# NAG Library Routine Document

## 1Purpose

g02bdf computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data.

## 2Specification

Fortran Interface
 Subroutine g02bdf ( n, m, x, ldx, xbar, std, sspz, rz, ldrz,
 Integer, Intent (In) :: n, m, ldx, ldsspz, ldrz Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(ldx,m) Real (Kind=nag_wp), Intent (Inout) :: sspz(ldsspz,m), rz(ldrz,m) Real (Kind=nag_wp), Intent (Out) :: xbar(m), std(m)
#include <nagmk26.h>
 void g02bdf_ (const Integer *n, const Integer *m, const double x[], const Integer *ldx, double xbar[], double std[], double sspz[], const Integer *ldsspz, double rz[], const Integer *ldrz, Integer *ifail)

## 3Description

The input data consists of $n$ observations for each of $m$ variables, given as an array
 $xij, i=1,2,…,nn≥2 , j=1,2,…,m m≥2,$
where ${x}_{ij}$ is the $i$th observation on the $j$th variable.
The quantities calculated are:
(a) Means:
 $x-j=1n∑i=1nxij, j=1,2,…,m.$
(b) Standard deviations:
 $sj=1n- 1 ∑i= 1n xij-x-j 2, j= 1,2,…,m.$
(c) Sums of squares and cross-products about zero:
 $S~jk=∑i=1nxijxik, j,k=1,2,…,m.$
(d) Correlation-like coefficients:
 $R~jk=S~jkS~jjS~kk , j,k= 1,2,…,m.$
If ${\stackrel{~}{S}}_{jj}$ or ${\stackrel{~}{S}}_{kk}$ is zero, ${\stackrel{~}{R}}_{jk}$ is set to zero.

None.

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations or cases.
Constraint: ${\mathbf{n}}\ge 2$.
2:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 2$.
3:     $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of ${x}_{\mathit{i}\mathit{j}}$, the $\mathit{i}$th observation on the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
4:     $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02bdf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
5:     $\mathbf{xbar}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{xbar}}\left(\mathit{j}\right)$ contains the mean value, ${\stackrel{-}{x}}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
6:     $\mathbf{std}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the standard deviation, ${s}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
7:     $\mathbf{sspz}\left({\mathbf{ldsspz}},{\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{sspz}}\left(\mathit{j},\mathit{k}\right)$ is the cross-product about zero, ${\stackrel{~}{S}}_{\mathit{j}\mathit{k}}$, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
8:     $\mathbf{ldsspz}$ – IntegerInput
On entry: the first dimension of the array sspz as declared in the (sub)program from which g02bdf is called.
Constraint: ${\mathbf{ldsspz}}\ge {\mathbf{m}}$.
9:     $\mathbf{rz}\left({\mathbf{ldrz}},{\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{rz}}\left(\mathit{j},\mathit{k}\right)$ is the correlation-like coefficient, ${\stackrel{~}{R}}_{\mathit{j}\mathit{k}}$, between the $\mathit{j}$th and $\mathit{k}$th variables, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
10:   $\mathbf{ldrz}$ – IntegerInput
On entry: the first dimension of the array rz as declared in the (sub)program from which g02bdf is called.
Constraint: ${\mathbf{ldrz}}\ge {\mathbf{m}}$.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 2$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ldrz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldrz}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldsspz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldsspz}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
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

g02bdf does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large $n$.

## 8Parallelism and Performance

g02bdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by g02bdf depends on $n$ and $m$.
The routine uses a two-pass algorithm.

## 10Example

This example reads in a set of data consisting of five observations on each of three variables. The means, standard deviations, sums of squares and cross-products about zero, and correlation-like coefficients for all three variables are then calculated and printed.

### 10.1Program Text

Program Text (g02bdfe.f90)

### 10.2Program Data

Program Data (g02bdfe.d)

### 10.3Program Results

Program Results (g02bdfe.r)