# NAG Library Routine Document

## 1Purpose

g05pyf generates a random correlation matrix with given eigenvalues.

## 2Specification

Fortran Interface
 Subroutine g05pyf ( n, d, eps, c, ldc,
 Integer, Intent (In) :: n, ldc Integer, Intent (Inout) :: state(*), ifail Real (Kind=nag_wp), Intent (In) :: d(n), eps Real (Kind=nag_wp), Intent (Inout) :: c(ldc,n)
#include nagmk26.h
 void g05pyf_ (const Integer *n, const double d[], const double *eps, Integer state[], double c[], const Integer *ldc, Integer *ifail)

## 3Description

Given $n$ eigenvalues, ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n}$, such that
 $∑i=1nλi=n$
and
 $λi≥ 0, i= 1,2,…,n,$
g05pyf will generate a random correlation matrix, $C$, of dimension $n$, with eigenvalues ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n}$.
The method used is based on that described by Lin and Bendel (1985). Let $D$ be the diagonal matrix with values ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n}$ and let $A$ be a random orthogonal matrix generated by g05pxf then the matrix ${C}_{0}=AD{A}^{\mathrm{T}}$ is a random covariance matrix with eigenvalues ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n}$. The matrix ${C}_{0}$ is transformed into a correlation matrix by means of $n-1$ elementary rotation matrices ${P}_{i}$ such that $C={P}_{n-1}{P}_{n-2}\dots {P}_{1}{C}_{0}{P}_{1}^{\mathrm{T}}\dots {P}_{n-2}^{\mathrm{T}}{P}_{n-1}^{\mathrm{T}}$. The restriction on the sum of eigenvalues implies that for any diagonal element of ${C}_{0}>1$, there is another diagonal element $\text{}<1$. The ${P}_{i}$ are constructed from such pairs, chosen at random, to produce a unit diagonal element corresponding to the first element. This is repeated until all diagonal elements are $1$ to within a given tolerance $\epsilon$.
The randomness of $C$ should be interpreted only to the extent that $A$ is a random orthogonal matrix and $C$ is computed from $A$ using the ${P}_{i}$ which are chosen as arbitrarily as possible.
One of the initialization routines g05kff (for a repeatable sequence if computed sequentially) or g05kgf (for a non-repeatable sequence) must be called prior to the first call to g05pyf.

## 4References

Lin S P and Bendel R B (1985) Algorithm AS 213: Generation of population correlation on matrices with specified eigenvalues Appl. Statist. 34 193–198

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the dimension of the correlation matrix to be generated.
Constraint: ${\mathbf{n}}\ge 1$.
2:     $\mathbf{d}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the $n$ eigenvalues, ${\lambda }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Constraints:
• ${\mathbf{d}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$;
• $\sum _{i=1}^{n}{\mathbf{d}}\left(i\right)=n$ to within eps.
3:     $\mathbf{eps}$ – Real (Kind=nag_wp)Input
On entry: the maximum acceptable error in the diagonal elements.
Suggested value: ${\mathbf{eps}}=0.00001$.
Constraint:  (see Chapter X02).
4:     $\mathbf{state}\left(*\right)$ – Integer arrayCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization routines g05kff or g05kgf.
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
5:     $\mathbf{c}\left({\mathbf{ldc}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: a random correlation matrix, $C$, of dimension $n$.
6:     $\mathbf{ldc}$ – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which g05pyf is called.
Constraint: ${\mathbf{ldc}}\ge {\mathbf{n}}$.
7:     $\mathbf{ifail}$ – 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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, an eigenvalue is negative.
On entry, the eigenvalues do not sum to n.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{eps}}=〈\mathit{\text{value}}〉$.
Constraint: .
${\mathbf{ifail}}=4$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=5$
The diagonals of the returned matrix are not unity, try increasing the value of eps, or rerun the code using a different seed.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{ldc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldc}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
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

The maximum error in a diagonal element is given by eps.

## 8Parallelism and Performance

g05pyf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05pyf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
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 g05pyf is approximately proportional to ${n}^{2}$.

## 10Example

Following initialization of the pseudorandom number generator by a call to g05kff, a $3$ by $3$ correlation matrix with eigenvalues of $0.7$, $0.9$ and $1.4$ is generated and printed.

### 10.1Program Text

Program Text (g05pyfe.f90)

### 10.2Program Data

Program Data (g05pyfe.d)

### 10.3Program Results

Program Results (g05pyfe.r)

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