G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01SCF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G01SCF returns a number of lower or upper tail probabilities for the ${\chi }^{2}$-distribution with real degrees of freedom.

## 2  Specification

 SUBROUTINE G01SCF ( LTAIL, TAIL, LX, X, LDF, DF, P, IVALID, IFAIL)
 INTEGER LTAIL, LX, LDF, IVALID(*), IFAIL REAL (KIND=nag_wp) X(LX), DF(LDF), P(*) CHARACTER(1) TAIL(LTAIL)

## 3  Description

The lower tail probability for the ${\chi }^{2}$-distribution with ${\nu }_{i}$ degrees of freedom, $P=\left({X}_{i}\le {x}_{i}:{\nu }_{i}\right)$ is defined by:
 $P = Xi≤xi:νi = 1 2 νi/2 Γ νi/2 ∫ 0.0 xi Xi νi/2-1 e -Xi/2 dXi , xi ≥ 0 , νi > 0 .$
To calculate $P=\left({X}_{i}\le {x}_{i}:{\nu }_{i}\right)$ a transformation of a gamma distribution is employed, i.e., a ${\chi }^{2}$-distribution with ${\nu }_{i}$ degrees of freedom is equal to a gamma distribution with scale parameter $2$ and shape parameter ${\nu }_{i}/2$.
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector parameters 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.
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

## 5  Parameters

1:     LTAIL – INTEGERInput
On entry: the length of the array TAIL.
Constraint: ${\mathbf{LTAIL}}>0$.
2:     TAIL(LTAIL) – CHARACTER(1) arrayInput
On entry: indicates whether the lower or upper tail probabilities are required. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{LTAIL}},{\mathbf{LX}},{\mathbf{LDF}}\right)$:
${\mathbf{TAIL}}\left(j\right)=\text{'L'}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({X}_{i}\le {x}_{i}:{\nu }_{i}\right)$.
${\mathbf{TAIL}}\left(j\right)=\text{'U'}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({X}_{i}\ge {x}_{i}:{\nu }_{i}\right)$.
Constraint: ${\mathbf{TAIL}}\left(\mathit{j}\right)=\text{'L'}$ or $\text{'U'}$, for $\mathit{j}=1,2,\dots ,{\mathbf{LTAIL}}$.
3:     LX – INTEGERInput
On entry: the length of the array X.
Constraint: ${\mathbf{LX}}>0$.
4:     X(LX) – REAL (KIND=nag_wp) arrayInput
On entry: ${x}_{i}$, the values of the ${\chi }^{2}$ variates with ${\nu }_{i}$ degrees of freedom with ${x}_{i}={\mathbf{X}}\left(j\right)$, .
Constraint: ${\mathbf{X}}\left(\mathit{j}\right)\ge 0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{LX}}$.
5:     LDF – INTEGERInput
On entry: the length of the array DF.
Constraint: ${\mathbf{LDF}}>0$.
6:     DF(LDF) – REAL (KIND=nag_wp) arrayInput
On entry: ${\nu }_{i}$, the degrees of freedom of the ${\chi }^{2}$-distribution with ${\nu }_{i}={\mathbf{DF}}\left(j\right)$, .
Constraint: ${\mathbf{DF}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{LDF}}$.
7:     P($*$) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array P must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{LTAIL}},{\mathbf{LDF}},{\mathbf{LX}}\right)$.
On exit: ${p}_{i}$, the probabilities for the ${\chi }^{2}$ distribution.
8:     IVALID($*$) – INTEGER arrayOutput
Note: the dimension of the array IVALID must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{LTAIL}},{\mathbf{LDF}},{\mathbf{LX}}\right)$.
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, ${x}_{i}<0.0$.
${\mathbf{IVALID}}\left(i\right)=3$
 On entry, ${\nu }_{i}\le 0.0$.
${\mathbf{IVALID}}\left(i\right)=4$
The solution has failed to converge while calculating the gamma variate. The result returned should represent an approximation to the solution.
9:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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, because for this routine the values of the output parameters may be useful even if ${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ 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

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Note: G01SCF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, at least one value of X, DF or TAIL was invalid, or the solution failed to converge.
${\mathbf{IFAIL}}=2$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LTAIL}}>0$.
${\mathbf{IFAIL}}=3$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LX}}>0$.
${\mathbf{IFAIL}}=4$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LDF}}>0$.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.

## 7  Accuracy

A relative accuracy of five significant figures is obtained in most cases.

For higher accuracy the transformation described in Section 3 may be used with a direct call to S14BAF.

## 9  Example

Values from various ${\chi }^{2}$-distributions are read, the lower tail probabilities calculated, and all these values printed out.

### 9.1  Program Text

Program Text (g01scfe.f90)

### 9.2  Program Data

Program Data (g01scfe.d)

### 9.3  Program Results

Program Results (g01scfe.r)
{"ft_lang_label":"__label__en","ft_lang_prob":0.7178132,"math_prob":0.9989065,"size":4047,"snap":"2021-31-2021-39","text_gpt3_token_len":997,"char_repetition_ratio":0.123917885,"word_repetition_ratio":0.10349854,"special_character_ratio":0.24413146,"punctuation_ratio":0.17010309,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998505,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-31T19:33:43Z\",\"WARC-Record-ID\":\"<urn:uuid:e7bcfc3e-5f62-431c-aa4c-284671568606>\",\"Content-Length\":\"25285\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6009746a-d8fb-496f-aac2-c7ff5a1020dc>\",\"WARC-Concurrent-To\":\"<urn:uuid:4625e252-7f61-43a0-b668-a698710febfc>\",\"WARC-IP-Address\":\"78.129.168.4\",\"WARC-Target-URI\":\"https://www.nag.com/numeric/nl/nagdoc_24/nagdoc_fl24/html/g01/g01scf.html\",\"WARC-Payload-Digest\":\"sha1:FBDJ7I3QDZBGOH2M3MGIDCEXUSWG5UN7\",\"WARC-Block-Digest\":\"sha1:HUWZ6FXM3NTIMLOIO2JIQYOQZC3VNGQS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046154099.21_warc_CC-MAIN-20210731172305-20210731202305-00398.warc.gz\"}"}