NAG Library Routine Document

g01saf  (prob_normal_vector)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{ltail}$ – IntegerInput

On entry: the length of the array tail.

Constraint: $\mathbf{ltail}>0$.

2:     $\mathbf{tail}\left(\mathbf{ltail}\right)$ – Character(1) arrayInput

On entry: indicates which tail the returned probabilities should represent. Letting $Z$ denote a variate from a standard Normal distribution, and $z_i=\frac{x_i-{\mu }_i}{{\sigma }_i}$, then for $j=\left(\left(\mathit{i}-1\right)\mathrm{ mod }\mathbf{ltail}\right)+1$, for $\mathit{i}=1,2,\dots ,\max \left(\mathbf{lx},\mathbf{ltail},\mathbf{lxmu},\mathbf{lxstd}\right)$:

$\mathbf{tail}\left(j\right)=\text{'L'}$

The lower tail probability is returned, i.e., $p_i=P\left(Z\le z_i\right)$.

$\mathbf{tail}\left(j\right)=\text{'U'}$

The upper tail probability is returned, i.e., $p_i=P\left(Z\ge z_i\right)$.

$\mathbf{tail}\left(j\right)=\text{'C'}$

The two tail (confidence interval) probability is returned, i.e., $p_i=P\left(Z\le \left|z_i\right|\right)-P\left(Z\le -\left|z_i\right|\right)$.

$\mathbf{tail}\left(j\right)=\text{'S'}$

The two tail (significance level) probability is returned, i.e., $p_i=P\left(Z\ge \left|z_i\right|\right)+P\left(Z\le -\left|z_i\right|\right)$.

Constraint: $\mathbf{tail}\left(\mathit{j}\right)=\text{'L'}$, $\text{'U'}$, $\text{'C'}$ or $\text{'S'}$, for $\mathit{j}=1,2,\dots ,\mathbf{ltail}$.

3:     $\mathbf{lx}$ – IntegerInput

On entry: the length of the array x.

Constraint: $\mathbf{lx}>0$.

4:     $\mathbf{x}\left(\mathbf{lx}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $x_i$, the Normal variate values with $x_i=\mathbf{x}\left(j\right)$, $j=\left(\left(i-1\right)\mathrm{ mod }\mathbf{lx}\right)+1$.

5:     $\mathbf{lxmu}$ – IntegerInput

On entry: the length of the array xmu.

Constraint: $\mathbf{lxmu}>0$.

6:     $\mathbf{xmu}\left(\mathbf{lxmu}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: ${\mu }_i$, the means with ${\mu }_i=\mathbf{xmu}\left(j\right)$, $j=\left(\left(i-1\right)\mathrm{ mod }\mathbf{lxmu}\right)+1$.

7:     $\mathbf{lxstd}$ – IntegerInput

On entry: the length of the array xstd.

Constraint: $\mathbf{lxstd}>0$.

8:     $\mathbf{xstd}\left(\mathbf{lxstd}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: ${\sigma }_i$, the standard deviations with ${\sigma }_i=\mathbf{xstd}\left(j\right)$, $j=\left(\left(i-1\right)\mathrm{ mod }\mathbf{lxstd}\right)+1$.

Constraint: $\mathbf{xstd}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,\mathbf{lxstd}$.

9:     $\mathbf{p}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array p must be at least $\max \left(\mathbf{lx},\mathbf{ltail},\mathbf{lxmu},\mathbf{lxstd}\right)$.

On exit: $p_i$, the probabilities for the Normal distribution.

10:   $\mathbf{ivalid}\left(*\right)$ – Integer arrayOutput

Note: the dimension of the array ivalid must be at least $\max \left(\mathbf{lx},\mathbf{ltail},\mathbf{lxmu},\mathbf{lxstd}\right)$.

On exit: $\mathbf{ivalid}\left(i\right)$ indicates any errors with the input arguments, with

$\mathbf{ivalid}\left(i\right)=0$

No error.

$\mathbf{ivalid}\left(i\right)=1$

On entry,

invalid value supplied in tail when calculating $p_i$.

$\mathbf{ivalid}\left(i\right)=2$

On entry,

${\sigma }_i\le 0.0$.

11:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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, at least one value of tail or xstd was invalid.
Check ivalid for more information.

$\mathbf{ifail}=2$

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

$\mathbf{ifail}=3$

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

$\mathbf{ifail}=4$

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

$\mathbf{ifail}=5$

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

$\mathbf{ifail}=-99$

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

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

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

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (g01safe.f90)

10.2

Program Data

Program Data (g01safe.d)

10.3

Program Results

Program Results (g01safe.r)