NAG Library Function Document

nag_multi_normal_pdf_vector (g01lbc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     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:     k – IntegerInput

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

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

3:     n – IntegerInput

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

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

4:     x[$\mathit{dim}$] – 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:     pdx – IntegerInput

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

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

6:     xmu[n] – const doubleInput

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

7:     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:     sig[$\mathit{dim}$] – 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:     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:   pdf[$\mathbf{k}$] – 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:   rank – Integer *Output

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

12:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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.

NE_MAT_NOT_POS_DEF

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

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)