# NAG Library Routine Document

## 1Purpose

g01taf returns a number of deviates associated with given probabilities of the Normal distribution.

## 2Specification

Fortran Interface
 Subroutine g01taf ( tail, lp, p, lxmu, xmu, xstd, x,
 Integer, Intent (In) :: ltail, lp, lxmu, lxstd Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: p(lp), xmu(lxmu), xstd(lxstd) Real (Kind=nag_wp), Intent (Out) :: x(*) Character (1), Intent (In) :: tail(ltail)
#include nagmk26.h
 void g01taf_ (const Integer *ltail, const char tail[], const Integer *lp, const double p[], const Integer *lxmu, const double xmu[], const Integer *lxstd, const double xstd[], double x[], Integer ivalid[], Integer *ifail, const Charlen length_tail)

## 3Description

The deviate, ${x}_{{p}_{i}}$ associated with the lower tail probability, ${p}_{i}$, for the Normal distribution is defined as the solution to
 $PXi≤xpi=pi=∫-∞xpiZiXidXi,$
where
 $ZiXi=12πσi2e-Xi-μi2/2σi2, ​-∞
The method used is an extension of that of Wichura (1988). ${p}_{i}$ is first replaced by ${q}_{i}={p}_{i}-0.5$.
(a) If $\left|{q}_{i}\right|\le 0.3$, ${z}_{i}$ is computed by a rational Chebyshev approximation
 $zi=siAisi2 Bisi2 ,$
where ${s}_{i}=\sqrt{2\pi }{q}_{i}$ and ${A}_{i}$, ${B}_{i}$ are polynomials of degree $7$.
(b) If $0.3<\left|{q}_{i}\right|\le 0.42$, ${z}_{i}$ is computed by a rational Chebyshev approximation
 $zi=sign⁡qi Citi Diti ,$
where ${t}_{i}=\left|{q}_{i}\right|-0.3$ and ${C}_{i}$, ${D}_{i}$ are polynomials of degree $5$.
(c) If $\left|{q}_{i}\right|>0.42$, ${z}_{i}$ is computed as
 $zi=sign⁡qi Eiui Fiui +ui ,$
where ${u}_{i}=\sqrt{-2×\mathrm{log}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({p}_{i},1-{p}_{i}\right)\right)}$ and ${E}_{i}$, ${F}_{i}$ are polynomials of degree $6$.
${x}_{{p}_{i}}$ is then calculated from ${z}_{i}$, using the relationsship ${z}_{{p}_{i}}=\frac{{x}_{i}-{\mu }_{i}}{{\sigma }_{i}}$.
For the upper tail probability $-{x}_{{p}_{i}}$ is returned, while for the two tail probabilities the value ${x}_{i{{p}_{i}}^{*}}$ is returned, where ${{p}_{i}}^{*}$ is the required tail probability computed from the input value of ${p}_{i}$.
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

## 4References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Wichura (1988) Algorithm AS 241: the percentage points of the Normal distribution Appl. Statist. 37 477–484

## 5Arguments

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 supplied probabilities represent. Letting $Z$ denote a variate from a standard Normal distribution, and ${z}_{i}=\frac{{x}_{{p}_{i}}-{\mu }_{i}}{{\sigma }_{i}}$, then for , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{lxmu}},{\mathbf{lxstd}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability, i.e., ${p}_{i}=P\left(Z\le {z}_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability, i.e., ${p}_{i}=P\left(Z\ge {z}_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'C'}$
The two tail (confidence interval) probability, 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, 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:     $\mathbf{lp}$ – IntegerInput
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4:     $\mathbf{p}\left({\mathbf{lp}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${p}_{i}$, the probabilities for the Normal distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left(j\right)$, .
Constraint: $0.0<{\mathbf{p}}\left(\mathit{j}\right)<1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lp}}$.
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 ${\mu }_{i}={\mathbf{xmu}}\left(j\right)$, .
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 ${\sigma }_{i}={\mathbf{xstd}}\left(j\right)$, .
Constraint: ${\mathbf{xstd}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lxstd}}$.
9:     $\mathbf{x}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lxmu}},{\mathbf{lxstd}},{\mathbf{lp}}\right)$.
On exit: ${x}_{{p}_{i}}$, the deviates for the Normal distribution.
10:   $\mathbf{ivalid}\left(*\right)$ – Integer arrayOutput
Note: the dimension of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lxmu}},{\mathbf{lxstd}},{\mathbf{lp}}\right)$.
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 ${x}_{{p}_{i}}$.
${\mathbf{ivalid}}\left(i\right)=2$
 On entry, ${p}_{i}\le 0.0$, or ${p}_{i}\ge 1.0$.
${\mathbf{ivalid}}\left(i\right)=3$
 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, at least one value of tail, xstd or p was invalid.
${\mathbf{ifail}}=2$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lp}}>0$.
${\mathbf{ifail}}=4$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lxmu}}>0$.
${\mathbf{ifail}}=5$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lxstd}}>0$.
${\mathbf{ifail}}=-99$
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.

## 7Accuracy

The accuracy is mainly limited by the machine precision.

## 8Parallelism and Performance

g01taf is not threaded in any implementation.

None.

## 10Example

This example reads vectors of values for ${\mu }_{i}$, ${\sigma }_{i}$ and ${p}_{i}$ and prints the corresponding deviates.

### 10.1Program Text

Program Text (g01tafe.f90)

### 10.2Program Data

Program Data (g01tafe.d)

### 10.3Program Results

Program Results (g01tafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017