NAG Library Function Document

nag_multi_normal_pdf_vector (g01lbc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{ilog}$ – Nag_BooleanInput

On entry: the value of ilog determines whether the logarithmic value is returned in PDF.

$\mathbf{ilog}=\mathrm{Nag\_FALSE}$

$f\left(X:\mu ,\Sigma \right)$, the probability density function is returned.

$\mathbf{ilog}=\mathrm{Nag\_TRUE}$

$\log \left(f\left(X:\mu ,\Sigma \right)\right)$, the logarithm of the probability density function is returned.

2:    $\mathbf{k}$ – IntegerInput

On entry: $k$, the number of points the PDF is to be evaluated at.

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

3:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of dimensions.

Constraint: $\mathbf{n}\ge 2$.

4:    $\mathbf{x}\left[\mathit{dim}\right]$ – const doubleInput

Note: the dimension, dim, of the array x must be at least $\mathbf{pdx}\times \mathbf{k}$.

Where $\mathbf{X}\left(i,j\right)$ appears in this document, it refers to the array element $\mathbf{x}\left[\left(j-1\right)\times \mathbf{pdx}+i-1\right]$.

On entry: $X$, the matrix of $k$ points at which to evaluate the probability density function, with the $i$th dimension for the $j$th point held in $\mathbf{X}\left(i,j\right)$.

5:    $\mathbf{pdx}$ – IntegerInput

On entry: the stride separating matrix row elements in the array x.

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

6:    $\mathbf{xmu}\left[\mathbf{n}\right]$ – const doubleInput

On entry: $\mu$, the mean vector of the multivariate Normal distribution.

7:    $\mathbf{iuld}$ – Nag_MatrixTypeInput

On entry: indicates the form of $\Sigma$ and how it is stored in sig.

$\mathbf{iuld}=\mathrm{Nag\_LowerMatrix}$

sig holds the lower triangular portion of $\Sigma$.

$\mathbf{iuld}=\mathrm{Nag\_UpperMatrix}$

sig holds the upper triangular portion of $\Sigma$.

$\mathbf{iuld}=\mathrm{Nag\_DiagonalMatrix}$

$\Sigma$ is a diagonal matrix and sig only holds the diagonal elements.

$\mathbf{iuld}=\mathrm{Nag\_LowerFactored}$

sig holds the lower Cholesky decomposition, $L$ such that $L{L}^{\mathrm{T}}=\Sigma$.

$\mathbf{iuld}=\mathrm{Nag\_UpperFactored}$

sig holds the upper Cholesky decomposition, $U$ such that $U^{\mathrm{T}}U=\Sigma$.

Constraint: $\mathbf{iuld}=\mathrm{Nag\_LowerMatrix}$, $\mathrm{Nag\_UpperMatrix}$, $\mathrm{Nag\_DiagonalMatrix}$, $\mathrm{Nag\_LowerFactored}$ or $\mathrm{Nag\_UpperFactored}$.

8:    $\mathbf{sig}\left[\mathit{dim}\right]$ – const doubleInput

Note: the dimension, dim, of the array sig must be at least $\mathbf{pdsig}\times \mathbf{n}$.

Where $\mathbf{SIG}\left(i,j\right)$ appears in this document, it refers to the array element $\mathbf{sig}\left[\left(j-1\right)\times \mathbf{pdsig}+i-1\right]$.

On entry: information defining the variance-covariance matrix, $\Sigma$.

$\mathbf{iuld}=\mathrm{Nag\_LowerMatrix}$ or $\mathrm{Nag\_UpperMatrix}$

sig must hold the lower or upper portion of $\Sigma$, with ${\Sigma }_{ij}$ held in $\mathbf{SIG}\left(i,j\right)$. The supplied variance-covariance matrix must be positive semidefinite.

$\mathbf{iuld}=\mathrm{Nag\_DiagonalMatrix}$

$\Sigma$ is a diagonal matrix and the $i$th diagonal element, ${\Sigma }_{ii}$, must be held in $\mathbf{SIG}\left(1,i\right)$

$\mathbf{iuld}=\mathrm{Nag\_LowerFactored}$ or $\mathrm{Nag\_UpperFactored}$

sig must hold $L$ or $U$, the lower or upper Cholesky decomposition of $\Sigma$, with $L_{ij}$ or $U_{ij}$ held in $\mathbf{SIG}\left(i,j\right)$, depending on the value of iuld. No check is made that $L{L}^{\mathrm{T}}$ or $U^{\mathrm{T}}U$ is a valid variance-covariance matrix. The diagonal elements of the supplied $L$ or $U$ must be greater than zero

9:    $\mathbf{pdsig}$ – IntegerInput

On entry: the stride separating matrix row elements in the array sig.

Constraints:

• if $\mathbf{iuld}=\mathrm{Nag\_DiagonalMatrix}$, $\mathbf{pdsig}\ge 1$;
• otherwise $\mathbf{pdsig}\ge \mathbf{n}$.

10:  $\mathbf{pdf}\left[\mathbf{k}\right]$ – doubleOutput

On exit: $f\left(X:\mu ,\Sigma \right)$ or $\log \left(f\left(X:\mu ,\Sigma \right)\right)$ depending on the value of ilog.

11:  $\mathbf{rank}$ – Integer *Output

On exit: $r$, rank of $\Sigma$.

12:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 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_ARRAY_SIZE

On entry, $\mathbf{pdsig}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{iuld}=\mathrm{Nag\_DiagonalMatrix}$, $\mathbf{pdsig}\ge 1$.

On entry, $\mathbf{pdsig}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{iuld}\ne \mathrm{Nag\_DiagonalMatrix}$, $\mathbf{pdsig}\ge \mathbf{n}$.

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

NE_BAD_PARAM

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

NE_DIAG_ELEMENTS

On entry, at least one diagonal element of $\Sigma$ is less than or equal to $0$.

NE_INT

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

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 2$.

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 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_MAT_NOT_POS_DEF

On entry, $\Sigma$ is not positive definite and eigenvalue decomposition failed.

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_NOT_POS_SEM_DEF

On entry, $\Sigma$ is not positive semidefinite.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (g01lbce.c)

10.2

Program Data

Program Data (g01lbce.d)

10.3

Program Results

Program Results (g01lbce.r)