NAG FL Interface
g01saf (prob_​normal_​vector)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{ltail}$ – Integer Input

On entry: the length of the array tail.

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

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

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 $\mathit{i}=1,2,\dots ,\max (\mathbf{lx},\mathbf{ltail},\mathbf{lxmu},\mathbf{lxstd})$:

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

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

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

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

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

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

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

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

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: $\mathbf{lx}$ – Integer Input

On entry: the length of the array x.

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

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

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

5: $\mathbf{lxmu}$ – Integer Input

On entry: the length of the array xmu.

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

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

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

7: $\mathbf{lxstd}$ – Integer Input

On entry: the length of the array xstd.

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

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

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

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) array Output

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

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

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

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

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: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

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

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results