}{{(π v)}^{m / 2} Γ (ν / 2) {|C|}^{\frac{1}{2}}} {[1 + \frac{{(x - a)}^{T} C^{- 1} (x - a)}{ν}]}^{\frac{- (ν + m)}{2}}

where

m

is the number of dimensions,

ν

is the degrees of freedom,

a

is the vector of means,

x

is the vector of positions and

\frac{ν}{ν - 2} C

is the covariance matrix.

The function returns the value

x = a + \sqrt{\frac{ν}{s}} z

where

z

is generated by nag_rand_normal (g05skc) from a Normal distribution with mean zero and covariance matrix

C

and

s

is generated by nag_rand_chi_sq (g05sdc) from a

χ^{2}

-distribution with

ν

degrees of freedom.

One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_matrix_multi_students_t (g05ryc).

4 References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford

5 Arguments

1: $order$ – 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

order = Nag_RowMajor

. See Section 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $mode$ – Nag_ModeRNGInput

On entry: a code for selecting the operation to be performed by the function.

$mode = Nag_InitializeReference$: Set up reference vector only.
$mode = Nag_GenerateFromReference$: Generate variates using reference vector set up in a prior call to nag_rand_matrix_multi_students_t (g05ryc).
$mode = Nag_InitializeAndGenerate$: Set up reference vector and generate variates.

Constraint:

mode = Nag_InitializeReference

Nag_GenerateFromReference

Nag_InitializeAndGenerate

3: $n$ – IntegerInput

On entry:

n

, the number of random variates required.

Constraint:

n \geq 0

4: $df$ – IntegerInput

On entry:

ν

, the number of degrees of freedom of the distribution.

Constraint:

df \geq 3

5: $m$ – IntegerInput

On entry:

m

, the number of dimensions of the distribution.

Constraint:

m > 0

6: $xmu [m]$ – const doubleInput

On entry:

a

, the vector of means of the distribution.

7: $c [\dim]$ – const doubleInput

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

pdc \times m

The

(i, j)

th element of the matrix

C

is stored in

$c [(j - 1) \times pdc + i - 1]$ when $order = Nag_ColMajor$ ;
$c [(i - 1) \times pdc + j - 1]$ when $order = Nag_RowMajor$ .

On entry: matrix which, along with df, defines the covariance of the distribution. Only the upper triangle need be set.

Constraint: c must be positive semidefinite to machine precision.

8: $pdc$ – IntegerInput

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

Constraint:

pdc \geq m

9: $r [lr]$ – doubleInput/Output

On entry: if

mode = Nag_GenerateFromReference

, the reference vector as set up by nag_rand_matrix_multi_students_t (g05ryc) in a previous call with

mode = Nag_InitializeReference

Nag_InitializeAndGenerate

On exit: if

mode = Nag_InitializeReference

Nag_InitializeAndGenerate

, the reference vector that can be used in subsequent calls to nag_rand_matrix_multi_students_t (g05ryc) with

mode = Nag_GenerateFromReference

10: $lr$ – IntegerInput

On entry: the dimension of the array r. If

mode = Nag_GenerateFromReference

, it must be the same as the value of lr specified in the prior call to nag_rand_matrix_multi_students_t (g05ryc) with

mode = Nag_InitializeReference

Nag_InitializeAndGenerate

Constraint:

lr \geq m \times (m + 1) + 2

11: $state [\dim]$ – IntegerCommunication Array

Note: the dimension,

\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.

12: $x [\dim]$ – doubleOutput

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

$\max (1, pdx \times m)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

Where

X (i, j)

appears in this document, it refers to the array element

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the array of pseudorandom multivariate Student's

t

vectors generated by the function, with

X (i, j)

holding the

j

th dimension for the

i

th variate.

13: $pdx$ – IntegerInput

On entry: the stride used in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq m$ .

14: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $df = 〈value〉$ .
Constraint: $df \geq 3$ .

On entry, lr is not large enough, $lr = 〈value〉$ : minimum length required $= 〈value〉$ .

On entry, $m = 〈value〉$ .
Constraint: $m > 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdc = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdc \geq m$ .

On entry, $pdx = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdx \geq m$ .

On entry, $pdx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdx \geq n$ .
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 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation 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 $m = 〈value〉$ and $m = 〈value〉$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

nag_rand_matrix_multi_students_t (g05ryc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_rand_matrix_multi_students_t (g05ryc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

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.

9 Further Comments

The time taken by nag_rand_matrix_multi_students_t (g05ryc) is of order

n m^{3}

It is recommended that the diagonal elements of

C

should not differ too widely in order of magnitude. This may be achieved by scaling the variables if necessary. The actual matrix decomposed is

C + E = L L^{T}

, where

E

is a diagonal matrix with small positive diagonal elements. This ensures that, even when

C

is singular, or nearly singular, the Cholesky factor

L

corresponds to a positive definite covariance matrix that agrees with

C

within machine precision.

10 Example

This example prints ten pseudorandom observations from a multivariate Student's

t

-distribution with ten degrees of freedom, means vector

[\begin{matrix} 1.0 \\ 2.0 \\ - 3.0 \\ 0.0 \end{matrix}]

and c matrix

[\begin{matrix} 1.69 & 0.39 & - 1.86 & 0.07 \\ 0.39 & 98.01 & - 7.07 & - 0.71 \\ - 1.86 & - 7.07 & 11.56 & 0.03 \\ 0.07 & - 0.71 & 0.03 & 0.01 \end{matrix}],

generated by nag_rand_matrix_multi_students_t (g05ryc). All ten observations are generated by a single call to nag_rand_matrix_multi_students_t (g05ryc) with