The function may be called by the names: g05rzc, nag_rand_multivar_normal or nag_rand_matrix_multi_normal.
When the covariance matrix is nonsingular (i.e., strictly positive definite), the distribution has probability density function
where is the number of dimensions, is the covariance matrix, is the vector of means and is the vector of positions.
Covariance matrices are symmetric and positive semidefinite. Given such a matrix , there exists a lower triangular matrix such that . is not unique, if is singular.
g05rzc decomposes to find such an . It then stores , and in the reference vector which is used to generate a vector of independent standard Normal pseudorandom numbers. It then returns the vector , which has the required multivariate Normal distribution.
It should be noted that this function will work with a singular covariance matrix , provided is positive semidefinite, despite the fact that the above formula for the probability density function is not valid in that case. Wilkinson (1965) should be consulted if further information is required.
One of the initialization functions g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to g05rzc.
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley
Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
2: – Nag_ModeRNGInput
On entry: a code for selecting the operation to be performed by the function.
Set up reference vector only.
Generate variates using reference vector set up in a prior call to g05rzc.
Set up reference vector and generate variates.
, or .
3: – IntegerInput
On entry: , the number of random variates required.
4: – IntegerInput
On entry: , the number of dimensions of the distribution.
5: – const doubleInput
On entry: , the vector of means of the distribution.
6: – const doubleInput
Note: the th element of the matrix is stored in
On entry: the covariance matrix of the distribution. Only the upper triangle need be set.
must be positive semidefinite to machine precision.
7: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
8: – doubleCommunication Array
On entry: if , the reference vector as set up by g05rzc in a previous call with or .
On exit: if or , the reference vector that can be used in subsequent calls to g05rzc with .
9: – IntegerInput
On entry: the dimension of the array r. If , it must be the same as the value of lr specified in the prior call to g05rzc with or .
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, lr is not large enough, : minimum length required .
On entry, .
On entry, .
On entry, and .
On entry, and .
On entry, and .
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.
On entry, state vector has been corrupted or not initialized.
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.
On entry, the covariance matrix is not positive semidefinite to machine precision.
m is not the same as when r was set up in a previous call.
Previous value of and .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g05rzc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05rzc 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 g05rzc is of order .
It is recommended that the diagonal elements of should not differ too widely in order of magnitude. This may be achieved by scaling the variables if necessary. The actual matrix decomposed is , where is a diagonal matrix with small positive diagonal elements. This ensures that, even when is singular, or nearly singular, the Cholesky factor corresponds to a positive definite covariance matrix that agrees with within machine precision.
This example prints ten pseudorandom observations from a multivariate Normal distribution with means vector
and covariance matrix
generated by g05rzc. All ten observations are generated by a single call to g05rzc with
The random number generator is initialized by g05kfc.