# NAG Library Routine Document

## 1Purpose

g01kqf returns a number of values of the probability density function (PDF), or its logarithm, for the Normal (Gaussian) distributions.

## 2Specification

Fortran Interface
 Subroutine g01kqf ( ilog, lx, x, lxmu, xmu, xstd, pdf,
 Integer, Intent (In) :: ilog, lx, lxmu, lxstd Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: x(lx), xmu(lxmu), xstd(lxstd) Real (Kind=nag_wp), Intent (Out) :: pdf(*)
#include nagmk26.h
 void g01kqf_ (const Integer *ilog, const Integer *lx, const double x[], const Integer *lxmu, const double xmu[], const Integer *lxstd, const double xstd[], double pdf[], Integer ivalid[], Integer *ifail)

## 3Description

The Normal distribution with mean ${\mu }_{i}$, variance ${{\sigma }_{i}}^{2}$; has probability density function (PDF)
 $f xi,μi,σi = 1 σi⁢2π e -xi-μi2/2σi2 , σi>0 .$
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.
None.

## 5Arguments

1:     $\mathbf{ilog}$ – IntegerInput
On entry: the value of ilog determines whether the logarithmic value is returned in PDF.
${\mathbf{ilog}}=0$
$f\left({x}_{i},{\mu }_{i},{\sigma }_{i}\right)$, the probability density function is returned.
${\mathbf{ilog}}=1$
$\mathrm{log}\left(f\left({x}_{i},{\mu }_{i},{\sigma }_{i}\right)\right)$, the logarithm of the probability density function is returned.
Constraint: ${\mathbf{ilog}}=0$ or $1$.
2:     $\mathbf{lx}$ – IntegerInput
On entry: the length of the array x.
Constraint: ${\mathbf{lx}}>0$.
3:     $\mathbf{x}\left({\mathbf{lx}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${x}_{i}$, the values at which the PDF is to be evaluated with ${x}_{i}={\mathbf{x}}\left(j\right)$, , for $i=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{lxstd}},{\mathbf{lxmu}}\right)$.
4:     $\mathbf{lxmu}$ – IntegerInput
On entry: the length of the array xmu.
Constraint: ${\mathbf{lxmu}}>0$.
5:     $\mathbf{xmu}\left({\mathbf{lxmu}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mu }_{i}$, the means with ${\mu }_{i}={\mathbf{xmu}}\left(j\right)$, .
6:     $\mathbf{lxstd}$ – IntegerInput
On entry: the length of the array xstd.
Constraint: ${\mathbf{lxstd}}>0$.
7:     $\mathbf{xstd}\left({\mathbf{lxstd}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\sigma }_{i}$, the standard deviations with ${\sigma }_{i}={\mathbf{xstd}}\left(j\right)$, .
Constraint: ${\mathbf{xstd}}\left(\mathit{j}\right)\ge 0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lxstd}}$.
8:     $\mathbf{pdf}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array pdf must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{lxstd}},{\mathbf{lxmu}}\right)$.
On exit: $f\left({x}_{i},{\mu }_{i},{\sigma }_{i}\right)$ or $\mathrm{log}\left(f\left({x}_{i},{\mu }_{i},{\sigma }_{i}\right)\right)$.
9:     $\mathbf{ivalid}\left(*\right)$ – Integer arrayOutput
Note: the dimension of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{lxstd}},{\mathbf{lxmu}}\right)$.
On exit: ${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
${\sigma }_{i}<0$.
10:   $\mathbf{ifail}$ – 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 xstd was invalid.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{ilog}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ilog}}=0$ or $1$.
${\mathbf{ifail}}=3$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lx}}>0$.
${\mathbf{ifail}}=4$
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.

Not applicable.

## 8Parallelism and Performance

g01kqf is not threaded in any implementation.

None.

## 10Example

This example prints the value of the Normal distribution PDF at four different points ${x}_{i}$ with differing ${\mu }_{i}$ and ${\sigma }_{i}$.

### 10.1Program Text

Program Text (g01kqfe.f90)

### 10.2Program Data

Program Data (g01kqfe.d)

### 10.3Program Results

Program Results (g01kqfe.r)

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