RNMVN

Generates pseudorandom numbers from a multivariate normal distribution.

Required Arguments

RSIG — Upper triangular matrix, K by K, containing the Cholesky factor of the variance‑covariance matrix. (Input)
The variance‑covariance matrix is equal to the product of the transpose of RSIG and RSIG. RSIG can be obtained from the variance‑covariance matrix using routine CHFAC.

R — NR by K matrix containing the random multivariate normal vectors in its rows. (Output)

Optional Arguments

NR — Number of random multivariate normal vectors to generate. (Input)
Default: NR = size (R,1).

K — Length of the multivariate normal vectors. (Input)
Default: K = size (R,2).

LDRSIG — Leading dimension of RSIG exactly as specified in the dimension statement in the calling program. (Input)
Default: LDRSIG = size (RSIG,1).

LDR — Leading dimension of R exactly as specified in the dimension statement of the calling program. (Input)
Default: LDR = size (R,1).

FORTRAN 90 Interface

Generic: CALL RNMVN (RSIG, R [, …])

Specific: The specific interface names are S_RNMVN and D_RNMVN.

FORTRAN 77 Interface

Single: CALL RNMVN (NR, K, RSIG, LDRSIG, R, LDR)

Double: The double precision name is DRNMVN.

Description

Routine RNMVN generates pseudorandom numbers from a multivariate normal distribution with mean vector consisting of all zeroes and variance‑covariance matrix whose Cholesky factor (or “square root”) is RSIG; that is, RSIG is an upper triangular matrix such that the transpose of RSIG times RSIG is the variance‑covariance matrix. First, independent random normal deviates with mean 0 and variance 1 are generated, and then the matrix containing these deviates is postmultiplied by RSIG. The independent normals are generated into the columns of a matrix, which has NR rows; hence, if RNSET is called with different values of NR, the output is different even if the seed is the same in the calls.

Deviates from a multivariate normal distribution with means other than zero can be generated by using RNMVN and then by adding the vector of means to each row of R.

Comments

The routine RNSET can be used to initialize the seed of the random number generator. The routine RNOPT can be used to select the form of the generator.

Example

In this example, RNMVN is used to generate five pseudorandom multivariate normal vectors of length 2 with variance‑covariance matrix equal to

 

The routine CHFAC is first called to compute the Cholesky factorization of the variance‑covariance matrix.

 

USE RNMVN_INT

USE UMACH_INT

USE CHFAC_INT

USE RNSET_INT

 

IMPLICIT NONE

INTEGER I, IRANK, ISEED, J, K, LDR, LDRSIG, NOUT, NR

REAL COV(2,2), R(5,2), RSIG(2,2)

!

CALL UMACH (2, NOUT)

NR = 5

K = 2

LDRSIG = 2

LDR = 5

COV(1,1) = 0.5

COV(1,2) = 0.375

COV(2,1) = 0.375

COV(2,2) = 0.5

! Obtain the Cholesky factorization.

CALL CHFAC (COV, IRANK, RSIG)

! Initialize seed of random number

! generator.

ISEED = 123457

CALL RNSET (ISEED)

CALL RNMVN (RSIG, R)

WRITE (NOUT,99999) ((R(I,J),J=1,K),I=1,NR)

 

99999 FORMAT (' Multivariate normal random deviates: ', /, &

(1X,2F8.4))

END

Output

 

Multivariate normal random deviates:

1.4507 1.2463

0.7660 -0.0429

0.0584 -0.6692

0.9035 0.4628

-0.8669 -0.9334