# NAG CL Interfaceg01tac (inv_​cdf_​normal_​vector)

Settings help

CL Name Style:

## 1Purpose

g01tac returns a number of deviates associated with given probabilities of the Normal distribution.

## 2Specification

 #include
 void g01tac (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer lxmu, const double xmu[], Integer lxstd, const double xstd[], double x[], Integer ivalid[], NagError *fail)
The function may be called by the names: g01tac, nag_stat_inv_cdf_normal_vector or nag_deviates_normal_vector.

## 3Description

The deviate, ${x}_{{p}_{i}}$ associated with the lower tail probability, ${p}_{i}$, for the Normal distribution is defined as the solution to
 $P(Xi≤xpi)=pi=∫-∞xpiZi(Xi)dXi,$
where
 $Zi(Xi)=12πσi2e-(Xi-μi)2/(2σi2), ​-∞
The method used is an extension of that of Wichura (1988). ${p}_{i}$ is first replaced by ${q}_{i}={p}_{i}-0.5$.
1. (a)If $|{q}_{i}|\le 0.3$, ${z}_{i}$ is computed by a rational Chebyshev approximation
 $zi=siAi(si2) Bi(si2) ,$
where ${s}_{i}=\sqrt{2\pi }{q}_{i}$ and ${A}_{i}$, ${B}_{i}$ are polynomials of degree $7$.
2. (b)If $0.3<|{q}_{i}|\le 0.42$, ${z}_{i}$ is computed by a rational Chebyshev approximation
 $zi=sign⁡qi (Ci(ti) Di(ti) ) ,$
where ${t}_{i}=|{q}_{i}|-0.3$ and ${C}_{i}$, ${D}_{i}$ are polynomials of degree $5$.
3. (c)If $|{q}_{i}|>0.42$, ${z}_{i}$ is computed as
 $zi=sign⁡qi [(Ei(ui) Fi(ui) )+ui] ,$
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 function 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

NIST Digital Library of Mathematical Functions
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: $\mathbf{ltail}$Integer Input
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2: $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbability Input
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 $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{lxmu}},{\mathbf{lxstd}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability, i.e., ${p}_{i}=P\left(Z\le {z}_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability, i.e., ${p}_{i}=P\left(Z\ge {z}_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailConfid}$
The two tail (confidence interval) probability, i.e., ${p}_{i}=P\left(Z\le |{z}_{i}|\right)-P\left(Z\le -|{z}_{i}|\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailSignif}$
The two tail (significance level) probability, i.e., ${p}_{i}=P\left(Z\ge |{z}_{i}|\right)+P\left(Z\le -|{z}_{i}|\right)$.
Constraint: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$, $\mathrm{Nag_UpperTail}$, $\mathrm{Nag_TwoTailConfid}$ or $\mathrm{Nag_TwoTailSignif}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3: $\mathbf{lp}$Integer Input
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4: $\mathbf{p}\left[{\mathbf{lp}}\right]$const double Input
On entry: ${p}_{i}$, the probabilities for the Normal distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left[j\right]$, .
Constraint: $0.0<{\mathbf{p}}\left[\mathit{j}-1\right]<1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lp}}$.
5: $\mathbf{lxmu}$Integer Input
On entry: the length of the array xmu.
Constraint: ${\mathbf{lxmu}}>0$.
6: $\mathbf{xmu}\left[{\mathbf{lxmu}}\right]$const double Input
On entry: ${\mu }_{i}$, the means with ${\mu }_{i}={\mathbf{xmu}}\left[j\right]$, .
7: $\mathbf{lxstd}$Integer Input
On entry: the length of the array xstd.
Constraint: ${\mathbf{lxstd}}>0$.
8: $\mathbf{xstd}\left[{\mathbf{lxstd}}\right]$const double Input
On entry: ${\sigma }_{i}$, the standard deviations with ${\sigma }_{i}={\mathbf{xstd}}\left[j\right]$, .
Constraint: ${\mathbf{xstd}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lxstd}}$.
9: $\mathbf{x}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, 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: $\mathbf{ivalid}\left[\mathit{dim}\right]$Integer Output
Note: the dimension, dim, 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: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
On entry, invalid value supplied in tail when calculating ${x}_{{p}_{i}}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
On entry, ${p}_{i}\le 0.0$, or, ${p}_{i}\ge 1.0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
On entry, ${\sigma }_{i}\le 0.0$.
11: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lp}}>0$.
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ltail}}>0$.
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lxmu}}>0$.
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lxstd}}>0$.
NE_BAD_PARAM
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of tail, xstd or p was invalid.
Check ivalid for more information.

## 7Accuracy

The accuracy is mainly limited by the machine precision.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g01tac 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 (g01tace.c)

### 10.2Program Data

Program Data (g01tace.d)

### 10.3Program Results

Program Results (g01tace.r)