# NAG Library Function Document

## 1Purpose

nag_rand_corr_matrix (g05pyc) generates a random correlation matrix with given eigenvalues.

## 2Specification

 #include #include
 void nag_rand_corr_matrix (Integer n, const double d[], double eps, Integer state[], double c[], Integer pdc, NagError *fail)

## 3Description

Given $n$ eigenvalues, ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n}$, such that
 $∑i=1nλi=n$
and
 $λi≥ 0, i= 1,2,…,n,$
nag_rand_corr_matrix (g05pyc) 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 nag_rand_orthog_matrix (g05pxc) 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 functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_corr_matrix (g05pyc).

## 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:    $\mathbf{d}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the $n$ eigenvalues, ${\lambda }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Constraints:
• ${\mathbf{d}}\left[\mathit{i}-1\right]\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$;
• $\sum _{i=1}^{n}{\mathbf{d}}\left[i-1\right]=n$ to within eps.
3:    $\mathbf{eps}$doubleInput
On entry: the maximum acceptable error in the diagonal elements.
Suggested value: ${\mathbf{eps}}=0.00001$.
Constraint:  (see Chapter x02).
4:    $\mathbf{state}\left[\mathit{dim}\right]$IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
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:    $\mathbf{c}\left[{\mathbf{n}}×{\mathbf{pdc}}\right]$doubleOutput
On exit: a random correlation matrix, $C$, of dimension $n$.
6:    $\mathbf{pdc}$IntegerInput
On entry: the stride separating row elements of the matrix $C$ in the array c.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{n}}$.
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_DIAG_ELEMENTS
The diagonals of the returned matrix are not unity, try increasing the value of eps, or rerun the code using a different seed.
NE_EIGVAL_SUM
On entry, the eigenvalues do not sum to n.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}>0$.
NE_INT_2
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{n}}$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_NEGATIVE_EIGVAL
On entry, an eigenvalue is negative.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{eps}}=〈\mathit{\text{value}}〉$.
Constraint: .

## 7Accuracy

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

## 8Parallelism and Performance

nag_rand_corr_matrix (g05pyc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_rand_corr_matrix (g05pyc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by nag_rand_corr_matrix (g05pyc) is approximately proportional to ${n}^{2}$.

## 10Example

Following initialization of the pseudorandom number generator by a call to nag_rand_init_repeatable (g05kfc), 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 (g05pyce.c)

### 10.2Program Data

Program Data (g05pyce.d)

### 10.3Program Results

Program Results (g05pyce.r)

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