NAG Library Routine Document

G01HDF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{N}$ – INTEGERInput

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

Constraint: $1<\mathbf{N}\le 1000$.

2:     $\mathrm{TAIL}\left(\mathbf{N}\right)$ – CHARACTER(1) arrayInput

On entry: defines the calculated probability, set $\mathbf{TAIL}\left(i\right)$ to:

$\mathbf{TAIL}\left(i\right)=\text{'L'}$

If the $i$th lower limit $a_i$ is negative infinity.

$\mathbf{TAIL}\left(i\right)=\text{'U'}$

If the $i$th upper limit $b_i$ is infinity.

$\mathbf{TAIL}\left(i\right)=\text{'C'}$

If both $a_i$ and $b_i$ are finite.

Constraint: $\mathbf{TAIL}\left(\mathit{i}\right)=\text{'L'}$, $\text{'U'}$ or $\text{'C'}$, for $\mathit{i}=1,2,\dots ,\mathbf{N}$.

3:     $\mathrm{A}\left(\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: $a_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the lower integral limits of the calculation.

If $\mathbf{TAIL}\left(i\right)=\text{'L'}$, $\mathbf{A}\left(i\right)$ is not referenced and the $i$th lower limit of integration is $-\infty$.

4:     $\mathrm{B}\left(\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: $b_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the upper integral limits of the calculation.

If $\mathbf{TAIL}\left(i\right)=\text{'U'}$, $\mathbf{B}\left(i\right)$ is not referenced and the $i$th upper limit of integration is $\infty$.

Constraint: if $\mathbf{TAIL}\left(i\right)=\text{'C'}$, $\mathbf{B}\left(i\right)>\mathbf{A}\left(i\right)$.

5:     $\mathrm{NU}$ – REAL (KIND=nag_wp)Input

On entry: $\nu$, the degrees of freedom.

Constraint: $\mathbf{NU}>0.0$.

6:     $\mathrm{DELTA}\left(\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: $\mathbf{DELTA}\left(\mathit{i}\right)$ the noncentrality parameter for the $\mathit{i}$th dimension, for $\mathit{i}=1,2,\dots ,\mathbf{N}$; set $\mathbf{DELTA}\left(i\right)=0$ for the central probability.

7:     $\mathrm{ISCOV}$ – INTEGERInput

On entry: set $\mathbf{ISCOV}=1$ if the covariance matrix is supplied and $\mathbf{ISCOV}=2$ if the correlation matrix is supplied.

Constraint: $\mathbf{ISCOV}=1$ or $2$.

8:     $\mathrm{RC}\left(\mathbf{LDRC},\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: the lower triangle of either the covariance matrix (if $\mathbf{ISCOV}=1$) or the correlation matrix (if $\mathbf{ISCOV}=2$). In either case the array elements corresponding to the upper triangle of the matrix need not be set.

On exit: the strict upper triangle of RC contains the correlation matrix used in the calculations.

9:     $\mathrm{LDRC}$ – INTEGERInput

On entry: the first dimension of the array RC as declared in the (sub)program from which G01HDF is called.

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

10:   $\mathrm{EPSABS}$ – REAL (KIND=nag_wp)Input

On entry: ${\epsilon }_a$, the absolute accuracy requested in the approximation. If EPSABS is negative, the absolute value is used.

Suggested value: $0.0$.

11:   $\mathrm{EPSREL}$ – REAL (KIND=nag_wp)Input

On entry: ${\epsilon }_r$, the relative accuracy requested in the approximation. If EPSREL is negative, the absolute value is used.

Suggested value: $0.001$.

12:   $\mathrm{NUMSUB}$ – INTEGERInput

On entry: if quadrature is used, the number of sub-intervals used by the quadrature algorithm; otherwise NUMSUB is not referenced.

Suggested value: $350$.

Constraint: if referenced, $\mathbf{NUMSUB}>0$.

13:   $\mathrm{NSAMPL}$ – INTEGERInput

On entry: if quadrature is used, NSAMPL is not referenced; otherwise NSAMPL is the number of samples used to estimate the error in the approximation.

Suggested value: $8$

Constraint: if referenced, $\mathbf{NSAMPL}>0$.

14:   $\mathrm{FMAX}$ – INTEGERInput

On entry: if a number theoretic approach is used, the maximum number of evaluations for each integrand function.

Suggested value: $1000\times \mathbf{N}$

Constraint: if referenced, $\mathbf{FMAX}\ge 1$.

15:   $\mathrm{ERREST}$ – REAL (KIND=nag_wp)Output

On exit: an estimate of the error in the calculated probability.

16:   $\mathrm{IFAIL}$ – 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

$\mathbf{IFAIL}=1$

On entry, $\mathbf{N}=〈\mathit{\text{value}}〉$.
Constraint: $1<\mathbf{N}\le 1000$.

$\mathbf{IFAIL}=2$

On entry, $\mathbf{TAIL}\left(k\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{TAIL}\left(\mathit{k}\right)=\text{'L'}$, $\text{'U'}$ or $\text{'C'}$.

$\mathbf{IFAIL}=4$

On entry, $k=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{B}\left(k\right)>\mathbf{A}\left(k\right)$ for a central probability.

$\mathbf{IFAIL}=5$

On entry, $\mathbf{NU}=〈\mathit{\text{value}}〉$.
Constraint: degrees of freedom $\mathbf{NU}>0.0$.

$\mathbf{IFAIL}=8$

On entry, $\mathbf{ISCOV}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ISCOV}=1$ or $2$.

$\mathbf{IFAIL}=9$

On entry, the information supplied in RC is invalid.

$\mathbf{IFAIL}=10$

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

$\mathbf{IFAIL}=12$

On entry, $\mathbf{NUMSUB}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{NUMSUB}\ge 1$.

$\mathbf{IFAIL}=13$

On entry, $\mathbf{NSAMPL}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{NSAMPL}\ge 1$.

$\mathbf{IFAIL}=14$

On entry, $\mathbf{FMAX}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{FMAX}\ge 1$.

$\mathbf{IFAIL}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.8 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.7 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-999$

Dynamic memory allocation failed.

See Section 3.6 in the Essential Introduction for further information.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (g01hdfe.f90)

10.2  Program Data

Program Data (g01hdfe.d)

10.3  Program Results

Program Results (g01hdfe.r)