NAG Library Routine Document

G01LBF

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

1:   ILOG – INTEGERInput

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

$\mathbf{ILOG}=0$

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

$\mathbf{ILOG}=1$

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

Constraint: $\mathbf{ILOG}=0$ or $1$.

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(LDX,$*$) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array X must be at least $\mathbf{K}$.

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:   LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G01LBF is called.

Constraint: $\mathbf{LDX}\ge \mathbf{N}$.

6:   XMU(N) – REAL (KIND=nag_wp) arrayInput

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

7:   IULD – INTEGERInput

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

$\mathbf{IULD}=1$

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

$\mathbf{IULD}=2$

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

$\mathbf{IULD}=3$

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

$\mathbf{IULD}=4$

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

$\mathbf{IULD}=5$

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

Constraint: $\mathbf{IULD}=1$, $2$, $3$, $4$ or $5$.

8:   SIG(LDSIG,$*$) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array SIG must be at least $\mathbf{N}$.

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

$\mathbf{IULD}=1$ or $2$

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.

$\mathbf{IULD}=3$

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

$\mathbf{IULD}=4$ or $5$

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:   LDSIG – INTEGERInput

On entry: the first dimension of the array SIG as declared in the (sub)program from which G01LBF is called.

Constraints:

• if $\mathbf{IULD}=3$, $\mathbf{LDSIG}\ge 1$;
• otherwise $\mathbf{LDSIG}\ge \mathbf{N}$.

10:   PDF($\mathbf{K}$) – REAL (KIND=nag_wp) arrayOutput

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 – INTEGEROutput

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

12:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

$\mathbf{IFAIL}=11$

On entry, $\mathbf{ILOG}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{ILOG}=0$ or $1$.

$\mathbf{IFAIL}=21$

On entry, $\mathbf{K}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{K}\ge 0$.

$\mathbf{IFAIL}=31$

On entry, $\mathbf{N}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{N}\ge 2$.

$\mathbf{IFAIL}=51$

On entry, $\mathbf{LDX}=⟨\mathit{\text{value}}⟩$ and $\mathbf{N}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{LDX}\ge \mathbf{N}$.

$\mathbf{IFAIL}=71$

On entry, $\mathbf{IULD}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{IULD}=1$, $2$, $3$, $4$ or $5$.

$\mathbf{IFAIL}=81$

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

$\mathbf{IFAIL}=82$

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

$\mathbf{IFAIL}=83$

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

$\mathbf{IFAIL}=91$

On entry, $\mathbf{LDSIG}=⟨\mathit{\text{value}}⟩$.
Constraint: if $\mathbf{IULD}=3$, $\mathbf{LDSIG}\ge 1$.

$\mathbf{IFAIL}=92$

On entry, $\mathbf{LDSIG}=⟨\mathit{\text{value}}⟩$.
Constraint: if $\mathbf{IULD}\ne 3$, $\mathbf{LDSIG}\ge \mathbf{N}$.

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

Program Text (g01lbfe.f90)

9.2 Program Data

Program Data (g01lbfe.d)

9.3 Program Results

Program Results (g01lbfe.r)